Richard Brauer Collected Papers I-III
 0262021579, 9780262021579

Table of contents :
Richard Brauer - Collected Papers. Volume I. Theory of Algebras, and Finite Groups
Richard Brauer 1901-1977
Contents
Preface
J. A. Green, Richard Dagobert Brauer [Bull. London Math. Soc., 10 (1978), 317-342]
Bibliography of Richard Brauer
Theory of Algebras
Introduction by O. Goldman
[3]Über Zusammenhänge zwischen arithmetischen und invariantentheoretischen Eigenschaften von Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1926), 410-416.
[4]Über minimale Zerfällungskörper irreduzibler Darstellungen (with E. Noether), Sitzungsber. Preuss. Akad. Wiss. (1927), 221-228.
[5]Untersuchungen über die arithmetischen Eigenschaften von Gruppen linearer Substitutionen I, Math. Z. 28 (1928), 677-696.
[7]Über Systeme hyperkomplexer Zahlen, Math. Z. 30 (1929), 79-107.
Corrected by Richard Brauer
Corrected by Richard Brauer
[11]Zum Irreduzibilitätsbegriff in der Theorie der Gruppen linearer homogener Substitutionen (with I. Schur), Sitzungsber. Preuss. Akad. Wiss. (1930), 209-226.
[12]Untersuchungen über die arithmetischen Eigenschaften von Gruppen linearer Substitutionen II, Math. Z. 31 (1930), 733-747.
[13]Über die algebraische Struktur von Schiefkörpern, J. Reine Angew. Math. 166 (1932), 241-252.
[14]Beweis eines Hauptsatzes in der Theorie der Algebren (with H. Hasse and E. Noether), J. Reine Angew. Math. 167 (1931), 399-404.
[15]Über die Konstruktion der Schiefkörper, die von endlichem Rang in bezug auf ein gegebenes Zentrum sind, J. Reine Angew. Math. 168 (1932), 44-64.
[16]Über den Index und den Exponenten von Divisionalgebren, Tôhoku Math. J. 37 (1933), 77-87.
[22]Algebra der hyperkomplexen Zahlensysteme (Algebren), accepted for publication (1936) in Enzyklopädie der Mathematischen Wissenschaften, vol. IB 8, Teubner, Leipzig. Not published, because of political considerations, until Math. J. Okahama U. 21 (1979), 53-89.
[28]On the regular representations of algebras (with C. Nesbitt), Proc. Nat. Acad. Sci. U.S.A. 23 (1937), 236-240.
Corrected by Richard Brauer
[30]On normal division algebras of index 5, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 243-246.
[31]On modular and p-adic representations of algebras, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 252-258
Corrected by Richard Brauer
[36]On sets of matrices with coefficients in a division ring, Trans. Amer. Math. Soc. 49 (1941), 502-548.
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
[41]On the nilpotency of the radical of a ring. Bull. Amer. Math. Soc. 48 (1942), 752-758.
[44]On hypercomplex arithrnetic and a theorem of Speiser, in Festschrift for 60th Birthday of Andreas Speiser, Orell Füssli Verlag, Zürich (1945), 233-245.
[50]On splitting fields of simple algebras, Ann. of Math. 48 (1947), 79-90.
[56]Representations of groups and rings, Amer. Math. Soc. Colloquium Lectures (1948). 21 pp.
[57]On a theorem of H. Cartan, Bull. Amer. Math. Soc. 55 (1949), 619-620.
[69]Some remarks on associative rings and algebras, in Linear Algebras. Nat. Acad. Sci. Nat. Res. Coun. Publication 502, (1957), pp. 4-11.
Finite Groups
Introduction by Paul Fong and Warren J. Wong
[18]Über die Darstellung von Gruppen in Galoisschen Feldern, Actualités Sci. Indust. 195 (1935), 15 pp.
[27]On the modular representations of groups of finite order I (with C. Nesbitt), Univ. Toronto Studies no. 4 (1937). 21 pp.
[32]On the representation of groups of finite order, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 290-295.
Corrected by Richard Brauer
[33]On the Cartan invariants of groups of finite order, Ann. of Math. 42 (1941), 53-61.
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
[34]On the modular characters of groups (with C. Nesbitt), Ann. of Math. 42 (1941), 556-590.
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
[37]On the connection between the ordinary and the modular characters of groups of finite order, Ann. of Math. 42 (1941), 926-935.
[38]Investigations on group characters, Ann. of Math. 42 (1941), 936-958.
[39]On groups whose order contains a prime number to the first power I, Amer. J. Math. 64 (1942), 401-420.
Corrected by Richard Brauer
[40]On groups whose order contains a prime number to the first power II, Amer. J. Math. 64 (1942), 421-440.
Corrected by Richard Brauer
[42]On permutation groups of prime degree and related classes of groups, Ann. of Math. 44 (1943), 57-79.
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
[43]On the arithmetic in a group ring, Proc. Nat. Acad. Sci. U.S.A. 30 (1944), 109-114.
Corrected by Richard Brauer
[46]On simple groups of finite order (with H. F. Tuan), Bull. Amer. Math. Soc. 51 (1945), 756-766.
[47]On the representation of a group of order g in the field of the g-th roots of unity, Amer. J. Math. 67 (1945), 461-471.
[48]On blocks of characters of groups of finite order, I, Proc. Nat. Acad. Sci. U.S.A. 32 (1946), 182-186.
[49]On blocks of characters of groups of finite order, II, Proc. Nat. Acad. Sci. U.S.A. 32 (1946), 215-219.
[51]On Artin's L-series with general group characters, Ann. of Math. 48 (1947), 502-514.
[53]Applications of induced characters, Amer. J. Math. 69 (1947), 709-716.
Corrected by Richard Brauer
[54]On a conjecture by Nakayama, Trans. Roy. Soc. Canada III(3) 41 (1947), 11-19.
[60]On the algebraic structure of group rings, J. Math. Soc. Japan 3 (1951), 237-251.
[61]On the representations of groups of finite order, Proc. Internat. Cong. Math. 1950, vol. II, pp. 33-36.
[62]A characterization of the charaders of groups of finite order, Ann. of Math. 57 (1953), 357-377.
[63]On the characters of finite groups (with J. Tate), Ann. of Math. 62 (1955), 1-7.
Richard Brauer - Collected Papers. Volume II. Finite Groups
Richard Brauer 1901-1977
Contents
Finite Groups (Continued)
[64]On groups of even order (with K. A. Fowler), Ann. of Math. 62 (1955), 565-583.
[65]Zur Darstellungstheorie der Gruppen endlicher Ordnung, Math. Z. 63 (1956), 406-444.
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
Corrected by Richard Brauer
[67]Number-theoretical investigations on groups of finite order, Proc. Internat. Symp. Algebraic Number Theory, Tokyo-Nikko, 1955, pp. 55-62.
[68]On the structure of groups of finite otder, Proc. Internat. Cong. Math. 1954, vol. I, pp. 209-217.
[70]A characterization of the one-dimensional unimodular projective groups over finite fields (with M. Suzuki and G. E. Wall), Illinois J. Math. 2 (1958), 718-745.
[71]On a problem of E. Artin (with W. F. Reynolds), Ann. of Math. 68 (1958), 713-720.
[72]On the number of irreducible characters of finite groups in a given block (with W. Feit), Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 361-365.
[73]Zur Darstellungstheorie der Gruppen endlicher Ordnung II, Math. Z. 72 (1959), 25-46.
[74]On finite groups of even order whose 2-Sylow group is a quaternion group (with M. Suzuki), Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1757-1759.
[76]On blocks of representations of finite groups, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1888-1890.
[77]Investigation on groups of even order, I, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1891-1893.
Corrected by Richard Brauer
Corrected by Richard Brauer
[78]On finite groups with an abelian Sylow group (with H. S. Leonard), Canad. J. Math. 14 (1962), 436-450.
[79]On groups of even order with an abelian 2-Sylow subgroup, Arch. Math. (Basel) 13 (1962), 55-60.
[80]On some conjectures concerning finite simple groups, in Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford (1962), pp. 56-61.
[81]On finite groups and their characters, Bull. Amer. Math. Soc. 69 (1963), 125-130.
[82]Representations of finite groups, in Lectures on Modern Mathematics, vol. I, Wiley (1963), pp. 133-175.
[83]On quotient groups of finite groups, Math. Z. 83 (1964), 72-84.
[84]A note on theorems of Burnside and Blichfeldt, Proc. Amer. Math. Soc. 15 (1964), 31-34.
[85]Some applications of the theory of blocks of characters of finite groups I, J. Algebra 1 (1964), 152-167.
[86]Some applications of the theory of blocks of characters of finite groups II, J. Algebra 1 (1964), 307-334.
[89]On finite Desarguesian planes I, Math. Z. 90 (1965), 117-123.
[90]On finite Desarguesian planes II, Math. Z. 91 (1966), 124-151.
[91]Investigation on groups of even order II, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 254-259.
Corrected by Richard Brauer
[92]Some applications of the theory of blocks of characters of finite groups III, J. Algebra 3 (1966), 225-255.
[93]A characterization of the Mathieu group M_{12} (with P. Fong), Trans. Amer. Math. Soc. 122 (1966), 18-47.
[94]An analogue of Jordan's theorem in characteristic p (with W. Feit), Ann. of Math. 84 (1966), 119-131.
Corrected by Richard Brauer
[95]Some results on finite groups whose order contains a prime to the first power, Nagoya Math. J. 27 (1966), 381-399.
[97]On simple groups of order 5・3^a・2^b, Mimeographed Notes, Harvard Univ. (1967). 49 pp.
[98]Über endliche lineare Gruppen von Primzahlgrad, Math. Ann. 169 (1967), 73-96.
[99]On a theorem of Burnside, Illinois J. Math. 11 (1967), 349-352.
[100]On blocks and sections in finite groups I, Amer. J. Math. 89 (1967), 1115-1136.
Corrected by Richard Brauer
[101]On pseudo groups, J. Math. Soc. Japan 20 (1968), 13-22.
[102]On blocks and sections in finite groups II, Amer. J. Math. 90 (1968), 895-925.
Corrected by Richard Brauer
Corrected by Richard Brauer
[104]On a theorem of Frobenius, Amer. Math. Monthly 76 (1969), 12-15.
[105]Defect groups in the theory of representations of finite groups, Illinois J. Math. 13 (1969), 53-73.
Richard Brauer - Collected Papers. Volume III. Finite Groups, Lie Groups, Number Theory, Polynomials and Equations; Geometry, and Biography
Richard Brauer 1901-1977
Contents
Finite Groups (Continued)
[106]On the order of finite projective groups in a given dimension, Nachr. Akad. Wiss. Göttingen 11 (1969), 103-106.
[109]On the first main theorem on blocks of characters of finite groups, Illinois J. Math. 14 (1970), 183-187.
[111]On finite Desarguesian planes III, Math. Z. 117 (1970), 76-82.
[112]Some applications of the theory of blocks of characters of finite groups IV, J. Algebra 17 (1971), 489-521.
[113]Types of blocks of representations of finite groups, Proc. Symp. Pure Math., 1971, vol. 21, pp. 7-11.
[114]Some properties of finite groups with wreathed Sylow 2-subgroup (with W. J. Wong), J. Algebra 19 (1971), 263-273.
[115]Character theory of finite groups with wreathed Sylow 2-subgroups, J. Algebra 19 (1971), 547-592.
[116]Blocks of characters, Proc. Internat. Cong. Math. 1970, vol. 1, pp. 341-345.
[118]Finite simple groups of 2-rank two (with J. L. Alperin and D. Gorenstein), Scripta Math. 29 (1973), 191-214.
[120]On the structure of blocks of characters of finite groups, Proc. 2nd Internat. Conf Theory of Groups (Canberra, 1973), pp. 103-130.
[121]Some applications of the theory of blocks of characters of finite groups V, J. Algebra 28 (1974), 433-460.
[122]On 2-blocks with dihedral defect groups, Symposia Math. (INDAM) 13 (1974), 367-393.
[123]On the centralizers of p-elements in finite groups (with P. Fong), Bull. London Math. Soc. 6 (1974), 319-324.
[125]On finite groups with cyclic Sylow subgroups, I, J. Algebra 40 (1976), 556-584.
[126]Notes on representations of finite groups I, J. London Math. Soc. (2) 13 (1976), 162-166.
[127]Blocks of characters and structure of finite groups, Bull. (new series) Amer. Math. Soc. 1 (1979), 21-38.
[128]On finite projective groups, in Contributions to Algebra, Academic Press, New York (1977), pp. 63-82.
[129]On finite groups with cyclic Sylow subgroups, II, J. Algebra 58 (1979), 291-318.
Lie Groups
[1]Über die Darstellung der Drehungsgruppe durch Gruppen linearer Substitutionen, Inaugural-Dissertation zur Erlangung der Doktorwürde, Friedrich-Wilhelms-Universität, Berlin. 71 pp.
[8]Die stetigen Darstellungen der komplexen orthogonalen Gruppe, Sitzungsber. Preuss. Akad. Wiss. (1929), 626-638.
[19]Spinors in n dimensions (with H. Weyl), Amer. J. Math. 57 (1935), 425-449.
[21]Sur les invariants intégraux des variétés représentatives des groupes de Lie simples clos, C. R. Aead. Sci. Paris 201 (1935), 419-421.
[25]On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857-872.
[26]Eine Bedingung für vollständige Reduzibilität von Darstellungen gewöhnlicher und infinitesimaler Gruppen, Math. Z. 41 (1936), 330-339.
[29]Sur la multiplication des caractéristiques des groupes continus et semisimples, C. R. Acad. Sci. Paris 204 (1937), 1784-1786.
[88]On the relation between the orthogonal group and the unimodular group, Arch. Rational Mech. Anal. 18 (1965), 97-99.
Number Theory
[52]On the zeta-functions of algebraic number fields, Amer. J. Math. 69 (1947), 243-250.
[58]On the zeta-functions of algebraic number fields II, Amer. J. Math. 72 (1950), 739-746.
[59]Beziehungen zwischen Klassenzahlen von Teilkörpern eines galoisschen Körpers, Math. Nachr. 4 (1951), 158-174.
[66]A note on the class-numbers of algebraic number fields (with N. C. Ankeny and S. Chowla), Amer. J. Math. 78 (1956), 51-61.
[87]On certain classes of positive definite quadratic forms, Acta Arith. 9 (1964), 357-364.
[119]A note on zeta-functions of algebraic number fields, Acta Arith. 24 (1973), 325-327.
[124]On the resolvent problem, Ann. Mat. Pura Appl. 102 (1975), 45-55.
[124-1]On the resolvent problem (ERRATUM), Ann. Mat. Pura Appl. 109 (1976), 391.
Polynomials and Equations; Geometry
[2]Über die Irreduzibilität einiger spezieller Klassen von Polynomen (with A. Brauer and H. Hopf), Jahresber. Deutsch. Math.-Verezn 35 (1926), 99-112.
[6]Über einen Satz für unitäre Matrizen, Tôhoku Math. J. 30 (1928), 72.
[9]Über einen Satz für unitäre Matrizen (with A. Loewy), Tôhoku Math. J. 32 (1930), 44-49.
[17]Über die Kleinsche Theorie der algebraischen Gleichungen, Math. Ann. 110 (1934), 473-500.
[20]Über Irreduzibilitätskriterien von I. Schur und G. Polya (with A. Brauer), Math. Z. 40 (1935), 242-265.
[23]Symmetrische Funktionen. Invarianten von linearen Gruppen endlicher Ordnung, accepted for publication (1936) in Enzyklopädie der Mathematischen Wissenschaften, vol. IB 12, Teubner, Leipzig. Not published, because of political considerations, until Math. J. Okahama U. 21 (1979), 91-113.
[24]A characterization of null systems in projective space, Bull. Amer. Math. Soc. 42 (1936), 247-254.
[35]A generalization of theorems of Schönhardt and Mehmke on polytopes (with H. S. M. Coxeter), Trans. Roy. Soc. Canada II1(3) 34 (1940), 29-34.
[45]A note on systems of homogeneous algebraic equations, Bull. Amer. Math. Soc. 51 (1945), 749-755.
[55]A note on Hilbert's Nullstellensatz, Bull. Amer. Math. Soc. 54 (1948), 894-896.
Biography
[96]Emil Artin, Bull. Amer. Math. Soc. 73 (1967), 27-43.
[117]On the work of John Thompson, Proc. Internat. Cong. Math. 1970, vol. 1, pp. 15-16.
Richard Brauer - Collected Papers. Not collected in Volume I-III
[75]Les groupes d'ordre fini et leurs caractères, in Sém. P. Dubreil, Dubreil-Jacotin et Pisot, Paris (1959), pp. 6-01 - 6-16.
[103]On simple groups of order 5・3^a・2^b, Bull. Amer. Math. Soc. 74 (1968), 900-903.
[110]Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups (with J. L. Alperin and D. Gorenstein), Trans. Amer. Math. Soc. 151 (1970), 1-261.

Citation preview

Richard Brauer: Collected Papers Volume I Theory of Algebras, and Finite Groups

Mathematicians of Our Time Gian-Carlo Rota, series editor Richard Brauer: Collected Papers Volume I Theory of Algebras, and Finite Groups edited by Paul Fong and Warren J. Wong [17] Richard Brauer: Collected Papers Volume II Finite Groups edited by Paul Fong and Warren J. Wong [18] Richard Brauer: Collected Papers Volume III Finite Groups, Lie Groups, Number Theory, Polynomials and Equations; Geometry, and Biography edited by Paul Fong and Warren J. Wong [19] Paul Erdös: The Art of Counting edited by Joel Spencer [5] Einar Hille: Classical Analysis and Functional Analysis Selected Papers of Einar Hille edited by Robert R. Kallman [11] Mark Kac: Probability, Number Theory, and Statistical Physic;s Selected Papers edited by K. Baclawski and M. D. Donsker [14] Charles Loewner: Theory of Continuous Groups notes by Harley Flanders and Murray H. Protter [l] Percy Alexander MacMahon: Collected Papers Volume I Combinatorics edited by George E. Andrews [13] George Polya: Collected Papers Volume I Singularities of Analytic Functions edited by R. P. Boas [7] Goerge Polya: Collected Papers Volume II Location of Zeros edited by R. P. Boas [8]

Collected Papers of Hans Rademacher Volume I edited by Emil Grosswald [3] Collected Papers of Hans Rademacher Volume II edited by Emil Grosswald [4] Stanislaw Ulam: Selected Works Volume I Sets, Numbers, and Universes edited by W. A. Bayer, J. Mycielski, and G.-C. Rota [9] Norbert Wiener: Collected Works Volume I Mathematical Philosophy and Foundations; Potential Theory; Brownian Movement, Wiener Integrals, Ergodic and Chaos Theories; Turbulence and Statistical Mechanics edited by P. Masani [10] Norbert Wiener: Collected Works Volume II Generalized Harmonie Analysis and Tauberian Theory; Classical Harmonie and Complex Analysis edited by P. Masani [15] Oscar Zariski: Collected Papers Volume I Foundations of Algebraic Geometry and Resolution of Singularities edited by H. Hironaka and D. Mumford [2] Oscar Zariski: Collected Papers Volume II Holomorphic Functions and Linear Systems edited by M. Artin and D. Mumford [6] Oscar Zariski: Collected Papers Volume III Topology of Curves and Surfaces, and Special Topics in the Theory of Algebraic Varieties edited by M. Artin and B. Mazur [12] Oscar Za,riski: Coflected Papers Volume IV Equisingularity on Algebraic Varieties edited by J. Lipman and B. Teissier [16] Note: Series number appears in brackets.

Richard Brauer

Collected Papers Volume I Theory of Algebras, and Finite Groups Edited by Paul Fong and Warren J. Wong

The MIT Press Cambridge, Massachusetts, and London, England

Publisher's note: Richard Brauer left an annotated s�t of hi-spublications. Following his wishes, these annotations have been incorporated by hand into these reprints. Publication of this volume was made possible in part by a grant from the International B usiness Machines Corporation. Copyright © 1980 by the Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. This book was printed and bound by The Murray Printing Company in the United States of America. Libra;y of Congress Cataloging in Publication Data Brauer, Richard, 1901Collected papers. (Mathematicians of our time; v. 17-19) Text in English and German. Bibliograph:y: p. CONTENTS: v. 1. Theory of algebras and finite groups.-v. 2. Finite groups.v. 3. Finite groups, Lie groups, number theory, polynomials and equations; geometry, and biography. 1. Mathematics-Collected works. I. Fong, Paul, 1933- II. Wong, Warren J. III. Series. 80-17622 510 QA3.B83 ISBN 0-262-02135-8 (v. 1) 0-262-02148-X (v. 2) 0-262-02149-8 (v. 3) 0-262-02157-9 (complete set)

TO ILSE whom I first met in November 1920 and whom I married in September 1925

)

Richard Brauer 1901-1977

Contents (Bracketed numbers are from the B~bliography) Preface XV

Richard Dagobert Brauer by J. A. Green XXI

Bibliography of Richard Brauer xlv

Volume I Theory of Algebras Introduction by 0. Goldman 3 [3] Über Zusammenhänge zwischen arithmetischen und invariantentheoretischen Eigenschaften von Gruppen linearer Substitutionen

5

[4] Über minimale Zerfällungskörper irreduzibler Darstellungen (with E. Noether)

12

[5] Untersuchungen über die arithmetischen Eigenschaften von Gruppen linearer Substitutionen I

20

[7] Über Systeme hyperkomplexer Zahlen

40

[11] Zum Irreduzibilitätsbegriff in der Theorie der Gruppen linearer homogener Substitutionen (with I. Schur)

69

[12] Untersuchungen über die arithmetischen Eigenschaften von Gruppen linearer Substitutionen II

88

[13] Über die algebraische Struktur von Schiefkörpern

103

[14] Beweis eines Hauptsatzes in der Theorie der Algebren (with H. Hasse and E. Noether)

115

[15] Über die Konstruktion der Schiefkörper, die von endlichem Rang in bezug auf ein gegebenes Zentrum sind

121

[16] Über den Index und den Exponenten von Divisionalgebren

142

[22] Algebra der hyperkomplexen Zahlensysteme (Algebren)

153

[28] On the regular representations of algebras (with C. Nesbitt)

190

[30] On normal division algebras of index 5

195

[31] On modular and 1)-adic representations of algebras

199

[36] On sets of matrices with coefficients in a division ring

206

[41] On the nilpotency of the radical of a ring

253

[44] On hypercomplex arithmetic and a theorem of Speiser

260

[50] On splitting fields of simple algebras

273

[56] Representations of groups and rings

285

[57] On a theorem of H. Cartan

306

[69] Some remarks on associative rings and algebras

308

Finite Groups Introduction by Paul Fong and Warren J. Wong 319 [18] Über die Darstellung von Gruppen in Galoisschen Feldern

323

[27] On the modular representations of groups of finite order I (with C. Nesbitt)

336

[32] On the representation of groups of finite order

355

[33] On the Cartan invariants of groups of finite order

361

[34] On the modular characters of groups (with C. Nesbitt)

370

vm

CONTENTS

[37] On the connection between the ordinary and the modular characters of groups of finite order

405

[38] Investigations on group characters

415

[39] On groups whose order contains a prime number to the first power I

438

[40] On groups whose order contains a prime number to the first power II

458

[42] On permutation groups of prime degree and related classes of groups

478

[43] On the arithmetic in a grou p ring

501

[46] On simple groups of finite order (with H. F. Tuan)

507

[47] On the representation of a group of order g in the field of the g-th roots of uni~y

518

i

[48] On blocks of characters of groups of finite order, I

529

[49] On blocks of characters of groups of finite order, II

534

[51] On Artin's L-series with general group characters

539

[53] Applications of induced characters

552

[54] On a conjecture by Nakayama

560

[60] On the algebraic structure of group rings

569

[61] On the representations of groups of finite order

584

[62] A characterization of the characters of groups of finite order

588

[63] On the characters of finite groups (with J. Tate)

609

1x

CONTENTS

Volume II Finite Groups (Continued) [64] On groups of even order (with K. A. Fowler)

3

[65] Zur Darstellungstheorie der Gruppen endlicher Ordnung

22

[67] Number theoretical investigations on groups of finite order

61

[68] On the structure of groups of finite order

69

[70] A characterization of the one-dimensional unimodular projective groups over finite fields (with M. Suzuki and G. E. Wall)

78

[71] On a problem of E. Artin (with W. F. Reynolds)

106

[72] On the number of irreducible characters of finite groups in a give? block (with W. Feit)

114

[73] Zur Darstellungstheorie der Gruppen endlicher Ordnung II

119

[74] On finite groups of even order whose 2-Sylow group is a quaternion group (with M. Suzuki)

141

[76] On blocks ofrepresentations of finite groups

144

[77] Investigation on groups of even order, I

14 7

[78] On finite groups with an abelian Sylow group (with H. S. Leonard)

150

[79] On groups of even order with an abelian 2-Sylow subgroup

165

[80] On some conjectures concerning finite simple groups

171

[81] On finite groups and their characters

177

[82] Representations of finite groups

183

x

CONTENTS

[83) On quotient groups of finite groups

226

[84) A note on theorems of Burnside and Blichfeldt

239

[85) Some applications of the theory of blocks of characters of finite groups. I

243

[86) Some applications of the theory of blocks of characters of finite groups. II

259

[89) On finite Desarguesian planes. I

287

[90) On finite Desarguesian planes. II

294

[91) Investigation on groups of even order, II

322

[92) Some applications of the theory of blocks of characters of finite groups, III

328

[93) A characterization of the Mathieu group M 12 (with P. Fong)

359

[94) An analogue of Jordan's theorem in characteristic p (with W. Feit)

389

[95) Some results on finite groups whose order contains a prime to the first power

402

[97) On simple groups of order 5 · 3a · 2b

421

[98) Über endliche lineare Gruppen von Primzahlgrad

471

[99) On a theorem of Burnside

495·

[100) On blocks and sections in finite groups, I

499

[101) On pseudo groups

521

[102) On blocks and sections in finite groups, II

531

[104) On a theorem of Frobenius

562

[105) Defect groups in ihe theory of representations of finite groups

566

XI

CONTENTS

Volume III Finite Groups (Continued) [106] On the order of finite projective groups in a given dimension

3

[109] On the first main theorem on blocks of characters of finite groups

7

[111] On finite Desarguesian planes. III

12

[112] Some applications of the theory of blocks of characters of finite groups IV

20

[113] Types of blocks of representations of finite groups

53

[114] Some properties of finite groups with wreathed Sylow 2-subgroup (with W. J. Wong)

58

[115] Character theory of finite groups with wreathed Sylow 2-subgroups

69

[116] Blocks of characters

115

[118] Finite simple groups of 2-rank two (with J. L. Alperin and D. Gorenstein)

120

[120] On the structure of blocks of characters of finite groups

144

[121] Some applications of the theory of blocks of characters of finite groups. V

172

[122] On 2-blocks with dihedral defect groups

200

[123] On the centralizers of p-elements in finite groups (with P. Fong)

227

[125] On finite groups with cyclic Sylow subgroups, I

233

[126] Notes on representations of finite groups, I

262

XII

CONTENTS

[127) Blocks of characters and structure of finite groups

267

[128) On finite projective groups

285

[129) On finite groups with cyclic Sylow subgroups, II

305

Lie Groups

[1] Über die Darstellung der Drehungsgruppe durch Gruppen

linearer Substitutionen

335

[8] Die stetigen Darstellungen der komplexen orthogonalen Gruppe

404

[19) Spinors in n dimensions (with H. Weyl)

418

[21] Sur les invariants integraux des varietes representatives des groupes de Lie simples clos

443

(

[25) On algebras which are connected with the semisimple continuous groups

446

[26) Eine Bedingung für vollständige Reduzibilität von Darstellungen gewöhnlicher und infinitesimaler Gruppen

462

[29) Sur la multiplication des caracteristiques des groupes continus et semi-simples

472

[88) On the relation between the orthogonal group and the unimodular group

475

. N umher T~eory

[521 On the zeta-functions of algebraic number fields

481

[58) On the zeta-functions of algebraic number fields II

489

[59) Beziehungen zwischen Klassenzahlen von Teilkörpern eines galoisschen Körpers

497

xiii

CONTENTS

[66] A note on the class-numbers of algebraic number fields (with N. C. Ankeny and S. Chowla)

514

[87] On certain classes of positive definite quadratic forms

525

[119] A note on zeta-functions of algebraic number fields

533

[124] On the resolvent problem

536

Polynomials and Equations; Geometry

[2] Über die Irreduzibilität einiger spezieller Klassen von Polynomen (with A. Brauer and H. Hopf)

549

[6] Über einen Satz für unitäre Matrizen

563

[9] Über einen Satz für unitäre Matrizen (with A. Loewy)

564

[17] Über die Kleinsche Theorie der algebraischen Gleichungen

570

[20] Über Irreduzibilitätskriterien von I. Schur und G. P6lya (with A. Brauer)

598

[23] Symmetrische Funktionen. Invarianten von linearen Gruppen endlicher Ordnung

62.2

[24] A characterization of null systems in projective space

645

[35] A generalization of theorems of Schönhardt and Mehmke on polytopes (with H. S. M. Coxeter)

653

[45] A note on systems of homogeneous algebraic equations

659

[55] A note on Hilbert's Nullstellensatz

666

Biography

[96] Emil Artin

671

[117] On the work of John Thompson

XIV

CONTENTS

688

Preface lt seems appropriate to describe in the preface to my collected papers the influences which led me to mathematics, and the course of my mathematical education. My interest in science in general, and in mathematics in particular, was awakened at a rather early age by my brother Alfred, who is seven years older than I. Of course, my own ideas were very immature at first. Since many new inventions came into use during the first ten years of this century, I dreamed at first of becoming an inventor. I learned that one had to know physics; I started to do simple experiments and to read books on science, which were often too difficult for me. However, what remained was the habit of doing things by myself. This remained so during most of my years in high school. I attended a Humanistisches Gymnasium in Berlin in which the emphasis was on Latin, Greek, and history, while modern languages and science were more or less neglected. Most of my teachers were not very competent. An exception was one teacher with whom I took an introductory calculus course in 1917. I learned later that he had taken a Ph.D. with Frobenius. I started the equivalent of my college education in February 1919 at the Technische Hochschule Berlin-Charlottenburg (now the Technical University of Berlin). I recognized soon that my own interests were more theoretical than practical, and I transferred to the University of Berlin after one term. My first decisive experience at the University of Berlin was the lectures of Erhard Schmidt. lt is not easy to describe their fascination. When Schmidt stood in front of a blackboard, he never used notes and was hardly ever well prepared. He gave the impression of developing the theory right there and then. In fact, he would stop in the middle of a proof and say, ".Let me start again. I see a better way of doing this." One of his publications originated from a proof he gave in a dass I attended some two years later. Occasionally Schmidt stopped in the middle of a proof and asked the dass, "Do you see what will come next? You should." He then waited patiently for an answer. We were thus forced to try to work out the details of a proof mentally in the dassroom. After one year I transferred again, this time to the University of Freiburg. lt was the custom of most German students at that time to spend at least one term in a different part of the country. The whole schedule of courses in Freiburg did not fit well into my education. Still, the term was not lost. Since I had a lot of free time, I started to read most of the first two volumes of H. Weber's

Lehrbuch der Algebra (Braunschweig, 1898 and 1899), which is still a dassic in 1977. There was one course in Freiburg which became important for my later development. This was a course of Oscar Bolza on the theory of invariants. Bolza had retired to Freiburg where occasionally he taught at the university, after teaching for a long time at the University of Chicago. In Freiburg I also first came in contact" with Alfred Loewy, one of the two professors in Freiburg. The course I took with him was on differential geometry, not his specialty and not very memorable. He also conducted a history seminar. I had to report on the topic "Could Gauss in 1800 Have Proved That the General Equation of the Fifth Degree Could Not Be Solved by Radicals?" This is a question which cannot be answered now, nor is it likely that it ever will be answered. I gave a report in which I worked out an elementary proof which in no way resembles the methods used in Galois theory. Loewy disagreed. He believed that Gauss was already aware of the importance of symmetry. lt was typical of Loewy that he remained in correspondence with this young student for years. In the fall semester of 1920 I returned to the University of Berlin, where I remained until I received my Ph.D. in 1925. I was now ready to take the advanced courses on algebra and number theory of I. Schur. Schur was quite different from Erhard Schmidt as a teacher. He was very well prepared for his dasses, anä he lectured very fast. If one did not pay the utmost attention to his words, one was quickly lost. There was hardly any time to take notes in dass; one had to write them up at home. At the end of each chapter of his course, he gave a survey of related developments and advised, "When you have the time, read Hilbert's Zahlbericht, Hilbert's work on invariants, Takagi on dass field theory .... " He conducted weekly problem hours, and almost every time he proposed a difficult problem. Some of the problems had already been used by his teacher Frobenius, and others originated with Schur. Occasionally he mentioned a problem he could not solve himself. One of the difficult problems was solved by Heinz Hopf and also by my brother Alfred and myself. We saw immediately that by combining our methods, we could go a step further than Schur. Our joint paper [2] in the list below originated this way. In the fall of 1922 I started to attend the seminar of Schur and the seminar conducted jointly by Erhard Schmidt and Ludwig Bieberbach. At the seminars a list of important publications was presented, and the participants then volunteered to report on those which looked interesting to them. In the fall of 1922 I reported on G. D. Birkhoff's work on dosed curves which form solutions of the Euler equations of certain variational problems. In preparing my talk I found what I believed to be a simpler proof of one of the theorems. At the condusion of my report, Bieberbach asked me whether he could use this proof

XVI

PREFACE

in the book he was writing. lt then appeared in a slightly modified form in L. Bieberbach, Differentialgleichungen: Grundlehren der Mathematischen Wissenschaften (Julius Springer, Berlin, 1923). In a way, this one-page proof represents my first publication. During a later term, I reported in the Schmidt-Bieberbach seminar on a chapter in Hilbert's book on integral equations, but I started to become more and more attracted by the Schur seminar. I first reportedjointly with my brother on C. L. Siegel's paper on what is now known as the Thue-Siegel--Roth theorem. More important for my own development were the reports the following year. I reportedjointly with Heinz Hopf on Schur's "Neue Begründung der Theorie der Gruppencharaktere." Then I reported on Schur's "Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen" and "Neue Anwendungen der Integralrechnung auf Probleme der Invariantentheorie, 1. Mitteilung" (cf. the Collected Papers of I. Schur, Springer-Verlag, Berlin, Heidelberg, and New York, 1973). My Ph.D. thesis [1] dealt with the same problem as the second part of the "Neue Anwendungen der Integralrechnung" quoted above. Schur had suggested that I give a more algebraic approach and answer a question which he himself had not bee"n able to settle. The same question was answered a little later by H. Weyl; his methods were entirely different and his investigation was undertaken in a much more general context (cf. H. Weyl, chapter II, §5, of "Theorie der Darstellung halbeinfacher Gruppen," Mathematische Zeitschrift 23 (1925)). As was customary, the last page of [1] contains my "Lebenslauf" (vita) and the list of courses I attended during my college days. The 1920s were a brilliant time at the U niversity of Berlin, and many of the names of the lecturers are still famous today. Among the physicists whose classes I attended were the Nobel prize laureates Planck, Einstein, and von Laue. My first academic position was that of Assistant at the Mathematical Institute of the University of Königsberg. In 1927, I became Privatdozent, that is, I was given the venia legendi, the right to give lectures. The University of Königsberg had been outstanding during the nineteenth century, but had then fallen into neglect. After Hilbert and Minkowski left, only second-rate mathematicians stayed for any length of time. Now the Prussian Minister of Education tried to improve conditions. There were two chairs for mathematics during my years at Königsberg, and these were occupied by K. Reidemeister and G. Szegö. In addition, there was one Privatdozent with title of Professor, W. Rogosinski. As far as it was possible, we four had to cover the whole of mathematics. During my eight years at Königsberg, I taught practically all the usual courses, but none in the fields in which I was working at the time. In attending mathematical meetings, I soon met mathematicians from other

xvii

PREFACE

universities who had similar interests. I mention Emmy Noether, with whom I collaborated on [4] in 1927, and H. Hasse, with whom Emmy Noether and I collaborated on [14] in 1931. The paper [ 11] was written in collaboration with I. Schur in 1930. During one of my visits to Berlin, Schur surprised me by suggesting that we should write a bookjointly on all aspects of the representation theory of groups, many of which were still unexplored at that time. My papers [14] and [18] originated in this way. A year or so later Schur told me that, with all his other duties and interests, he simply did not have the time to work on a book. The project would now have to indude chapters on the work of E. Wigner, which had appeared in the meantime, on the application of representation theory to quantum mechanic:s. He suggested that now I should write a hook on group representations with a young physicist whom he knew. Since Hitler seized power not long afterward, and I had to leave Germany, the project had tobe dropped. I lost my position in Königsberg in the spring of 1933 after Hitler became Reichskanzler of Germany. In the fall of 1933, I received an invitation for a year's visit from the University of Kentucky, which I accepted. During the academic year 1934-1935 I was assistant to Hermann Weyl at the Institute for Advanced Study in Princeton. I had hoped since the days of my Ph.D. thesis to get in contact with him some day; this dream was now fulfilled. \ We soon collaborated on [19]. Carl Ludwig Siegel was at the Institute during the spring term 1935. lt seems superfluous to describe the stimulating influence on all who came to know him. I mention one example which became important for my later work. In 1934 Heilbronn had proved Gauss's famous conjecture that the dass number of an imaginary quadratic number field with the discriminant -d tends to infinity with d. In a talk in the colloquium Siegel spoke about his own paper in Acta Arithmetica 1 ( 1935), 83-86, in which he gave an extremely short proof of a refinement of the theorem. Siegel also stated a conjecture dealing with algebraic number fields of arbitrary degree of which his theorem for quadratic fields is a special case. Siegel's lecture became very important for me about 10 years later. In the meantime I had often been with E. Artin, and discussions with him had familiarized me with many of his ideas about L-series and dass field theory. I had succeeded in 1946 in proving part of one of his conjectures (cf. [51]). When I prepared a talk about this for the Symposium on Problems of Mathematics held in Princeton in December 1946 in commemoration of the bicentennial of Princeton University, it occurred to me that I was now able to prove Siegel's conjecture (cf. [52]).

xvm

PREF ACE

I consider my year at the Institute in Princeton as the last part of my mathematical education. In the fall of 1935 I became Assistant Professor at the University of Toronto. Richard Brauer Belmont, Massachusetts February, 1977

xix

PREFACE

RICHARD DAGOBERT BRAUER J. A. GREEN

Richard Dagobert Brauer, Emeritus Professor at Harvard University and one of the foremost algebraists ofthis century, died on April 17, 1977, in Boston, Massachusetts. He had been an Honorary Member of the Society since 1963. Richard Brauer was born on 10 February, 1901, in Berlin-Charlottenburg, Germany; he was the youngest of three children of Max Brauer and his wife Lilly Caroline. Max Brauer was an influential and wealthy businessman in the wholesale leather trade, and Richard was brought up in an affluent and cultured home with his brother Alfred and his sister Alice. Richard Brauer's early years were happy and untroubled. He attended the Kaiser-Friedrich-Schule in Charlottenburg from 1907 until he graduated from there in 1918. He was already interested in science and mathematics as a young boy, an interest which owed much to the influence ofhis gifted brother Alfred, who was seven years older than Richard. His youth saved him from service with the German army during the first W orld War. He graduated from high school in September 1918, and he and his classmates were drafted for civilian service in Berlin. Two months later the War ended, andin February 1919 he was able to enrol at the Technische Hochschule in Berlin-Charlottenburg (now the Technische Universität Berlin). The choice of a technical curriculum had been the result of Richard's boyhood ambition to become an inventor, but he soon realised that, in this own words, his interests were " more theoretical than practical ", and he transferred to the U niversity of Berlin after one term. He studied there for a year, then spent the summer semester of 1920 at the University of Freiburg -it was a tradition among German students to spend at least one term in a different university-and returned that autumn to the University of Berlin, where he remained until he took his Ph.D. degree in 1925. The University of Berlin contained many brilliant mathematicians and physicists in the nineteen-twenties. During his years as a student Richard Brauer attended lectures and seminars by Bieberbach, Caratheodory, Einstein, Knopp, von Laue, von Mises, Planck, E. Schmidt, I. Schur and G. Szegö, among many others. In the customary postscript to his doctoral dissertation [l], Brauer mentions particularly Bieberbach, von Mises, E. Schmidt and I. Schur. There is no doubt that the profoundest influence among these was that of Issai Schur. Schur had been a pupil of G. Frobenius, and had gradulated at Berlin in 1901; he had been "ordentlicher Professor" (füll professor) there since 1919. His lectlires on algebra and number theory were famous for their masterly structure and polished delivery. Richard Brauer's first published paper arose from a problem posed by Schur in a seminar on number theory in the winter semester of 1921. Alfred Brauer also participated in this seminar. He was less fortunate than Richard, in that his studies were seriously interrupted by the War; he had served for four years with the army and been very badly wounded. Tue Brauer brothers succeeded in solving Schur's problem in one week, and in the same week a completely different solution was found by Heinz Hopf. The Brauer proof was published in the book by Polya and Szegö (1925; p. 137, [BULL. LONDON MATH,

Soc., 10 (1978), 317-342]

318

RICHARD DAGOBERT BRAUER

pp. 347-350), and some time later the Brauers and Hopf combined and generalized their proofs in their joint paper [2]. Richard Brauer also participated in seminars conducted by E. Schmidt and L. Bieberbach on differential equations and integral equatfons-a proof which he gave in a talk at this seminar in 1922 appears, with suitable acknowledgment, in Bieberbach's book on differential equations (1923; p. 129). But Brauer became more and more involved in Schur's seminar. As a participant in this, he reported on the first part of Schur's paper " Neue Anwendungen der Integralrechnung auf Probleme der Invariantentheorie " (1924), which shows how Hurwitz's method of group integration can be used for the study of the linear representations of continuous linear groups. In the second part of this work, Schur applied his method to determine all the irreducible (continuous, finite-dimensional) representations of the real orthogonal and rotation groups. He suggested to Brauer that it might also be possible to do this in a more algebraic way. This became Brauer's doctoral thesis [1], for which he was awarded his Ph.D. summa cum laude on March 16, 1926. On September 17, 1925 Richard Brauer married Ilse Karger, a fellow-student whom he had first met in November 1920 at Schur's lecture course on number theory. Use Karger was the daughter of a Berlin physician. She studied experimental physics and took her Ph.D. in 1924, but she realized during the course of her studies thät she ~as more interested in mathematics than in physics, and she took mathematics courses with the idea of becoming a school-teacher. In fact she subsequently held instructorships in mathematics at the Universities of Toronto and Michigan and at Brandeis University, and she eventually became assistant professor at Boston University. The marriage of Ilse and Richard Brauer was a long and very happy one. Their two sons George Ulrich (born 1927) and Fred Günther (born 1932) both became active research mathematicians, and presently hold chairs at, respectively, the U niversity of Minnesota, Minneapolis, and the University of Wisconsin, Madison. Brauer's first academic post was at the U niversity of Königsberg (now Kaliningrad), where he was offered an assistantship by K. Knopp. He started there in the autumn of 1925, became Privatdozent (this is the grade which confers the right to give lectures) in 1927, and remained in Königsberg until 1933. The mathematics department at that time had two chairs, occupied by G. Szegö and K. Reidemeister (Knopp left soon after Brauer arrived), with W. Rogosinski, Brauer and T. Kaluza in more junior positions. The Brauers enjoyed the friendly social life of this small department, and Richard Brauer enjoyed the varied teaching which he was required to give. During this time he also met mathematicians from other universities with whom he had common interests, particularly Emmy Noether and H. Hasse. This was the time when Brauer made his fundamental contribution to the algebraic theory of simple algebras. In [4], he and Emmy Noether characterized Schur's " splitting fields " of a given irreducible representation r of a given finite- dimensional algebra, in terms of the division algebra associated to r. Brauer developed in [3], [5] and [7] a theory of central division algebras over a given perfect field, and showed in [13] that the isomorphism classes of these algebras can be used to form a commutative group whose properties give great insight into the structure of simple algebras. This group became known (to its author's embarrassment!) as the "Brauer group ", and played an essential part in the proof by Brauer, Noether and Hasse [14] of the longstanding conjecture that every rational division algebra is cyclic over its centre. Early in 1933 Hitler became Chancellor of the German Reich, and by the end of March had established himself as dictator. In April the new Naziregime began to

xxii

BIOGRAPHY

RICHARD DAGOBERT BRAUER

319

implement its notorious antisemitic policies with a series of laws designed to remove Jews from the" intellectual professions "such as the civil service, the law and teaching. All Jewish university teachers were dismissed from their posts. Later some exemptions were made-it is said at the request of Hindenburg, the aged and by now virtually powerless President of the Reich-to allow those who had held posts before the first World War, and those who had fought in that War, to retain thefr jobs. Richard Brauer came into neither_ of these categories, and was not reinstated. lt is tragically weil known that the " clemency " extended to those who were allowed to remain at their posts was short-lived. Alfred Brauer, whose war service exempted him from dismissal in 1933, eventually came to the United States in 1939. Their sister Alice stayed in Germany and died in an extermination camp during the second World War. The abrupt dismissal of Jewish intellectuals in Germany in 1933 evöked shock and bewilderment abroad. Committees were set up and funds raised, particularly in Great Britain and the United States, to find places for these first refugees from Nazism. Through the agency of the Emergency Committee for the Aid of Displaced German Scholars, which had its headquarters in New York, and with the help of the Jewish community in Lexington, Kentucky, enough money was raised to offer Richard Brauer a visiting professorship for one year at the University of Kentucky. He arrived in Lexington in November 1933, speaking very little English, but already with a reputation as one of the most promising young rnathematicians of his day. His arrival was greeted wit~ sympathetic curiosity; the local paper reported an interview with the newcomer, conducted through an interpreter, and recorded Brauer's first impressions of American football. Ilse Brauer and the two children, who had stayed behind in Berlin, followed three months later. The friendly welcome which the Brauers found in Lexington, and their own adaptability, made the transition to life in the United States an easy one. In that same academic year 1933-34 the Institute for Advanced Study at Princeton came into füll operation. Among its first permanent professors was Hermann Weyl. Brauer did not know Weyl personally, but had always hoped to do so from the time when he had been writing his thesis on the rotation group; Weyl's classic papers, in which he combined the infinitesimal methods of Lie and E. Cartan with Schur's group integration method to determine the characters of all compact semisimple Lie groups, appeared in 1925-26. lt was therefore the fulfilment of a dream for Brauer to be invited to spend the year 1934-35 at the Institute as Weyl's assistant. Brauer's great admiration and respect for Weyl were returned. Many years later Weyl wrote that working with Brauer had been the happiest experience of scientific collaboration which he had ever had in his life. The famous joint paper on spinors [19] was written during this year, and also Brauer's paper [21] ,on the Betti numbers of the classical Lie groups. Pontrjagin had recently determined these numbers by topological means (1935), and Brauer, in response to a question by Weyl, was able to give in a few · weeks an alternative purely algebraic treatment based on invariant theory. The references to Brauer in Weyl's book The Classical Groups (1939) make evident the esteem in which he held hls younger colleague. Brauer collaborated with N. Jacobson, who had been Weyl's assisfant during the second halfof 1933-34, in writing up notes of Weyl's lectures on Lie groups, and of some ofthe seminar talks which followed. These appeared under the title The Structure and Representation of Continuous Groups (Princeton, 1934-1935). The year at Princeton was. very productive of new mathematical contacts for Brauer. The Institute was already a brilliant centre for mathematics. Besides its

xxiii

BIOGRAPHY

320

RICHARD DAGOBERT BRAUER

permanent professors (J. W. Alexander, A. Einstein, J. von Neumann, 0. Veblen and Weyl) there were in the School of Mathematics that year four assistants and thirty-four " workers" (i.e. visiting members). Among the 1 0) of a compact semisimple Lie group G (considered as a manifold) is equal to the dimension vP of the space of invariant differential forms m of order p on G. These m are determined by their

°

XXX

BIOGRAPHY

RICHARD DAGOBERT BRAUER

327

behaviour at the identity element of G, and correspond to those elements of the pth exterior (alternating) power EP = g* A ... Ag* (g is the Lie algebra of G) which are invariant under the action of G on EP which derives from the adjoint action of Gon g. Thus the problem of finding the vP-that is of calculating the Poincare polynomial 1 +v 1 t+v 2 t 2 + ... of G~reduces to a problem of algebraic invariant theory. Brauer solved this problem for the classical groups (unitary, symplectic, orthogonal); an outline of the proof appears in [21], and the complete proof for the unitary group is given by Weyl in his book (1946, pp. 232-238). These Poincare polynomials had also been calculated by direct topological methods by Pontrjagin (1935). The other compact groups G, corresponding to the "exceptional" simple Lie algebras, were treated by Chevalley (1950). [25] was Brauer's last substantial paper on continuous groups, and gives a glimpse of a general representation theory of continuous grou_ps, based on invariant theory, and of a strictly " algebraic " nature. Unfortunately a promised sequel ([25, p. 858]) never appeared. Many of the ideas in [25] appeared, with generous acknowledgment, in Weyl's book (1939). 2. Simple algebras and splitting fields Brauer's researches on simple algebras had their origin in Schur's "arithmetic" theory of irreducible groups of matrices. Let K be a fixed ground field, K. an algebraically closed extehsion of K, andf a positive integer. We write R1 for the ring of all Jxf matrices over a given ring or algebra R. Let f> be an irreducible subset of K I which is also a semigroup, i.e. f> is multiplicatively closed and contains the identity matrix. f> is said to be rationally representable over a field L (K s L s K.) if there exists sbme matrix Re GL(f, K) such that R- 1 f>R s L 1 . Then L certainly contains the character x of f>, that is, x(H) = trace(H) lies in L, for all Hin f>. From now on we shall assume that the ground field K contains X, and also that f> is rationally representable over some L of finite degree (L : K). Such a field L is called a splitting field for f> (or for x) over K, and the minimal degree (L: K) ofall these splitting fields is Schur's index m = mK(f>) = mK(x). In two papers (1906, 1909) Schur proved the following theorems in the case K. = IC. 1. m divides J.

II. m divides the degree (L: K) of any splitting field L. III. If

then

f>Cml is the

semigroup of all mf-rowed matrices

f>Cml is rationally representable

over K.

Schur's ideas are often expressed in terms of linear algebras. Our assumptions imply that the K-linear closure A = Kf> of f> is a finite-dimensional central simple algebra over K (" central " or " normal " means that the centre of A contains only the scalar multiples of the identity). A given field L (we assume always K s L s K and (L : K) < oo) is a splitting field for f>, if and only if it is one for A. Moreover

xxxi

BIOGRAPHY

328

RICHARD DAGOBERT BRAUER

L f:, is isomorphic to L ® K A, which is a simple algebra over L, and it follows easily that L is a splitting field for A if and only if L®KA~Lf.

This condition depends only on the abstract structure of A as algebra over K; accordingly L can be described as a splitting field for this abstract algebra. Wedderburn's structure theorem (1907) says that A ~ D,, where t;,, 1 is an integer, and Dis a central division algebra (algebras are now assumed tobe over K), which is determined up to isomorphism by A. The splitting fields for A are the same as those for D, therefore these fields are characteristic ofthe algebra class [A] of A; two central simple algebras A, B are put into the same class if they determine isomorphic division algebras. In the late 1920's Brauer and Emmy N!)ether, working independently and using quite different methods, showed that Schur's theorems hold in arbitrary characteristic; moreover if A has Schur index m, then dimK D = m 2 , and the splitting fields L of degree (L : K) = m coincide, up to isomorphism, with the maximal subfields of D. After Brauer and Noether had become aware of each other's work, Brauer was able to improve this last theorem to IV. Every splitting field of degree mr (see II) is isomorphic to a maximal subfield of D,. Conversely, every maximal subfield L of D, is a splitting field, and (L: K) = ms for some divisor s of r. Brauer 'proved IV under the assumption that K was perfect; Noether was later able to remove this restriction. They announced this and other common discoveries in [4, (1927)]. Noether's proofs used her new structure theory of algebras (1929, 1933), and were based on the systematic use of representation modules. Brauer's proofs appeared in threepapers ([3, (1926)), [5, (1928)), [7, (1929))). They were based on his theory of f actor-sets of separable field extensions. Suppose L = K(0) is separable over K, and that {0a}a=l, ... ,, are the conjugates of 0 over K. To each central simple algebra A which has L as splitting field, Brauer associated a factor-set (capy)a, p, r, = 1 , ... , ,, whose values Capy are non-zero elements of the normal closure of L over K. The capy satisfy certain " cocycle " conditions (of course, the cohomological language was not used until much later), and the set of all such "cocycles ", taken modulo suitable "coboundaries ", forms a multiplicative group which we will denote HL(K). The main theme in [3, 5, 7) is that the correspondence A ~ (capy) induces an isomorphism BL(K) ~ HL(K); here B(K) is the "Brauer group ", whose elements are the classes [AJ of all central simple algebras A over K, multiplied by the rule [AJ[B] = [A ®KB], and BL(K) is the subgroup consisting of those [AJ for which L is a splitting field. The unit element of B(K) is [K], the class of all A which are isomorphic to some Kif> 1). The group B(K) did not appear explicitly until [13), which was concerned with Noether's noncommutative Galois theory (1933). But the results in the early papers [3, 5, 7] are proved by using the interplay between an algebra A and its factor sets. We mention here only one such theorem. The exponent l of A can be regarded as the order of [A] as element of B(K). Schur's theorem III can be read as [Ar= 1, hence l divides m. In [3], Brauer showed that every prime divisor of m also divides 1, by an argument which appeared later in the famous joint paper with Hasse and Noether [14] on central division algebras over an algebraic number field.

xxxii

BIOGRAPHY

RICHARD DAGOBERT BRAUER

329

At the heart of Brauer's theory is a cortstruction [5, 7] which shows how to make a central simple algebra A, with Las splitting field and having a prescribed factor-set (capy), When L is a Galois ( = normal and separable) extension of K, the algebra A reduces to a crossed-product algebra ( = verschränktes Produkt; this term is due to Noether), and the factor-set (cap 1 ) reduces to a Noether factor-set (rs, T) indexed by the elements S, T ofthe Galois group G = Gal (L/K) ([15]; see also the excellent account of the Noether and Brauer theories in van der Waerden (1937)). Each factor..:set (rs, T) determines a group-extension of G by the multiplicative group L * of L, and H L(K) can be identified with the usual cohomology group H 2 ( G, L *). But if it happens that all the rs, T are roots of unity, then one can make a finite group extension G 1 of G by the cyclic group generated by the rs, T- The study of these finite extensions led Brauer to some of his deepest work on the structure of division algebras ([15], [50]). Brauer's isomorphism H 2 (G, L*) ~ BL(K), together with. Hilbert's "theorem 90" (whose cohomological formulation is H 1 (G, L*) = 0), has formed the basis of Galois cohomology, which has had a great influence in number theory-particularly through Tate's work on class-field theory (Tate, see Cassels and Fröhlich 1967)-and, more recently, in the theory of commutative rings. Azuyama (1951) and Auslander and Goldman (1960) defined a Brauer group B(R) for an arbitrary commutative ring R; Auslander and Goldman gave a generalized version of the isomorphism H 2 (G, L*) ~ BL(K). A great deal of further generalization has followed-see particularly Chase, Harri~on and Rosenberg (1965), and for recent literature, see the proceedings ofa conferenl::e on Brauer groups held in 1975 at Evanston {Lecture Notes in Mathematics no. 549, Springer, Berlin 1976). Schur's original problem had been to calculate the Schur index mK(X), over a 1ield K of characteristic zero, of a given irreducible character x of a given finite group G. A related problem was to find splitting.fields for G, that is, fields K such that mK(X) = 1 for all irreducible characters x of G; In [47] Brauer verified a long-standing conjecture by proving V. Let e be a primitive JGJth root of unity, where JGJ is the order of G. Then Q(e) is a splitting field for G. The proof in [47] used modular characters. A quite different proof, and some sharper versions of V, resulted in [53] from the application of Brauer's "induction theorem "-we shall describe this below. Using the same ideas, Brauer gave in [60] a profound reduction of Schur's index problem: he showed that all the Schur indices for a given finite group G can be found, if the same can be done for all the "hyperelementary" subgroups Hof G. A group His hyper-elementary if, for some prime p, there is a cyclic normal subgroup H 0 of H such that H/H 0 is a p-group. Brauer first proved his induction theorem in his famous paper [51] on Artin's L-series (see p. 331). In [62] he proved the "characterization of characters ", and showed that this was equivalent to the induction theorem. Roquette (1952) gave a proof much simpler than those in [51] and [62], and this was further simplified by Brauer a~d Tate [63] to give the elegant proofwhich is now standard. None ofthese proofs uses modular methods, but they are all based on the idea of induction from elementary subgroups of G, and this idea appeared in Brauer's earliest paper [18] on modular representations. A finite group E is called elementary if E = A x B, where A is cyclic, and Bis a p-group for some prime p. We write R(G) for the set of all "generalized characters" of G, i.e. integral combinations z 1 Xi+ ... z.x. of the irreducible characters Xi, ... , x. of G.

xxxiii

BIOGRAPHY

330

RICHARD DAGOBERT BRAUER

Brauer's Induction Theorem. Every character X of G can be written as a linear combination x = I: c;i/Ji*, where each c; is an integer and i/Ji* is the character of G induced from a linear character i/1; of some elementary subgroup E; of G. The Characterization of Characters. Let 8 be a c~111plex-valued class-function on G. Then 8 lies in R(G) if and only if the restriction 81E lies in R(E) for every elementary subgroup E of G. These must be the most widely-quoted of all Brauer's theorems. He applied them himself to class-field theory, to the theory of Schur indices (as we have seen) and to modular and ordinary character theory. Ofthe many generalizations and applications made by others, we might mention particularly Swan's induction theorems for integral representations (1960), and Atiyah's paper (1961) on the connection between R(G) and the integral cohomology of G. Serre (1971) gives a very good discussion of the induction theorem and of its application to character theory.

3. Modular representations As early as 1902, L. E. Dickson showed that Frobenius's theory (I 896) of characters of a finite group G holds in an algebraically closed field k of prime characteristic p, provided p does not divide the order ICI of G. In later papers Dickson (1907a, b) considered the case where p divides IGI. In this case the group-algebra A = kG is not semisimple. A representation F: G ~ GL(n, k) is in general not completely reducible, and is very imperfectly described by its natural character XF( = trace F). Dickson found some interesting facts about such "modular" representations, but they did not amount to a general theory. The subject lay dormant until the middle 1930's, when Brauer laid the foundations of his modular representation theory in three fundamental papers [18], [27], [28]; the two last were written jointly with C. Nesbitt. [27], a short memoir published by the University of Toronto Press, contains in 21 pages all the main ingredients of the mature theory; the proofs are complete, except for some important theorems on the regular representations of algebras which were announced in [28] and proved by Nesbitt in his thesis (Nesbitt 1938). Nakayama (1938) gave alternative proofs for some of the theorems in [27] and [28]. Subsequent accounts of modular theory appeared in [34], [65] and [73]. Let G0 denote the set of all p'-elements of G (i.e. elements whose order is prime to p). A conjugacy class of Gis called a p'-class (or p-regular class) if it lies in G0 • The "modular character" c/>F of a representation F: G ~ GL(n, k) (since known as the "Brauer character ") is a complex-valued class-function on G0-it is a kind of "complexified" version of the natural trace function XF· lt was defined in [27]. If F 1 , ••• , F 1 is a füll set of irreducible modular representations, their Brauer characters cp 1 , ... , cp 1 are linearly independent. For any modular representation F, one has 4>F = I: n;(F) c/>;, where n;(F) is the multiplicity with which F; appears as a composition factor in F. This was used in [27] to prove

I. The number l of irreducible modular representations F; of G, is equal to the number of p'-classes of G. Brauer had already proved this beautiful theorem in [18] in a different way. For a third proof, see [65]. The most important and useful feature of modular theory is that it relates "ordinary" (characteristic zero) representations to modular ones. Let K be a field

XXXIV

BIOGRAPHY

RICHA,RD DAGOBERT BRAUER

331

of characteristiczero, which is a splitting field for G. Let R be a subring of K having K as quotient; we assume that R is a principal ideal domain, and that it has a prime ideal :p containing p. Identify R = R/:p with a subfield of k. Any ordinary character x of G can be realized by a repres~ntation X: g-+ X(g) by matrices X(g) all of whose coefficients lie in R. Taking these mod :p, we get a modular representation X: g-+ X(g) of G. The equivalence dass of Xis not uniquely determined by X, but its Brauer character is, and in a very simple way: 0 bilde man dann alle aus t 'z Matrizen m-ten Grades zusammengesetzten Matrizen T = (Z,~)x,x (~, Jl = 1 , 2 , . . . , t), wo die Zx~ ganz beliebige Matrizen a u s ~ sind. Ersichtlich bilden die Matrizen T:eine in K(v~) rationale, irreduzible Gruppe J des Grades t . m , deren Charakter zu K gehSrt, die in bezug auf K komplett ist und deren Faktorensystem cap r ist: Da der Grad aUer irreduziblen Gruppen mit diesem Faktorensystem dutch m teilbar, also yon der Form t . m ist, haben wir flit alle mSglichen Grade eine kompletee Gruppe m i t diesem Faktorensystem angegebe n. Beriicksichtigt man, dab man nach Satz I i I yore Index m des Faktorensystems spreehen kann, so erhMt man: S a t z IX. Dann Und nur dann gibe es zu einem gegebenen Faktoren. system %Pr yore Index m irreduzible Gruppen yore Grade f, wenn f dutch m tdlbar ist. Weiter iolgt unter Beriieksichtigung yon (9), (10) und der Si~tze V und VIII: Satz X. ~ sei eine irreduzible Gruppe f.ten Grades yore Index m in bezug au/ einen Zahlk6rTer K, dem der Charakter van ~ angehdrt, es sei f = t . m ; %Pr sei ein zum 'Faktorensystem van ~0 assoziiertes, das zu einem algebraischen K6rper K(O) van m~glichst kleinera Grad geh6rt. Dann hat K(v~) den Grad m. ,~ind 01, O~. . . . , t~ die Konjugierten zu ~, bedeutet R die Wronskisehe Matrix re,ten Grades (#~'-"),,a, so haben bei passender Wahl der Yariablen alle Matrizen van ,~ die Form

/ D { 1 1(1,1)~ 1)-1

(24)

/

9

~

9

~ l - - l , x ~ W,x 3.]L

*

.

9

.

.1, ,Lq[, /x,,~

D[ ~

*

:, . . . ~

.

~

9

1 70,t)~ 9

.[I, 1 ~ \Cx~, 1

~

~

.*

x,t/

Ix.j[

~--1\ 9

Jt~

o

|

9

. /, --

44*

692

R. Bmuer.

Dabei geniigen /iir jedes /este Paar a,p ( a , / ~ - I , 2 , . . . , ~ ) die m ~ galden l~ ~ den im Eatz VIII /~r die Zahlen ~ gestellten Bedin~ungen. gusatz: Man kann die Elemente ~on ~ darstdlen als Matrizen ~om Grade ~, deren Elemente sdbst Matrizen der irreduziblen kompletten Gruppe t~tn Grade m m i t dem Faktorensystem r sind, Kennt man also ein assoziiertes Faktorensystem, das zu einem K6rpez K(O) yon m6gliehst kleinem Grad gehfrt, so iibersieht man vollsfiindig, in weleher Weise bei passender Wahl der Variablen die Irrationalitiiten yon K(O) in ~3 au~treten. Aus dem friiher Gesagten ergibt sieh, da6 alle H in K(O) rational sind. w Im Anschlul~ an w 2 soll untersueht werden, wie man die Faktorensysteme fiber einem gegebenen Gr~ndk6rper K aufstellenkann. 8 a t z XI. Hat das Faktorensystem c , ~ den Exponenten l, s o gib~ es d n assoziiertes Faktorensystem~ dessen ~zhlen sdmtlich l-re Ein~eits. ~ r ~ l n 6nd. Beweis. c,# r geh6re zum KSrper K(ff). Da der Exponent des Systems Iist, gibt es naeh B Zahlen kap, die den Bedingungen (A) geniigen, und fiir die fiir alle a,/~, ~, (25)

kop ~pr = ct

ist~ Wir denken uns zu K(O) alle Konjugierten yon 0 und alle Werte yon ~ fiir "alle g und • adjungiert, der entstehende Normalk6rper sei P K(~). Die zu K(~) geh6rige Erweiterung cx~ yon ~aPz erhiilt man naeh w 3. Man iiberzeugt sich leicht, dg6 es Zahlen k~'~ gibt, die den Bedingungen (A) jetzt fiir den K6rper K(~) geniigen, fiir die (26)

k~ k'

ist, derart, da6 j~es k.'~ mit einem /c~p iibeminstimmt. Daher geh~izen alle Werte von ~ (27)

zu K(~). Man setze h~----- ~ ,

wo man iibez die/-ten Wurzeln nur so zu vediigen hat, daft die Zahlen h~x auch den Bedingungen (A) geniigen, was offenbar m/iglich ist.' Jet~t setze man tf.. CN~.~,, ~

CJ, . ~J.,. hxZhj.~

"

Gruppen linearer Subsfitutionen. L

693

Aus (26) und (27) folgt, daft dieses zu c ~ und also auch zu c~pr assoziierte Faktorensystem nur aus /-ten Einheitswurzeln besteht. Damit ist die Behauptung bewiesen. Offenhar ist die Betrachtung auch dann anwendbar, wenn l nicht der genaue Exponent ist, abet Gleichungen (25) bestehen. Will man entscheiden, ob zwei gegebene Faktorensysteme assoziiert mind, so hat man nut zu untersucl~en, ob der ,Quotient" der beiden Systeme zum Einheitesystem assoziiert ist. Da man annehmen kann, daft alle Zahlen dieses Quotienten l-re Einheitswurzeln sind, so hat man nut zu untersuehen, ob ein gegebenes Faktorensystem, dessen Zahlen Einheitswurzeln eines testen Grades, etwa n - r e Einheitswurzeln sind, zu dem Ein-. heitssystem assoziiert ist; n sei dabei m6glichst niedrig gew~hlt. Das betreffende Faktorensystem soi %pr und geh6re zum K~rper K (0) yore Grade r, den wit als Normalk6rper annehmen k6nnen. Dann und nut dann ist es dem Eiaheitssystem assoziiert, wenn es Zahlen k~p gibt, die den Bedingungen (A) geniigen, und fii~ die

k.=P'k"r

(28)

k~r

-

- - c~p~

ist. Dann wird aber

Daraus folgtl~), dab es eine Zald 9(29)

w

in K (O) gibt, fiir die

w= k=~= -~p

(=,p=1,2,

.. . , r )

ist, wenn w = w 1, w s , . . . , w r die Konjugierten zu w sind. Wit adjungieren einen Weft yon " ~ zu K(O); dann geh6ren alle Werte yon "}f~ zu dem entstehenden KSrper, da die n-ten Eiaheitswurzeln sich auch durch d i e r ausdriicken lassen und daher zu K(O) geh6ren. Ferner gehSren wegen (29) alle Werte yon ~ ( a = l , 2, . . . . r) zu dem entstehenden K6rper. Denkt man sich diese Adjunktion yon vornherein durchgefiihrt, so erkennt man, dal~ man ohne Einschr~nkung annehmen kann, dal3 alle Werte yon (r

. . . . , r ) ~-u K(O)geh6ren. Es sei

(so)

w= =

z:,

t~) Nach einer Methode yon He'rm Speiser (Sp. S. 8) set~e man n&mlich 9

9

tO=----~ue/~~ , WO u eine Zah] ~us K(O) mit den Konjugierten u = u l , ~ . . . . ~ur rot, e--1 und v e r ~ e ~ber # so, dab w ~ 0 wird, was stets mSglich is~.

694

R. Brauer.

und zwar seien die z,, so gew~ihlt; d~l~ sie die Konjugierten zu z = z~ sind. Setzen wir k~'p= k ~ .

Za

so geniigen aueh diese Zahlen den Be-

dingungen (A) und erfiJllen aueh (28). Wegen (29) und (30) sind sie alle n-re Einheitswurzeln. Ist also das System c, py dem Einheitssystem assoziiert, so ist es naeh Adjunktion der n-ten Wurzel aus einer geeigneten Zah.1 aus K(v~) mSg!ieh, die Zahlen k~p als n-te "Einheitswurzeln zu w~ililen. "Wir gehen yon einem festen Faktor~nsystem aus, dessen Zahlen n-re Einlaeitswurzeln sind, und adjungieren ~x-x, wo x eine ganz beliebige Zahl aus K(O) ist; in dem entstehenden KSrper K(~/) gehSrt zu cap r eine ganz bestimmte Erweiterung c',~./,. Um c:~,, naeh w 3 aufzustellen, braucht man nut die Galoissehe Gruppe yon K(~?) in bezug auf K und die zum TeilkSrper K(o~) gehSrige Untergruppe dieser Gruppe zu kennen. W~ihlt man fiir x der Reihe naeh alle Zahlen yon K(#), so erh~ilt m~n insgesamt nur endlieh vide MSgliehkeiten. Bei jedem festen K(~) hat man dann fiir die k,~ nut endlich viele MSgliehkeiten durehzuprobieren, dal~ alle k,~ n-te Einheitswurzeln sind. Man kann als0 sagen, daft bei lest gegebenem e,~y die Entseheidung, ob dieses System dem Einheitssystem assoziiert ist, nur davon abh~ingt, ob es KSrper gibt, die aus K (#) dureh Adjunktion einer n-ten Wurzel entstehen und in bezug auf K eine geeignete Galoissehe Gruppe haben. Es sei z.B. v~ V0m Grade 2 in bezug auf K, das Faktorensystem sei gegeben dureh cxg~~-e~g = -

1,

c,~ r = 1

fiir aUe anderen Tripel.

Dieses System ist dem Einheitssystem dann und nur dann assoziiert, wenn es zyklisehe KSrper 4. Grades gibt, die K (#) enthalten. Ist K der KSrper der rationalen Zahten, .~?= ~ mit ganzem quadratfreiem d, so ist das dann und ~nur dann der Fall, wenn d sieh als Summe yon zwei ganzen Quadraten darstellen liil~t, insbesondere also d ~ 0 ist, Man kann also sagen, dal3 die Behandlung der Frage nut yon algebraisehen Struktureigensehaften von K(v~) ab.h~ingt. Jetzt sei K ein algebraiseher Zi~hlkSrper, e,a r ein zu dem NormalkSrper K(#) geh6riges Faktorensystem, das aus n-ten Einheitswurzeln besteht. Es soll eine andere idealtheoretisehe Methode~zur Untersuehung der Frage; ob e,,~., dem Einheitssystem assoziiert ist, skizziert werden. k , a und w mSgen dieselbe Bedeutung wie in (28) und (29) haben. .~lan dad annehmen, dal~ w eine ganze algebraisehe Zahl ist. Ist a irgendein Ideal des KSrpers K(,~), so bezeiehne a~ dasildeal, das aus a entsteht, wenn man aUe Zahlen von a dureh ~ ausdriiekt und

Gruppen linearer Substitutionen. I.

695

c~urch ~a ( ~ 1, 2, . . . . r) ersetzt. Dabei seien wie friiher die ~a die zu 0 in bezug auf K als Grundk~rpor konjugierten Gr~13en. Man zerlege des Hauptideal w. o in Primideale, es sei

wo ~ ein Produkt yon Primidealpotenzen ist, deren Exponent < n ist. Dann i~t Da nun nach (29) w~ fiir alle ~ und fl die n-te Potenz einer Zahl sus wp K(O) ist, folgt, da~ ein in ~ au|gehendes Primidea] in derselben Potenz in jedem ~ aufgeht, und daher iSt ~,

~

(r

1' 2, . . . , r).

Daher ist (31)

wp.r: -----w~.r~'

(~, p ~ - l , 2 , . :,, f')..

Durchliiuit a slle Ideale einer Idealklassc K, so durchl~uft such a, alle Idesle einer Klasse, die mit K~ bezeichnct werden m6gc. K sei die Idealklesse, der r angeh6r~, dana folgt aus (31) (32)

K~ = K~*

Ca,/~ ----1, 2 , . . . , r )

Ist umgekehrt fiir eine Idealklasse K diese Bedingung effiillt und r ein Ideal aus K, so gibt cs Zahlen 9 ~ und %p, die den Bedingungen (A) - geniigen, so daft (33)

-~*

o

-~*

ist. Dutch (38) ist ,%~, 9 Qap bis auf eine Einheit bestimmt. Soil es also eine Zahl w geben, fiir die aul~erdem (81) erfii]lt ist, so mu~ {3~

w__~ oop ~ p=, ~p ~

(~,/~ = 1 , 2, .. . , r )

sein, wo die e~ ein System yon Einheiten sind, die den Bedingungeu CA) geniigen. Mit Hilfe eines Systems yon Fundamentaleinheiten ist es leicht mSglich zu untersuchcn, ob sich (84) erfiillen liil~t. Wie man leicht einsieht, geniigt es flit das Folgende, bei gegcbenem 1:, falls, es iiberhsupt L6sungen gibt, nut endlich viele angebbare Systeme e~ zu betrachten; man hat dann such fii~ w nu~ endlich vide M6glichkeiten. Danaeh b a r m a n bei festem w zu entsche~den, o b /~p ~- ~ , ,

r

anch w#

9

wirklich eine Zahl aus K ( ~ ) i s t (a, p f f i l , . 2 , . . r). Uber die Wurzeln ist dsbei nut so z u v~diigen, dal~ die /~p den Bedingungen (A)geniigen.

696

R. Br~uer. Gruppen linearer Substitutionen.

Das Faktorensystem cap r i s t durch (28) bestimmt; zu dem Ideal r geh~ren also gewisse angebbare Fakto~rensysteme capr. Geht man yon einem anderen Ideal der Klasse K aus, so wird man auf dieselben Faktorensysteme gefiihrt. Man fiihre die Betrachtung fiir jede der endlich vielen Idealklassen durch; es geniigt, sich anf diejenigen zu b~chr~nken, fiir die (32)ediiUt ist. Man erh~lt mit endlich vielen Schritten alle dem Einheitssystem assoziierten Faktorensysteme, deren Zahlen n-re Einheitswarzeln sind. Dah6r ist es fiir den t~all, daft K ein algebraischer KSrper ist, m#glich, auf diesem Wege zu entseheiden, ob zwei Faktorensysteme assoziiert mind. (Eingegangen am 22. Juli 1927.)

Uber Systeme hyperkomplexer Zahlen. Von

Richard Brauer in KOn~berg i. P~.

Die Theorie der Systeme hyperkomplexer Zatden tiber einem Grundk6rper K hiingt auis engste mit der Theorie der Gruppen ]inearer Substi, tutionen mit Koefllzienten aus einem gegebenen K~rper zussmmen, also mit denjenigen gruppentheoretischen Betrachtungen, die im allgemeinen als arithmetische Untersuchungen tiber Gruppen linearer Substitutionen bezeichnet werden. Dem Fall, daft K der K6rper aller Zshlen ist, entspricht die Theorie der Gmppen linearer Substitutionen mit bdiebigen Koeffizienten bier hat der erw~ihnte Zusammenhang yon Anfang an bei den Untersuchungen eine wesentliche Rolle gespielt. In tier neuen Theorie dagegen, bei der als Grundk6rper ein beliebiger K6rper zugelassen wh~, ist der Zusammenliang wohl durchweg unberiicksichtigt geblieben. Zweck der vorliegenden Arbeit ist es, die Systeme hyperlromplexer Gr~l~en mit Hilfe yon gruppentheoretischen S~itzen der genannten Art zu untersuchen. Dementsprechend werden yon der Theorie der hyperkomplexen GrS~en n ~ die Begriffe und die einfachsten S~tze vorausgesetzt, wiihrend gruppentheoretische S~itze in 9 Marie herangezogen werden sollen ~/. Als Grundk~rper K werden nicht nur beliebige Zahlk6rper, sondern a|lgemein vollkommene K6rper ~) zugelassen. Bei den Systemen hyperkomplexer Gr~flen beschr~inke ich reich im wesentlichen auf die Behandlung halbeinfacher (Dedekindscher) Systeme. Ich setze attflerdem voraus, daft der 1) Fiir die Theorie der hyperkomplexenZahlen vergleicheman etwa: L. E. Dickson, Algebren und ihre Zsidentheorie, Ziirich 1927 (umgearbeitete deutsche Ubersetzung y o n Algebras and their arithmetics, Chicago 1928). Ieh zitiere das Buch mit Diekson. Eine Reihe der verwendeten gruppentheoretischen S~tze findet man in w 1 der vorliegenden Arbeit zusammengestellt. ~) Vg]. E. Steinitz, Algebraische Theorie der K6rper, Journ. f. d. reine u. angew. Math. 187 (1909), S. 167-309. -- Wegen der Voraussetzung, dab K vollkommen ist, vgl. die in Anm. ') zitierte Arbeit yon Fr]. E. Noether.

80

R. Bmuer.

Rang in bezug auf K endlich istS/. Auf schon bekannte Siitze soil im iolgenden nut verhiiltnismiiflig kurz eingegangen werden. I m w 1 der vorliegenden Arbeit werden die verschiedenen Begrifle vom gruppentheoretischen Standpnnkt aus untersucht und die Reduktion der halbeinfachen Systeme auf einfache Systeme behandelt, Diese Betrachtungen sollen als Grundlage flit das Folgende dienen. Im w 2 wird zuniichst aus gruppentheoretischen S~tzen der Wedderburnache Satz tiber die Darstellbarkeit eines einfaehen Systems als direktes Produkt A X B hergeleitet; dabei bedeutet A ein System ohne Nullteiler, B besteht aus allen Mat~izen eines wohlbestimmten Grades mit Koeffio zienten aus K. Weiterhin ergibt sich als notwendige und hinreichende Bedingung daftir, daf ein einfaches System A keine Nuilteiler besitzt, daft ftir eine absolut irreduzible Darsteilung yon A der Schursche Index und der Grad iibereinstimmen. Bei Behandlung der Frage, in welchen algebraischen K6rpern tiber dem Grundk6rper K eine absolut irreduzible Darsteilung eines Systems A sieh rational maehen liift, kann man sich aui den Fall besehr~inken, daft A keine Nullteiler besitzt. Wie in w 3 gezeigt wird, kann man diese KSrper voilstiindig durch die grSflten Unterk6rper yon gewissen einfachen Systemen hyperkomplexer GrSflen charakterisieren. Es handelt sich dabei gerade um die Systeme, die bei Darstellung als direktes produkt nach dem Wedderburnschen Satz gerade das System A als nullteilerfreien Faktor enthalten. -- Frl. E. Noether hat diese Siitze unabhiingig auf ganz anderem Wege bewiesen und die Betrachtungen auch itir den Fall durchgefiihrt, dab der GrundkSrper nieht vollkommen ist. Wir haben dartiber gemeinsam ohne Beweise berichtet 4). Nach diesen Untersuchungen wende ich reich imw 4 zu der Bestimmung aller Systeme A ohne Nullteiler tiber einem gegebenen GrundkSrper K. Diese Aufgabe ist voilst~ndig mit der anderen iiquivalent, alle Faktorensysteme ~) 8) Gewisse andere Systeme sind in neuerer Zeit yon Herrn E. Artin in der Ab. handlung: Zur Theorie der hyperkomplexen Zahlen, AbhandL a. d. Math. Seminar Hamburg 5 (1927), S. 251--260 in die Theorie einbezogen worden. Ffir die vorliegende Arbeit ist die Beschr;~nkung auf Systeme, die yon endiichem Rang in bezug auf einen Grundk~rper sind, wesentlich. 4) R. Brauer und E. Noether, Uber minimale Zed~llungsk~rper irreduziblerDarstellungen, Berl. Sitzungsberichte 1927, S. 221--228. Der Noethersche Beweis wird in einer demn~chst in der Math. Zeitschr. erscheinenden Arbeit yon Frl. E. Noether durchgef~hrt. Im Spezisliall g = 1 waren die Siitze 4 und 5 der vorliegenden Arbeit Frl. Noether bereits wesentlich friiher a|s dem Verfasser bekannt. ~) Man'vgl. dazu R. Braucr, Untersuchungen iiber die arithmetischen Eigenschaften yon Gruppen linearer Substitutionen, Math. Zeitschr. 28 (1928), S. 677-696. Ich zitiere diese Arbeit im foigenden mit U.

Uber Systeme hyperkomplexer Zahlen.

81

in bezug auf diejenigen K~rper Z anzugeben, die algebraiseh (yon endlichem Grade) fiber K sind; Z wird dabei dem Zentrum yon A einstufig isomorph. Wie man aber atle Faktorensysteme zu Z. aufstellen kann, babe ich in der in Anm. 5) zitierten Arbeit angegeben. Man kann daher sagen, dab die Aufstellung aller Systeme ohne Nullteiler auf diesem Wege m6glich ist. Praktisch kSnnen dabei noch Schwierigkeiten auftretenS), da es notwendig ist, Struktureigenschaften der algebraischen KSrper fiber K, z.B. gewisse Eigenschaiten der Galoisschen Gruppe zu beherrschen. Immerbin dad man wohl derartige Fragen, die siimtlich K6rper, also elementarere GegenstRnde als Systeme hyperkomplexer Zahlen betreflen, flit unsere Untersuchungen als bekannt voraussetzen. Kennt man ein zu einem gegebenen Faktorensystem assoziiertes, das zu einem K6rper mSglichst niedrigen Grades geh6rt, so kann man das zugehSrige System hyperkomplexe~ Gr6l~en mit Hilfe einer Darstellung dutch Matrizen explizit angebenT). Ich erl~iutere den Sachverhalt an zwei einfachen Beispielen: 1. K sei der K6rper aller reellen Zahlen. Dann kommen nut drei wesentlieh verschiedene, sofort angebbare Faktoxensysteme in Betraeht; dem entspreehen als einzige Systeme ohne Nullteiler fiber K die beiden KSrper K und K (f) und das System der QuaternionenS). Gruppentheoretisch entsprechen den drei Faktorensystemen die drei Typen yon Gruppen linearer Substitutionen in bezug auf den reellen K6rper als Grundbereich. 2. K sei ein endlicher K6rper. Dann kommt, wie ich zeige, flit jeden algebraischen K6rper Z fiber K nur das Einheitsfaktorensystem in Betracht. Das ergibt einerseits den Satz yon Wedderburng), dab bei end]ichen KSrpern das kommutative Gesetz der Multiplikation eine Folge der anderen Axiome ist. Andererseits fmdet man daraus den Satz, da~ ein absolut irreduzibler Bestandteil einer Gruppe linearer Substitutionen mit Koeifizienten aus einem endlichen K6rper immer im KSrper seines Char a l ~ r s darstellbar ist. ~) Insbesondere soll nicht behauptet werden, da~ es stets m{~gliehist, mit unsern Hilfsmitteln unendlich viele Systeme ohne NuUteiler gleichzeitig zu iibersehen. 7) In gewissen ziemlich speziellen F~illen finden sieh die DarsteUungen schon bei Cecioni, Rend. de] Circ. Mat. di Palet~no 47 (1923), p. 203--254; vgl. such alas 3. Kapite] des Dicksonsohen Buches. 8) Das wurde bekanntlich zuerst von Frobenius im Journal f. d. reine u. angew. Math. 84 (1878), S. 59 bewiesen; sp~ter wurdenvon verschiedenvnAutoren eine ganze Reihe von Beweisen angegeben~ 9) Trans. of -the Am. M~ath. Soc. 6 p. 349, vgl. aueh Dickson S. 253. Ein besonders einfaeher Beweis finder sieh bei Herrn E. Artin, Uber einen Satz yon Herrn J. H. MaclaganWedderburn, Abhandl. a. do Math. Sere. Hamburg 5 ( 1927), S. 245--250. Setzt man diesen Satz als bekannt voraus, so kann man den folgenden Satz aus ihm herleiten. ~lathematigche Zeltschrift. 30.

82

R. Brauer.

Im w 5 behandle ich schliefllieh unter anderem einige Siitze fiber direkte Produkte yon Systemen. Dabei ergibt sich eine einfaehe Deutung flit den Exponenten l, den ich friiher bei anderen Untersuchungen verwandt habel~ -- Ich hebe hier nur einen Satz hervor: Ist A ein System h~perkomplexer Zahlen ohne Nullteiler, ist K -- wie man ohne Einschriinkung annehmen lcann -- das Zentrum yon A, ist sehliefllich m g der Rang n ) yon A und m ---- p~l p~J . . , p ~ so kann man A als direktes Produkt A1 X A~ ... A~ darstellen, wo A~ auch das Zentrum K hat, ebenfalls keine Nullteiler besitzt und den Rang (p~e)2 hat. w Ieh beginne damit, einige S~tze der Theorie der Gruppen linearer Substitutionen zu nennen, die im ~olgenden meI]rfach verwendet~werdenl~). In der Literatur sind diese S~itze nur f l i t den Fall behaudelt worden, daft der Grundk6rper K ein gewShnllcher ZahlkSrper ist. Die meisten yon ihnen sind abet auf den Fall eines vollkommenen KSrpers K zu iibertragen. Die Beweise miissen dabei manchmal etwas modifiziert werden, doch ~eten k e i n e wesentlichen Schwierigkeiten auf; i~h werde darauf noch an anderer Stelle eingehen. Eine Gruppe mit Koeffizienten aus~rgendeinem K6rper K tieil~t absolut irreduzibe], wenn sie in allen ErweiterangskSrpern yon K irreduzibel ist. Eine Gruppe ~heil3t absolut vollst~ndig reduzibel, wenn sie in einem ErweiterungskSrper vollst~indig in absolut irreduzible Bestandteile zerlegt werden kann. .~hnliehe (~iquivalente) Gruppen gelten als nicht.wesentlich versehieden. Die Grul0pe, die aus lauter~ Nullen besteht, werde als Nullgruppe bezeichnet. S a t z a: Dann und nut dann ist eine yon der Nullgruppe verschiedene Gruppe des Grades f absolut irreduzibel, wenn sie f9 linear unabh~ngige Matrizen enth~lt. 1o) R. Brauer, Uber Zusammenhiinge yon arithmetischen und invariantentheoretischen Eigensehaften yon Gruppen ]inearer Substitutionen, Sitzungsberiehte der Berliner Akademie 1926, S. 410. 11) Dieser Rang ist stets eine Quadratzahl. -- Unter dem Rang ist im folgenden stets die Maxhnalanzahl linear unabh~ngiger Elemente yon A in bezug auf K zu verstehen. lg) Man verg'~iche: W, Burnside, Proceedings of the London MatlL Soc. 2. Ser. Vo]. III (1905), p. 480 (Satz a); G. Frobenius und I. Schur, Uber die •quivalenz yon Gruppen linearer Substitutionen, Sitzungsberichte d. Berliner Akademie 1906, S. 209 (Satz fl und 7); I, Schur, Beitriige zur Theorie der Gruppen linearer Substitutionen, Transactions of the American Math. Soe., .2 .Ser.,Vol. XV (1909), p. 159--175 (Satz his ~). Satz ~ finder sieh bereits bei H. Taber, Comptes Rendus 142 (1906), S. 948 his 951. - - Ein Satz, der nieht mehr gilt, wenn die Charakteristik des Grundk6rpers von Null verschieden ist, ist z.B. der Satz HI der eben genannten Arbeit yon Frobenius und Schur.

!

über Systeme hyperkomplexer Zahlen.

83

Satz ß: m und ~ seien zwei absolut irreduzible Darstellungen einer Gruppe; alle Koeffizienten mögen einem gewissen Körper angehören. Dann und nur dann sind m und ~ ähnlich, wenn sie denselben Charakter haben. Satz y: Die Gruppe @ sei absolut vollständig reduzibel; die von der Nullgruppe und untereinander wesentlich verschiedenen/\irreduziblen Bestandteile seien @1 , @ 2 , ••• , @k mit den Graden fl' {2 , ••• , fk. Dann enthält @ genau {12 + {22 + fl linear unabhängige Elemente.

+ ...

Satz a: Eine in einem Körper K rationale und irreduzible Gruppe ist absolut vollständig reduzibel. Satz s: m und ~ seien beide im Körper K rational und irreduzibel und nicht ähnlich. Dann haben 2( und ~ auch keinen absolut irreduziblen Bestandteil gemeinsam. Satz (: Eine Gruppe @ 1 sei in dem algebraischen Körper K ( ß) über K rational, ß = ß 1 , ß2 , ••• , ßr seien die Konjugierten von ß in bezug auf K. ·Ersetzt man in allen Elementen von @ 1 die Größe ß durch ße, so geht @1 in eine isomorphe Gruppe @e über ( e = 1, 2, ... , r). Die Gruppe @, die vollständig in die Bestandteile @ 1 , @ 2 , ••• , @r zerfällt, ist dann in K rational darstellbar. Satz 17: Ist @ in K irreduzibel, sind @ 1 , @ 2 , ••• , @k die wesentlich verschiedenen absolut irreduziblen Bestandteile von @, so kommt jedes@,. in @ gleich oft, etwa m-mal vor. Durch Adjunktion des Charakters von @1 zu K entsteht ein algebraischer Körper Z vom Grade k über K; geht man von @2 , @3 , ••• , @k aus, so konimt man zu den konjugierten Körpern. @1 ist in einem algebraischen Körper vom Grade m über Z rational darstellbar. Ist @ 1 in K ( ß) rational und ersetzt man in @1 die Größe ß durch eine in bezug auf K konjugierte, so .erhält man ( abgesehen von Ähnlichkeit) eine der Gruppen @1 , @ 2 , , •• , @k und zwar bei passender Wahl der Konjugierten eine beliebig vorgegebene von ihnen. Satz ß: Ist bei den Bezeichnungen von Satz 17 die Gruppe @1 in einem Erweiterungskörper Q von K rational darstellbar, so ist Z in Q enthalten. Ist Q algebraisch vom Grade t über Z, so ist t durch m teilbar. - m werde als Schurscher Index von @ 1 in bezug auf den Grundkörper bezeichnet. Außer diesen Sätzen werden im folgenden die Ergebnisse meiner hier mit U. zitierten Arbeit aus der Math. Zeitschr. 28 weitgehend verwendet. Der Grundkörper werde im folgenden stets als vollkommener Körper vorausgesetzt. A sei ein System hyperkomplexer Zahlen ( eine Algebra) über K als Grundbereich. Man nennt ein System m von Matrizen eines festen 6*

44

THEORY OF ALGEBRAS

1 f

84

R. Braun.

Grades f m/t Koeffizienten aus einem ErweiterungskSrper yon K eine Darwenn den Elementen yon A eindeutig Elemente yon zugeordnet sind derart, dab Isomorphie bez'tigl/ch Addition und Multiplikation besteht, daI3" ferner der Multipllkation eines Elementes yon A mit einer Gr6/~e x von K die Multiplilmtion der zugeordneten Matrix mit entspricht. Die Ma~Tizen yon ~[ bilden eine Gruppe18). W/r wenden die Begriffe der Gruppentheorie, wie z.B. ~Imlichkeit, auch auf Darstellungen y o n Systemen hyperkomplexer Zahlen an; ghnliche Darstellungen gelten als nicht wesentlich versehieden. Ordnet man jedem Element yon A die Gr6~e 0 zu, so erh~lt man eine Darstellung, d i e Nulldarstellung. Diese w/rd abet im folgenden i m allgeme/nen ausgesehlossen. Bekannt|ieh k a n n man zu A immer eine einstufig isomorphe, in K rationale DarsteUung finden14).

stellung yon A,

Eine Or//l]e g yon A heiBt Wurzd der Null oder ni~otent, wenn eine Potenz yon ~ verschwindet. r heil3t Wurzelgr6~e oder eigentlich nilrpotent, wenn flit alle ~ aus A stets g~ Wurzel der Null ist. Enthglt A aul]er 0 keine Wurzelgr61]en, s o heil]t A ein Dedeklndsches oder ein halbein/achea

Eystera ~6). Es gilt dann der SatzlS)c Dann und nut dann ist eine einstu/ig i~omorphe, in K rationale DarsteUung ~I van A eine voUs~dndi9 reduzible Gru~ype, wean A halbein/ach ist. Aus dem in Anmerlmng ~s)angegebenen Grunde skizziere ich noch e i n e n neuen Beweis dieses Satzes: Man kann die Darstellung 9~ yon A v o n is) Be~ einer Gruppe yon Matrizen (oder linearen Substitutionen) verlangen wit nut, dab lhit je zwei Elementen auch immer das Produkt zum System geh~rt. I)agegen wild fiber die Existenz eines Einheitse]ementes oder yon Inversen niehts vor9ausgesetzt. 14) Vg]. z.B. Diekson, S. 33. $g

16) Sind ea,~,...,e~ die Basiselemente" von A und ist e~ep---~'araper, so ist, falls K die Charakteristik 0 hat, die notwendige und hinreichende Bedingung ffir #g

Halbeinfachheit, dad die Determinante der n s GriiBen dx~=.~aex~aoeo nicht vet@,o = 1 schwindet. Falls aber die Ch~rakteristik yon K yon 0 versckieden ist, kann dieses K~terium versagen; wit diirfen daher im fo]genden keinen Gebrauch yon ihm maohen. Io) Vgl. die in Anm. t~) zitjerte Arbeit yon Taber. Einen anderen Beweis f;ir 9 Hiilfte dibses Satzes findet man bei I. Schur, Uber Ringbereiche im Gebiet der ganzzahligen l/nearen Substitutionen, Sitzungsber. d. BerL Akademie (1923), S. 145--168 "und M. Herzberger, Uber Systeme hyperkomplexer Gr/~l]en, Dissertation (~Berlin 1923), Da bei diesen Beweisen abet das in der vorigen Anmerkung genannte Kriterium verwandt wird, reichen sie flit den Zweck der vor]iegenden Arbeit nicht vollst;~ndig aus. Naehtrgglich erkennt man voch, da~ die Voraussetzungen fiber ~ sich einsohrgnken lassen, was f~ir uns aber unwesentlich ist.

85

über Systeme hyperkomplexer Zahlen.

vornherein m K m irreduzible Bestandteile zerlegt annehmen, es sei

2! =

( 1)

ml ( m~l m2

~k~ ~k~

) ....

·m~ ,

wo die \ll„ in K rationale und irreduzible Darstellungen von A sind. Neben \ll betrachte man die Darstellung m*, die aus \ll entsteht, wenn man in (1) alle \ll„J. mit u > 2 durch Null ersetzt. Notwendig und hinreichend nicht vollständig reduzibel ist, ist, daß es in mehr linear dafür, daß unabhängige Elemente als in m* gibt1 '). Ist diese Bedingung erfüllt, so gibt es lineare von Null verschiedene Verbindungen von Elementen von \ll mit Koeffizienten aus K, für die die entsprechende Verbindung aus den zugeordneten Elementen von \ll* verschwindet. Es gibt also in \ll von Null verschiedene Elemente A, bei denen bei der Schreibweise (1) in der Hauptdiagonale lauter Nullen stehen. Ist Z ein beliebiges Element von \ll, so stehen auch in A Z in und unter der Hauptdiagonale lauter Nullen; daher verschwindet eine Potenz von A Z. A und wegen der einstufigen Isomorphie also auch das zugeordnete Element von A sind Wurzelgrößen. Daher ist A sicher nicht halbeinfach. Umgekehrt sei a Wurzelgröße, die zugeordnete Matrix sei A. In \ll„ entspreche A etwa A,, (u = 1, 2, ... , k). Für ein beliebiges Z„ aus \ll„ muß dann eine geeignete Potenz von A„Z" verschwinden. Nach Satz o ist \ll„ im absoluten Sinn vollständig reduzibel. Q3 sei einer der absolut irreduziblen Bestandteile von \ll", B sei die A„ in Q3 entsprechende Matrix. Dann muß für alle Y aus Q3 eine geeignete Potenz von B Y verschwinden. Daher sind alle charakteristischen Wurzeln und folglich auch die Spur von BY Null

m

m

x(BY) = 0.

(2)

17) Man beweist meist diese Tatsache mit Hilfe des hier zu beweisenden Satzes; man kann sie aber direkt mit den Hilfsmitteln der Theorie der Gruppen linearer Substitutionen nachweisen, so daß kein Zirkelschluß vorliegt. Ich werde darauf noch in einer andern Arbeit in der Math. Zeitschrift eingehen. Die durchgeführte Betrachtung läßt ohne Verwendung diese3 gruppentheoretischen Satzes erkennen, daß ein halbeinfaches System A eine in K rationale einstufig isomorphe"Darstellung besitzt; denn enthält 21: mehr linear unabhängige Elemente wie 21: *, so folgt aus dem Beweis, daß A nicht halbeinfach sein kann. Enthält aber 2( ebenso viele linear unabhängige Ele: mente wie 21: *, so ist 21: * eine einstufig isomorphe vollständig reduzible Darstellung von A. Bei dem Beweis der Umkehrung, daß ein nicht haJbeinfaches System nicht vollständig reduzible Darstellungen besitzt, wird der gruppentheoretische Satz nicht benützt. Nur diese in der Anmerkung genannten Tatsachen werden im folgenden verwendet.

46

THEORY OF ALGEBRAS

8(~

R. Brauer.

Bei festem B stellt (2)eine lineare homogene Relation ffir die Koeffizienten von Y dar, wobei Y ein beliebiges Element der absolut irreduziblen Gruppe ~) ist. Ist f der Grad yon ~ , so kann wegen dieser Relation nicht f~ linear unabh~ngige Elemente enthalten. Man erh~lt einen WidersPmch zu Satz ~, auger wenn B ~ 0 ist. Da ~ ein beliebiger Bestandteil der vollst~ndig reduziblen Gruppe ~[~ war, folgt A~-~ 0. Bei der Sehreibweise (1) stehen also in der Hauptdiagonale yon A lauter NuUen. Das zeigt nnmittelbar, dab ~[ nicht vollst~indig reduzibel ist, da sonst A zu 0 iihnlich, also A ~ 0 sein miil~te. Unsere Betrachtung ergibt auch noch, dal~ bei der Schreibweise (1) von 9~ ein Element dann und nur dann Wurzelgr6~e ist, wenn in der Hauptdiagonale lauter Nullen stehen. A sei i~n folgenden halbeinfach; eine in K rationale, einstufig isomorphe Darstellung kann man in der Form

~-~

~.

annehmen, wo die 9 ~ in K rational und irreduzibel sind. Ferner daft man ohne wesentliche Einsch~inknng annehmen, dab alle 9~x wesentlieh verschieden sind, und dab die Nulldarstel]ung unter ihnen nicht vorkommt. Nach den Siitzen $ und e sind alle Gnlppen 9~ absolut vollst~ndig reduzibel; fiir ~ + ~ haben ~ und ~I~ keinen absolut irreduzib~en Bestandteil gemeinsam. Es sei k ~ l . ~ sei ein fester der Werte 1, 2, ..'., k. Wit fragen: Wieviel linear unabh~ingige Elemente in 9/ gibt es, bei denen bei der Sehreibweise (3) aul~er in 9~e lauter Nullen stehen? Ersichtllch ist diese Anzahl gleich der Anzahl u~der linear unabh~ingigen Elemente yon vermindert um die Anzahl u* de~ linear unabh~ingigen Elemente derjenigen Gruppe ~*, die aus 9~ durch Fortlassen des Bestandteiles ~Q entsteht. Die Anzahl der linear unabhiingigen Elemente in ~ sei u~ ffir x ~.1, 2 , . . . , k. Wegen der eben erw~hnten Eigenschaften der Gruppe ~x foigt aus Satz 7

u = u ~ + u . 2 + . . . ~ - u k,

u*:=u~-u~§247247

Also ist u - u*----uQ. Folglich gibt es in ~ genau ebensoviel linear unabh~ngige~Elemente, bei denen nur an Stelle yon ~e etwas yon Null veto schiedenes steht, wie es in ~[e iiberhaupt linear unabh/ingige Elemente gibt. Daher mul} ~ alle Elemente yon "

Uber Systeme hyperkomplexer Zahlen.

87

~

t~ ) 0'~

we 92~ in der ~,ten Zeile, Q-ten Spalte stehen sell), enthalten. Das gilt fiir ~--~ 1, 2, ..J., k. Jedes Element yon 92 kann man eindeutig in der Form B 1 ~ B~ ~ - . . . -~ B~ mit B x ~ ~x daste]]en. ~ stellt ein invariantes Teilsystem is) yon 9/ dar. Muitipliziert man ein Element yon ~ , mit einem Element yon ~ , so erhiilt man flit ~ ~ ~ stets 0. Wegen dieses Sachverhaltes sagt ma~ 92 sei als d~ekte Summe is) der invarianten Teilsysteme ~1, ~s, ., ~ dargeste]]t. ~ ist z u 9I~ einstufig isomorph. Wit wollen ein System A y o n hyperkomplexen Gr~l~en e~n[ach nennen, wenn es eine in K rationale und irreduzible, einstufig isomorphe Darstellung besitzt19). Jedes einfache System ist also sicher halbeinfach. -- Bei unseren Betrachtungen stellt ~ ein einfaehes System dar, wit haben also bewiesen: Ein halbeinfaches System hy~erkomplexer Gr6[3en ld[3t sich al~

direlcte Summe ein/acher 8ysteme da~stdlen. Ein einfaehes System A besitzt iiberhaupt nut eine einzige in K irreduzible Darstellung. Denn ist 9/ die naeh Definition vorhandene einstufig isomorphe, in K irreduzible Darstellung; ist ~ eine zweite in K irreduzible und yon 9/ wesentlich verschiedene Darstellung, so bfldet auch

eine Darste]]ung. Die Anzabl der linear unabh~ingigen Elemente yon ~ ist aber wegen der S~itze r und ~ gr61~er als die entsprechende Anzahl bei 9/ und folglich auch bei A. Man hat also einen Widerspruch. Genau analog f~lgt, da~ A aul~er den absolut irreduziblen Bestandt~;len yon 92 keine absolu~irreduziblen Darstellungen besitzt. Satz~l:~ Ein tin/aches ~yslem hy~erkomplexer Zahten besitzt nut eine einzi~e ~n K ~rreduzible Dars~ellung. Dutch Zerlegun~ i n absolu~ irreduzible Bestandteile erhdl~ man alle absolu~ ~rreduz~blen Darstdlungen. is) Vgl. etws Diokson, S. 85--86. 19) Diese Definition, die yon der iiblichen verschieden ist, let fiir unsere Zweoke bequem. Gew~hnlieh definiert man sin einfaches System als ein System, das kein invariantes Teils~t~em besitzt (vgl. Dickson, S. 92). Naoh dieser9Definition mul3 man dann abet auch ~dM duroh (4) gegebene System als einfaeh bezeichnen, das nach unserer Definition nioht einfach ist; im fibrigen sind, wie sparer gezeigt wird, beide Definitionen iiquivalent. In iilteren Untersuchungen (Molien, Cartan, Frobenius) waren Systeme wie "(4) ausgeschl0s~n; es ist nicht zweckm~flig, es unter die einfaohen Systeme zu z~hlen, ds seine St~rktur vSllig anders ist.-

88

R. Br~uer.

A sei ein System hyperkomplexer GrSllen, das kein invariantes Teilsystem besitzt. Ist A halbeinfach, so mu~ es einfach sein, da es sicher nicht als direkte Summe darstellbar ist. I s t A nicht halbeinfach, so dad es nut aus WurzelgrSl~en bestehen, weil sonst die WurzelgrSilen ein invariantes Teflsystem, das sogenannte Radikal yon A, bilden, wie sieh aus der oben gegebenen Darstellung der WurzelgrSi~en unmittelbar ergibt. Man sehliellt claim weiter sehr einfaeh (vgl. Diekson, S. 102), daft A einstufig isomorph mit dem System aller Matrizen

ist, wo a eine beliebige gahl aus K sein dad. Umgekehrt besitzt ein einfaehes System kein invariantes Teilsystem. Der Beweis, der sieh auf versehiedene Weisen fiihren liillt, soil bier der Kiirze halber auf w 2 verschoben werden. Wenn er gefiihrt ist, ist gezeigt, da~ die bier gegebene Definition der einfachen Systeme hyperkomplexer GrSllen abgesehen yon dem dutch (4) gegebenen Ausnahmefall mit der iiblichen Definition iiquivalent ist (vgl. 19)). Ein System hyperkomplexer Grfil~en olme Nullteiler besitzt sicher keine WurzelgrSllen und ist daher halbeinfachl Da es sieh nicht sis direkte Summe mehrerer Systeme darstellen lassen kann, mug es einfaeh sein. w Eme Gruppe linearer Substitutionen ~ heille in bezug auf den KSrper K komplett, wenn aHe linearen Verbindungen yon Elementen yon mit Koeflizienten aus K zu ~ geh6ren. Die Matrizen jeder in bezug auf K rationalen und kompletten Gruppe lassen sich als'Elemente eines Systems hyperkomplexer GrStSen iiber K auffassen. H i l f s s a t z 1. ~ s d ~ absolut irre~t~ibler B e ~ l t e i l ei~r ia bezug au! K kompletten unit irre~uziblen Oru~ype ~ . Der dutch Adiunktion des Charakters vort ~ zst K entstehende algebraische KOvper "~ ~iber K heipe Z. Dann ist ~ auch in bezug au] Z komplett. Beweis. ~ ist naeh Satz ~ in einem algebraisehen Kfirper Q fiber Z rational darstellbar; wir k6nnen yon vornherein ~ Jn ~ rational annehmen. Zu ~ geh~rt eine wohlbestimmte in bezug auf Z komplette Gruppe ~ , die aus allen linearen Verbindungen yon endlieh vielen Elementen yon mit Koeffizienten aus Z besteht. Es ist aUCh ~ in Q rational. ~ is~ ebenfaUs in bezug auf K algebraiseh. -- Ist G ein Element aus q~, so seien go) Unter einem algebraischen K6rper ist im folgenden immer ein algebraischer K6rper end]/chen Relativgrades gemeint.

Uber Systeme hyperkomplexer Zah.len.

89

G-----G~,G~, .... G, die algebraisch zu G in bezug auf K konjugierten Elemente; analog seien F ~ F~, F~, ..., F~ die zu einer Matrix F < ~ algebraiseh konjugierten Matrizen. Wit setzen

0,

)

~---

O, Bildet man flit alle G aus (~ das Element (~, so erh~lt man eine einstufig zn (~ isomorphe Gruplae ~ . Dieso ist nach Sa~z ~ in K rational darstellbar, es sei ( ~ * ~ Q - ~ Q in K rational. Bildet man t~ nut fiir alle G aus ~, so erh~lt man eine Untergruppe ~ yon ~ ; Q - ~ Q ~ * ist als Untergruppe yon ~ * auch in K rational. Die yon allen Matrizen Ge bei festem ~ gebildete Gruppe heine t~e; ent~prechend sei ~e die ans allen Matrizen Fo gebildete Gruppe. ~e und ~o sind ei~tufig isomorph, ebenso ~e und ~o fiir ~, a----1, 2,..., r. Da ~ und daher aueh ~ irreduzibel find, sind auch ~ und t~e irreduzibel. Dann und nut dann sind ~e und ~o ~ihnlieh, wenn~~ und ~o iihnlieh sinci~). Aus (5) ergibt fieh, daI~ die absolut irreduziblen Bestand~ile yon ~ * die Oruppen (~, r q~ find; die Bestandteile von ~ * sind analog ~1, ~ , . . . , ~ - Die Anzahl der wesentlieh verschiedenen Bestandteile ist in beiden Fiillen gleich, aUe Bestandteile haben denselben Grad. Nach dem Satz 7 besitzen q~* und ~* dieselbe Anzahl yon linear unabhiingigen Elementen. Wegen ~ * ~ qS* kann man daher jedes Element yon r als lineare Verbindung von Elementen yon ~* darstellen..Da ferner ~* und ~ * in K rational find, kann man die dabei auftretenden Koeffizienten als Zahlen yon K w~ihlen. Nun ist aber mit 9I aueh ~ in K komplett. Das gleiche gilt dann auch flit ~* und daher gehSrt jede lineare Verbindung v0n Elementen yon ~* mit Koeffizienten aus K zu ~*. Es folg~ also ~* ~ @*. Daraus ergibt sieh ~ --~ C~ und ~ ----(~. Folglich ist ~ wirklich in Z komplett. A sei j etzt ein einfaches System hyperkomplexer Gr6flen iiber dem Grundk6rper K; 91 sei die in K irredazible Darstellung yon A, die nach w eindeutig bestimmt und einstufig isomorph ist. Naeh Satz ~ mad ~1) Ist ~e zu ~a iihnlioh, so haben beide denselben Chsmkter X~ = Za. Adjunktion dieses Charakters zu K ergibt einen zu Z konjuglerten KSrper Z e = Za. Ein beliebiges (~e ist als lineare Verbindung yon Elementen yon ~e mit Koeffizienten aus Z e darstellbar, der Charakter geh~rt also auch zu Z e. Das zugehSrige algebrsisch konjugierte 6~ hat infolgedessen denselben Charakt~r, also sind r e und (~a naeh Satz p Rhnlich. Aus der Ahnlichkeit yon 6_)5 und ~ folgt unmittelbar die Ahnlichkeit yon ~0 und ~o-

90

R. Brauer.

ist 9~ absolui~ vollst~indigreduzibel, seine absolut irreduziblen Bestandtei!e lassen sich als algebraisch in bezug auf K konjugierte Gruppen schreiben und sind daher untereinander und zu 9~ einstufig isomorph. ~ sei einer dieser Bestandteile, dann bildet auch ~ eine einstufig isomorphe Darstellung yon A. Ist ~ eine absolut irreduzibleGruppe, die in bezug auf K komplett und in einem algebraischen KSrper fiber K rational ist, so ist ~ nach Satz ~ irreduziblerBestandteil einer in bezug auf K rationalen, irreduziblen und kompletten Gruppe 9~. Nach dem eben durchgefiihrten SchluB sind 9~ und ~ einstufig isomorph; man kann daher die Matrizen yon ~ auch als Elemente eines einfachen Systems hYPerkomplexer Zahlen fiber K auffassen. Dutch Adjunktion des Charakters yon ~ zu K entsteht ein algebraischer KSrper Z fiber K.; ist der zugehSrige Sehursche Index m, so kann man ~ in einem KSrper Z (v~) vow. Grade m fiber Z rational darstellen. Die Konjugierten zu # seien ~9 ~- ~i, tg.~..... v~m. Das Faktorensystem yon ~.~s) in bezug auf Z als GrundkSrper sei ca#~ (~,/~,~ ~ I, 2, ...,m). Wir bilden dann alle Systeme yon m s Zahlen /ap mit folgenden Eigenschaften: 1. l,~ ist eine Zahl aus Z(~?~, ~p) fiir ~, fl----I,2,...,m. 2. Eine Permutation der Galoissehen Gruppe yon Z(zgj, ~, ...,~?m) in bezug auf Z als GrundkSrper, bei der ~?~ in ~9, v~B in ~9~ fibergeht, ffihrt l~ in Ir~ fiber. Ffir alle derartigen Systeme l~p bilden wit die Matrizen (6)

1 I~,) D-----(-~

(~ Zeilen-, 2 Spaltenindex; ~ , 2 = 1 , 2 , . .

., m).

Die Gesamtheit dieser Matrizen bildet naeh U . sS) eine absolut irreduzible Gruppe ~ , deren Charakter zu Z gehSrt. Das Faktorensystem ist ebenfalls capr (genauer zu e~a v assoziiert) und der zugeh{irige Schursche Index daher m. ~) ist ferner in bezug auf Z komplett. Der Grad f yon ~ ist ein Vielfaches yon ra, es sei f----t.m. Die Ges~mtheit aller Matrizen F vom Grade t, deren Koefllzienten beliebige Matrizen aus ~) sind, (7)

F=(D**)

(n, 2 = l , 2, ..., t; D,~ < ~))

bildet eine absolut irreduzible, in bezug auf Z komplette Gruppe ~* vom Grade rat, deren Charakter zu Z gehSrt und die das FalC~orensystem c~ar ~'~) Vgl. etws U. w 1. ,a) Vgl. insbesondere den Beweis yon Satz IV von U. Dort wird yon einer zu ~hnllchen Gruppe R~)R -~ gezeigt, dal3 ihr Faktorensystem r ist; nach U. Satz VIII 9ist daher das Faktorensystem yon ~ zu caar assoziiert.

Uber Systeme hyperkomplexer Zshlen.

91

besitzt. Dutch diese Eigenschaften ist aber ~ * naeh U. Satz V bis auf Ahnlichkeitstransformation eindeutig bestimmt. "~ und ~ * sind also ~ihn]ich und daher nicht wesentlich verschieden. Wir ersetzen im folgenden durch ~ * u n d schreiben fiir ~ * wieder ~. S a t z 2. Die notwendige und hinreichende Bedingung da]iir, daft ein

tin/aches System hyperkompl~er Gr6flen A keine Nullteiler besitzt, i st, dap /iir eine absolut irreduzible Darstellung der Grad und der Schursche Index, iibereinstimmen. B e w e i s . Jede absolut irreduzible Darstellung yon A ist n a c h w 1 ein Bestandteil yon 92, wir kSnnen annehm'en: daft es gerade ~ ist. Ist n u n f : > m, also t > 1 , so kann man leicht zwei yon Null versehiedene Elemente F x und F~ angeben, deren Produkt versehwindet. Man setze z.B. in F z naeh (7) Dll + 0, alle anderen D x x ~ 0 ; in F 9 abet D ~ - = 0, alle anderen Dx~@ 0. Da A und ~ einstufig isomorph sind, hat aueh A NuUteiler. Ist dagegen f = m, also t = 1, so wird ~ = ~ . Hat ein Element F yon ~3 die Determinante 0, so ist fiir alle X < ~ ebenfalls IXF[ ~ 0 , also 0 eharakteristisehe Wurzel von X F. Nach U. Satz V I I und Anmerkung ~6) yon U. ist 0 dann m-faehe Wurzel yon X F , also fiberhaupt die einzige Wurzel. Daher verschwindet d i e S p u r 7.(XF) fiir alle X < t~Wie in w 1 folgt daraus F = 0. Fiir alle F ~ 0 ist also auch I F I ~= 0. Daher besitzt ~ und damit auch A keine Nullteiler, womit alles bewiesen ist. Im allgemeinen Fall t ~ 1 stellt das durch (6) .definierte ~ stets ein System A v o n hyperkomplexen GrSfen ohne Nullteiler dar. Denn ~ ist absolut irreduzibel, in einem algebraischen KSrper fiber K rational und in bezug auf K komplett; fiir ~ stimmen der Grad und der Schursehe Index iiberein. Nach (7) ist das ein]aehe System A einstu]ig ~somorph zu dem

System aller Matrizen, deren Koe]]izienten beliebige Elemente aus A sind; wobd A ein System ohne Nullteiler bedeutet. Das ist ein bekannter Satz yon Wedderbum *"). ~4) Wedderburn, On hyperoomplex numbers, Proceedings of the London Math. Society 6 (1907), p. 77-118. -- Man kann diesen Satz auch so formulieren: Ist A ein einfaches System, so kann man A als direktes Produkt A • M darsteUen ; A besitzt dabei keine Nulheiler nnd M ist dem System aller Matrizen vom Grade t mit Kocffizienten aus K einstufig isomorph. ~ Man zeigt aueh noch leioht, dab A dutch A eindeutig festgelegt ist; denn soll A = A • M sein, so muf das Faktorensystem einer absolut irreduziblen DarsteUung yon A auch das Faktorensystem einer absolut irFeduziblen Darstellung yon A, also das yon ~ oder einer algebraisch konjugierten Gruppe Bein. In~olgedessen muff es sieh um ein zu caBr algebraisch in bezug auf K konjugiertes Faktorensystem handeln; das liefert dann abet ein einstufig zu ~ ~isomorphes System.

92

R. Bmuer.

Eine ganz einiache Betrachtung, die man z.B. bei Dickson S. 121 finder, zeigt, dab jedes auf diese Weiso darstellbare, also jedes einfache System A kein invariantes Teilsystem besitzt (vgl. w 1). Setzt man in (6) la~-~l, lap-----0 fiir a ~ f l , so erh~ilt man fiic D wegen r = 1 (vgl. U. w 2) die Einheitsmatrix E m..5). In (7) setze man D,~ ffi E~, D ~ - ~ 0 flit ~ + 1. Man erkennt, da~ aueh ~ die Einheitsmatrix enth~lt. Daher besitzt ein einfaehes System A ein Einheitselement. - - Soll ~ eine Matrix c E enthalten, so muG, wie aus (6) und (7) folg$, c eine Zald ans Z sein. A sei ein einfaehes System, ~ eine absolut irreduzible Darstellung. Der Charakter yon ~ gehiire zu K, naeh w 4 bedeutet diese Voraussetzung keine wesentliche Einschr~nkung: - Nach Satz 1 und Satz ~ ist dann die einzige absolut ir~eduzible Darstellung yon A. Ist a < A, A die zugeordnete Matrix von ~, so bezeichnen wir die Determinante [A[ als Norm N(a) yon a. Offenbar ~ndert sich N ( a ) nicht, wenn man ~ dutch eine ~hnliehe Darstellung ersetzt. Nach Definition gilt =

Ist N ( a ) + 0, so existiert A -1. Es sei (8)

~o(A) ----A~-~- a~ A ~-x + . . . + ak_ ~A + a~ E = 0

die Gleir niedrigsten Grades mit Koeffizienten aus K, der A geniigt. Dann ist ak + 0 , da sonst A -1 ~v(A) = 0 eine Gleichung niedrigeren Grades ergeben wiirde. Dann folgt abet A_I = __ ak 1 (A1,_1 + a l A ~ _ ~ + . . .

+ aT,-t E ) .

Da die reehte Seite zu ~- gehSrt, gehiirt aueh A -1 zu ~. Infolgedessen enth~ilt A, falls N(a)4ffi 0 ist, ein zu a inverses Element. Ist N ( u ) - ~ 0, so ist in (8) a k = 0, well sonst die letzte Formel ein zu A inverses Element liefern wiirde. (8) kann man abet sehreiben A ( A k - i + al A~-~ + . . . + a~_ 1E) ~ O. Die Klammer ist hier wegen der Minimaleigenschaft yon k yon 0 versehieden. Daher ist A und folglich anch u Nullteiler. Es sou noeh ein Hilfssatz bewiesen ,werden, der sp~iter niitzlich ist. H i l f s s a t z 2. ~ aei eine absolut irreduzible Darstdlung eines ein/achen Systems; ,ihr Grad sei f. ~ und ~ seien zwei dhnliche Unter. gruppen yon ~, d.h. es gebe eine Matrix P yore Grade f mit van N*dl ~) Al]gemein bezeiehnea wit die Einheitsmatrix n-ten Grades mit E~.

Uber Systeme hyperkomplexer Zalden.

93

~rsd~'edener Determinan~e,/~r die (9) ist.

P-~ ~ P =

Dann kann man P als Element yon ~ u~len.

B e w e i a K sei ein unendlicher K6rpert6), ~ ' s e i die Gruppe aller Transponierten yon Matrizen yon ~. Ist Z wieder der K6rper, der aus K dutch Adjunktion des Charakters yon ~ entsteht, so kann man nsch U. Sstz II das Kmneckersche Produkt ~ x ~ ' in Z rational darstellen; Q - x ( ~ x ~ ' ) Q = ~t sei in Z rational. ~ enth;tlt die beiden Untergmppen (I0)

~=Q-a(~xEf)Q,

~=O-l(Efxq~')Q.

}R seinerseits enth~ilt als Untergruppen (11)

1I = Q - I ( ~ x E/) Q,

Setzt man noch T f Q - * ( P x E f ) Q ,

~8=Q-*(~xEf)Q. so ist I T I + 0 .

Aus (9), (10)

.nd (11) folgt (12)

IIT= T~8,

~T=

T~.

Umgekehrt sei T eine Matrix, die (12) erfiiUt und flit die IT[ _4=0 ist. T ist dann wegen der Vertauschbarkeit mit ~ yon der Form Q- 1 ( p x El) Q, cla nach U. w 1 die ,nit Ef x ~' vertauschbare Matrix Q T Q-1 die Form P x Ef hat. P ist dabei yore Grade f, es ist [ Pt + 0. Aus der ersten Gleichung (12) folgt dann aber wegen (11) wieder P-I?SP= ~. ~ , 6 , II und ~ sind als Untergruppen yon (~ in Z rational. Die Gleichungen (12) stellen daher fiir die Koeffizienten einer unbekannten Matrix T eine Reihe yon linearen Gleichungen mit Koefllzienten aus Z dar. Da wir wissen, da~ diese Gleichungen eine L6sung haben, flit die I T [ + 0 ist, und da Z unendlich viele Elemente enth~ilt, gibt es anch eine in Z rationale L6sung T m i t [ T I + 0. Kollstruieren wir aus diesem T wie eben eine Matrix P, so erfiillt P die Gleichung (9). Wir behaupten, da~ dieses P zu ~ geh6rt. Da ns ~ absolut irreduzibel ist, enth~ilt es (nach Satz ~) fs linear unabh~ingige Elemente; man kann daher P als lineare Verbindung yon Elementen yon ~ darstellen; wegen (10) also T = Q-I(Px Ef)Q als lineare Verbindung yon Elementen yon ~. Da T und 9~ in Z rational sind, kann m a n dabei die Koeffizienten aus Z w~ihlen, folglich auch bei der Darstellung von P durch Elemente yon ~. D a aber ~ in bezug anf Z komplett is% ist wirklich P < ~. ~) Ist K ein endlicher K6rper, so versagt der bier gegebene Beweis. Wio sich aus w 4 ergibt,gilt der Hilfssatz auch noch in diesem FMIe.

94

R. Brauer.

w A sei ein einfaehes SyStem, ~ die in K irreduzible Darstellung. 9/ enth a l t e r wesentlich verschiedene, absolut irreduzible Bestandteile; der Grad derselben, der fiir alle naeh Satz ~ derselbe ist, sei f. Nach den Sgtzen ~ und' 7 enthglt dann 9~ genau r f ~ linear unabh~ingige Elemente; der Rang yon A ist folglieh r f ~. Dabei l~isst r sich auch als Relativgrad desjenigenKSrpets Z deuten, der aus K dutch Adjfinktion des Charakters yon ~ entsteht, wo ~ wie friiher eine abs01ut irreduzible Darstellung yon A bezeichne. Das System derjenigen GrSssen yon A, die mit allen GrSBen yon A vertauschbar sind, bezeichnen wit als das Zentrum yon A. Ist ~ eine GrSSSe des Zentrums yon A, so muSS die Zugeordnete Matrix yon ~ m i t allen Elementen der absolut irredtLziblen Gruppe-~ vertauschbar und daher eine Multiplikation cEf sein. Wie aus der Darstellung (6) und (7) yon f01gt, mu~ dabei c eine Zahl aug Z sein (vgl. w 2). Fiir alle c < Z enthglt ~ umgekehrt c E t, d a ~ in bezug auf Z komplett ist und E t enth~ilt. Das Zentrum von A wird. also dutch die Elemente c E~ dargestellt und ist daher zu Z einstufig isomorph. Satz 3. Ist A d n ein/aches System, ~ eine der absolut irreduziblen Darstellungen, und ist Z der dutch Ad~unktion des Charakters yon ~ zu K entstehende K(~rper, s o ist Z zum Zentrum yon ~ einstu/ig isomorph. Ist r der Relativgrad yon Z in bezug au/ K und hat ~ den Grad f, so ist der Rang yon A in bezug au/ K genau r f ~. A sei im folgenden ein festes System hyperkomplexer GrSl~en ohne .Nullteiler, ~ wie oben eine absolut irreduzible Darstellung. Ist t > 0 eine ganze rationale ZahI, so bilden die Matrizen des Grades t, deren Koeffizienten beliebige Elemente aus J sind, ein einfaehes System; es werde mit At bezeiehnet. Man erhglt so alle einfachen Systeme, denen bei Zerlegung nach dem Wedderburnschen Satz gerade A als nultteiIerfreies System zugeordnet ist. Die Elemente yon ~ kann man wieder dureh (6) gegeben annehmen, (7) liefert eine absolut irreduzible Darstellung yon At, die j etzt deutlicher mit ~t bezeichnet werde. Da ~) und ~t naeh w2 das gleiehe Faktorensystem haben, sind die KSrper, in denen ~) und ~t rational dargestellt werden kSnnen, fiir beide Gruppen die gleichen; wit wollen sie die Darstellungsk6rper yon ~t nennen ~6~). Sie enthaltenalle Z. Die Zentren aller 20a), In der in Anmerkung ') zitierten Arbeit werden diese KSrper u n d ihre Konjugierten in bezug auf Z als Zer/~llungsk6rper der in K rationalen irreduziblen Darstellung yon At bezeichnet. Abweichend yon der genannten Arbeit sollen in, folgenden unter K6r~e~ immer kommutative K6rper, also KSrper im gew6hnlich6n Sinne verstanden werden.

Uber Systeme hyperkomplexer Zahlen.

95 84

Systeme At sind nach Satz 3 zu Z isomorph; wit kSnnen direkt das dutch c E dargestellte Element des Zentrums yon At m i t c identifizieren Und dementsprechend ohne Gefahr einer Verwechslung das Zentrum yon At ebenfalls mit Z bezeichnen. Unter einem Teilk6rper /1 yon At verstehen wit ein Teilsystem, das einen KSrper bildet und Z enthiilt; Z selbst ist auch ein TeilkSrper yon At. Ein TeilkSrper heiflt maximal, wenn er in keinem andern enthalten ist. Ist F ' ein Teilk6rper yon A,, so soll eine Abbildung yon F auf F ' nu~ dann isomorph genannt werden, wenn sie einstufig ist~ und die Elemente Yon Z sieh-selbst entsprechen. Ist F ein TeilkSrper yon At, so ist F ein K6rper endlichen Ranges fiber Z. Da mit K auch Z vollkommen ist, entsteht /~ aus Z dutch Adjunktion eines Elementes 7, das in Z einer irreduziblen Gleiehung f(x) ~ 0 geniigt. Der hSchste Koeffizient yon f ( x ) s e i 1, der Grad heiBe k. 0 sei die 7 in ~t entsprechende Matrix, ~0(x)--~ 0 sei die charakteristische Gleichung yon 0. Da man nach Satz ~ E ~ x ~ t und damit auch E ~ x C in Z rational machen kann, ist die charakteristische Gleichung yon E~ x O in Z rational; das ist aber ~0(x)m= 0. Da Z vollkommen ist, ist daher auch 90(x) selbst in Z rational. Wegen f ( C ) = 0 sind alle Wurzeln yon (x)-----0 auch Wurzeln yon f(x)-----0, und wegen der Irreduzibilitiit yon f(x) ist ~(x) eine Potenz yon f(x). Nun hat q~(x) naeh (6) nnd (7) den Grad t . m , also folgt b I t m t.Tt$

(13)

~ ( z ) = f(x) k

F * sei ein zu F isomorpher Teilk6rper yon At. 7" sei die ? entsprechende Or6ge, C* die zugeordnete Matrix yon ~t. Dann ist aueh f(G*) = 0 . Da f ( x ) = 0 in Z irreduzibel ist und daher nut einfaehe Wurzeln hat, sind C und C* iihnlieh. Naeh Hilfssatz 2 gibt es dann in 9 o 7 ~t ein Element P, so dab C * = P - " C P 1st). Ist ~z das ent~preehende Element yon At, so ist 7" ---- ~r- 1 ~ ~r und daher F * = ~r- 1 Fz~.~ Ist F in einem Teilk5rper B enthalten, so ist F * < ~ - l B ~ und ist daher gleich, falls nicht maximal. H i l f s s a t z 3. Der Relativgrad k eines TeillMrpers F yon A t in bezug au[ Z iat ein Teiler yon t m . Sind F u n d F* isomorphe Teilk6rper yon At, so gibt ea ein Element ~ in A t , so daft 1 ~ * ' = ~ - 1 F ~ ist. F u n d F * sind dann gleichzeitig maximal. , sei ein Teiler yon t, etwa z s = t. F sei j etzt ein TeilkSrper yon A,, die entsprechende Untergruppe yon ~,. Wegen (6) und (7) kalm man t~) Ist K eln endlicher KSrper, so gilt nach w4 Hflfssatz 2 auch noch.

96

It. Brauer.

~j aucb als Gesamtheit der Mat~zen yore Grade z deuten, deren Koeffizienten selbst Elemente aus ~ , sind: (14)

P = (G~)

(~, ~ = 1, ~ . . . . . z; G ~
ist, wie behauptet. Für den Fall, daß l:J eine unendliche Primstelle ist und nicht der triviale Fall ™i> = 1 vorliegt, ist ™iJ = 2 und Ai> die Quaternionenalgebra über dem reellen ZahlDaß AK'.ll, = (A1>)K'.ll, "'K'.1!1 ist, ist dann gleichbedeutend damit, daß K'.1!1 körper

Qi>.

der komplexe Zahlkörper ist, d. h. mit n'.ll,

=

f '.1!1 = 2

=

mi, (e!ll, tritt hier nicht auf),

was auch hier wieder die Behauptung ergibt (n'.ll, ist wie mi, nur der Werte 1 oder 2 fähig). 7. Zu bedeutungsvollen Resultaten kommt man, wenn man die in Satz 3 beantwortete Fragestellung nach den sämtlichen Zerfällungskörpern K einer festen Algebra A umdreht, nämlich bei festem algebraischem Körper K über Q die Gesamtheit der von K zerfällten Algebren A über Q betrachtet. Diese Algebren A bilden eine durch K bestimmte Untergruppe ~ der R. Brauerschen Gruppe 21: aller Algebren (genauer aller Klassen ähnlicher Algebren) über Q Da). Setzt man Kals galoissch über Q voraus, so kommt man so zu Sätzen, die als Verallgemeinerung von Hauptsätzen der Klassenkörpertheorie (Theorie der relativ-abelschen Zahlkörper) auf allgemeine relativ-galoissche Zahlkörper anzusehen sind : Satz 4. (Zerlegungssatz). Der Relativgrad f der Prirnteiler in K eines nicht in der Relativdiskriminante von K aufgehenden Primideals l:J r>on Q ist gleich dem frühesten Exponenten, für den Ai ,..., Qil wird für alle Algebren A aus der K zugeordneten Grz~ppe ~Beweis. Da f der (für alle Primteiler von l:J in K übereinstimmende) l)-Grad von K ist, während andererseits der l:J-lndex von A gleich dem Index, also Exponenten von Ai, ist, so ist nach Satz 3 f jedenfalls ein Multiplum jenes frühesten Exponenten, und es genügt zum Beweise noch zu zeigen, daß es in der Gruppe ~ Algebren A mit dem genauen l:J-lndex f gibt. Nach dem Frobeniusschen Dichtigkeitssatz gibt es nun sicher noch ein weiteres nicht in der Relativdiskriminante von K aufgehendes Primideal l:J' in Q, dessen Primteiler in K den Relativgradf haben. Es sei dann Zein zyklischer Körper über Q, dessen l)-Grad und l:J'-Grad den Wert f hat. Ferner sei IX eine Zahl in Q, für die die Normenrestsymbole

(IX~Z)

und

(IX~,z)

reziproke Werte der Ordnung

übrigen Teiler q des Führers von

Z gilt (IX'qZ)

f haben,

während für alle

= 1. Nach dem verallgemeinerten

Satz von der arithmetischen Progression . kann IX zudem so gewählt werden, daß es außer ev. l:J, l:J' und den q nur noch ein einziges Primideal t von Q genau in der ersten Potenz enthält. Für dieses ist dann nach dem Produktheorem für das Normenrestsymbol (Reziprozitätsgesetz) ebenfalls

(IX, Z) =

(IX•/) = 1 [ H, 3. 8-10].

A hat dann den l)-lndex und l:J'-lndex

Jede Algebra

f, dagegen für alle anderen Primstellen

•a) R. Brauer, Über Systeme hyperkomplexer Größen, Jahresber. d. D. M.-V. 38 (1929), S. 47/48. auch H, 13. 1.

öl*

119

REPRINT OF [14]

Siehe

404

Brauer, Hasse und Noether, Beweis eines Hauptsatzes in der Theorie der AlgelJren.

von Q den Index i[H, 17.7]. Nach Satz 3 folgt daraus mit Rücksicht auf die Voraussetzung über .1J und die Wahl von .!J', daß K Zerfällungskörper für Aist. Damit sind in der Tat in der Gruppe se Algebren A vom .!J-lndex f nachgewiesen. Satz 5. (Eindeutigkeits- und Anordnungssatz.) Ist K ~ K', so ist se ~ se' und umgekehrt. Insbesondere ist also die Zuordnung der Algebrengruppen se zu den galoisschen Körpern K eine umkehrbar eindeutige. Beweis. a) Aus K ~ K' folgt trivialerweise se ~ se'; denn jede von K zerfällte Algebra A ist a fortiori eine von K' zerfällte Algebra. A'. b) Sei umgekehrt se ~ se'. Dann ist nach dem Zerlegungssatz für jeden Nichtteiler .1J der Relativdiskrimiante 1'9 1 /~. Insbesondere ist demnach f'9 = 1 für fast alle .1J mit f{, = 1. Daraus folgt aber nach dem geläufigen analytischeh SchluPverfahren 10 ) K ~K'.

8. Schließlich sei noch ausgeführt, daß der Hauptsatz eine wesentliche Förderung gibt für die von 1. Schur 11 ) behandelte Frage nach den Zahlkörpern, in denen die absolutirreduziblen Darstellungen einer endlichen Gruppe möglich sind: Satz 6. Die absolut-irreduziblen Darstellungen einer endlichen Gruppe @ sind sämtlich in Kreiskörpern möglich, z.B. jedenfalls stets im Körper der nh-ten Einheitswurzeln, wenn n di,e Ordnung von @, und h hinreichend groß ist. Beweis. Geht man zum rationalzahligen Gruppenring G von @ über, so werden die absolut-irreduziblen Darstellungen r, von @ zu den absolut-irreduziblen Darstellungen der einfachen Bestandteile G, der halbeinfachen Algebra G, und ihre Zentren sind jeweils die Körper Qi der zugehörigen Charaktere 12 ), wegen der Endlichkeit von @ also jedenfalls Kreiskörper, und zwar sicherlich Teilkörper des Körpers der n-ten Einheitswurzeln. l\Taeh Sat.io; ,l iqt nun ein zyklisch:Jl· Körper ?..: 'iJ...cr Q, Zerfö.11„ngskörp"r fiir C;, wenn für jede Primstelle .1J von Q, sein .!J-Grad lli.lJ ein Multiplum des .!J-lndex m,'9 von G1, ist. mi.'9 ist von 1 verschieden lediglich für die Primteiler des Grundideals von G. bezüglich Q., also sicherlich lediglich für die Primteiler der absoluten Diskriminante von G; da diese in n" aufgeht - n" ist die Diskriminante einer (nicht-maximalen) Ordnung in G 13 ) - , somit höchstens für die Primteiler .lJ von n. Um n.'9 für diese .lJ zum Multiplum von m,'9 zu machen, genügt es vorzuschreiben, daß die fraglichen .1J in Z, von einer jeweils durch mi.'9 teilbaren Ordnung verzweigt sind. Das leistet aber, wie man sich leicht überlegt, der Körper der nh-ten Einheitswurzeln für hinreichend hohes h. Im letzten Satz der angeführten Arbeit stellt 1. Schur fest, daß man in allen bisher bekannten Fällen ·schon mit dem Körper der n-ten Einheitswurzeln auskommt. Ob dies immer zutrifft, und ob die hier entwickelten Methoden ausreichen, um diese Frage zu entscheiden, bleibt weiteren Untersuchungen vorbehalten. Siehe dazu etwa H.. Hasse [l. c. Anm. 6], § 25, III. I. Schur, Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen, Berl. Akad.Ber. 1906. 12 ) Siehe dazu: a) R. Brauer und E. Noether, Über minimale Zerfällungskörper irreduzibler Darstellungen, Berl. Akad.-Ber. 1927; § 1. b) R. Brauer, Über Systeme hyperkomplexer Zahlen, Math. Zeitschr. 30 (1929); Satz 3. c) E. Noether, Hyperkomplexe Größen und Darstellungstheorie, Math. Zeitschr. SO (1929); §§ 21, 24, 26. 13) Siehe E. Noether [l. c. Anm. 7 c], § 26. 10 )

11 )

Eingegangen 11. November 1931.

120

THEORY OF ALGEBRAS

Über die Konstruktion der Schiefkörper, die von endlichem Rang in bezug auf ein gegebenes Zentrum sind. Von Richard Brauer in Königsberg i. Pr.

Die Sätze von Herrn Maclagan Wedderburn gestatten, die Aufstellung aller halbeinfachen Systeme hyperkomplexer Größen über einem gegebenen Körper K auf die Konstruktion aller Divisionsalgebren 6. über K zurückzuführen. Diese Systeme A werden von denjenigen Schiefkörpern 1 ) gebildet, die K im Zentrum enthalten und von endlichem Rang in bezug auf K sind. Offenbar bedeutet es dabei keine wesentliche Einschränkung, wenn man annimmt, daß K selbst das Zentrum von A ist. Mit der so entstehenden Aufgabe, alle Schiefkörper A von endlichem Rang über einem gegebenen Körper K aufzustellen, will sich die vorliegende Arbeit befassen; der Körper K wird dabei als vollkommen vorausgesetzt. Die Aufgabe ist, wie sich zeigen wird, mit gewissen Fragen der „kommutativen" Algebra äquivalent; vielleicht dürften deshalb auch diese Untersuchungen dazu .beitragen, die Untersuchungen der „nichtkommutativen" Algebra vom kommutativen Standpunkt aus interessant erscheinen zu lassen. In der Hauptsache handelt es sich um spezielle Probleme der folgenden Art: Gegeben ist ein Normalkörper K( ,& ) über K mit der Galoisschen Gruppe @, ferner eine endliche Gruppe S) und eine homomorphe Abbildung von S) auf @; kann man K(-&) in einen Normalkörper K(-&*) über K derart einbetten, daß K(-&*) die Galoissche Gruppe S) in bezug auf K besitzt und die aus der Galoisschen Theorie wegen K(-&) < K(-&*) entspringende homomorphe Beziehung von S) auf @ gerade die gegebene Abbildung ist? Ob diese Frage zu bejahen oder zu verneinen ist, hängt natürlicli ganz von den gegebenen Stücken ab. Wir wollen von einem „Einbettungsproblem" für den Normalkörper K(,&) sprechen. Zur Aufstellung der Schiefkörper A verwenden wir die Zerfällungskörper von A; das sind die maximalen, in vollständigen Matrixalgebren mit Koeffizienten aus A enthaltenen (kommutativen) Teilkörper 2 ). Unter allen Zerfällungskörpern von A werden gewisse ausgezeichnet und als reguläre Zerfällungskörper bezeichnet; sie unterliegen unter anderem der Bedingung, Normalkörper über K zu sein. Es gibt dann nur endlich viele Schiefkörper A mit dem Zentrum K, die einen gegebenen Normalkörper K(-&) .als regulären Zerfällungskörper besitzen. Alle diese A lassen sich mit Hilfe von gewissen endlichen Gruppen S) kennzeichnen, die homomorph auf die Galoissche Gruppe @ von 1 ) Die Terminologie ist im folgenden anders wie in dem Buch: van der Waerden, Modeme Algebra, Berlin 1930/31. Die dort (Band 1, S. 40) als Körper bezeichneten Systeme bezeichnen wir als Schiefkörper, während wir das Wort Körper wie sonst meist üblich für die kommutativen Körper in der van der Waerdenschen Bezeichnungsweise reservieren. 2 ) Vgl. R. Brauer und E. Noether, Über minimale Zerfällungskörper irreduzibler Darstellungen, Sitzungsberichte d. Preußischen Akademie d. Wissensch. 1927, S. 221-228.

121

REPRINT OF [15]

Brauer, Konstruktion

s,, (a)

=

I; c,1,_,a, . >

Elemente s = I; z,e, des Zentrum von A sind durch R(s)=S(s) gekennzeichnet ; man erhält daraus lineare homogene Gleichungen für die z,. Die beiden Determinanten J R(a) J =N1(a) und J S(a) J =N2 (a) heißen diebeidenNormenvon a, es ist N1,_(aß)=N,.(a)N1,_(ß) für l=l,2. Ist

158

THEORY OF ALGEBRAS

ALGEBREN

59

r=

~ c,e, in A gegeben, und soll E = ~ x,e, aus der Gleichung Ea = r bestimmt werden, so hat man die linearen Gleichungen

für die x, aufzulösen. Ist N1 (a) = 0, so ist a ein Rechtsnullteiler, d. h. das Proukt Ea verschwindet für ein E =/= 0. Dagegen hat im Fall N1 (a) =/= 0 die Gleichung Ea = r genau eine Lösung E. Ist nicht jedes a ein Rechtsnullteiler, so gibt es ein rechtsseitiges l-Element e, für das Ee = E für alle E aus A gilt. Analoge Betrachtungen gelten für die Gleichung aE = ß, wenn man N2 anstelle von N1 verwendet und überall rechts und links vertauscht. Algebren mit 1-Element sind dadurch charakterisiert, daß weder Ni(a) noch N2(a) für alle a verschwindet. Jeder Rechtsnullteiler ist dann Linksnullteiler und umgekehrt. Ist a nicht Nullteiler, so gibt es ein a- 1 mit aa- 1 = a- 1a = 1. Sowohl R(a) wie S(a) sind dann einstufig1\ Es braucht nicht N1 (a) = N2 (a) zu sein. Es sei E die Einheitsmatrix. Die characteristischen Polynome /1(x) = /2(x) =

(11)

\

+ (-l)nN1(a), O",(a)xn-t+ ... + (-1)nN2(a)

xE- R(a) \ = xn- l11(a)xn- 1+ ...

1 xE-

S(a) 1 =

X

n-

von R(a) und S(a) heißen die charakteristischen Polynome von a, sie sind unabhängig von der speziellen Wahl der Basis e,. Die Koeffizienten sind homogene Polynome der Komponenten a, von a. Insbesondere sind O"i(a) und O"i(a), die beiden Spuren von a, lineare Fuktionen der a,. Hat A 1-Element, so gilt nach I B 2 (Henke) /1(a) = 0,

(12)

Es ist also a Wurzel von Gleichungen n-ten Grades mit Koeffizienten aus K. Man kann auch leicht die Gleichung niedrigsten Grades mit Koeffizienten aus K aufstellen, der a genügt. Ersetzt man die Komponenten a, von a durch Unbestimmte, so wird das Polynom niedrigsten Grades, das für x = a verschwindet, als das Gradpolynom f(x) der Algebra bezeichnet; es ist ein Teiler von /1 (x) und /2 (x ), die Koeffizienten sind Polynome in den Unbestimmten a,. /(x) heißt auch die Hauptgleichung von A. Ein Algebra A ohne 1-Element ist in Algebra Ä mit 1-Element als Unteralgebra enthalten. Die regulären Darstellungen von Ä liefern einstufige Darstellungen von A. Darstellungen von Algebren werden 6)

Für Einstufigkeit von R(a) und S(a) in andern Fällen vgl. R. Brauer,

159

REPRINT OF [22]

R. BRAUER

60

allgemein in 14 (Deuring) behandelt7). 6. Isomorphe und homomorphe Ringe. Wir betrachten Abbildungen eines Ringes A auf einen Ring A* mit demselben Operatorenbereich K, bei denen jedes Element von A* als Bildelement auftritt. Eine solche Abbildung a ~ a* heißt ein Homomorphismus, wenn für alle a, /3 aus A und t aus K die Beziehungen (13a)

(a

+ /3)* =

a*

+ /3*,

(13b) (. Die Definition der Ideale, der Quotientenalgebra, der Einfachheit ist dieselbe wie im assoziativen Fall. Stets ist A/ A' eine abelsche Algebra. Jede Lie-Algebra enthält ein eindeutig bestimmtes maximales auflösbares Ideal L, das Radikal. Ist L = 0, so heißt A halbeinfach. Jede einfache Algebra (außer der Algebra von der Ordnung 1) ist auch halbeinfach. Hat der Grundkörper K die Charakteristik 0, so ist auch hier eine halbeinfache Algebra direkte Summe einfacher Algebren. Die Gesamtheit aller einfachen Algebren ist von H. Cartan 86 > für den Fall aufgestellt worden, daß K ein algebraisch abgeschlossener oder reell abgeschlossener Körper ist. Auch der Fall eines beliebigen Grundkörpers K der Charakteristik O ist in letzter Zeit weitgehend untersucht worden8n. Man hat eine Darstellung einer Lie-Algebra, wenn jedem Element a von A eine Matrix Ma derart zugeordnet ist, daß für alle a, ß aus A und alle t aus K (50)

Ma+ß = M.

+ Mß, Mi.=

tM., Maß= M.Mß - MßMa

gilt. Analog zu der regulären Darstellung (Nr. 5) kann man auch hier Darstellungen erhalten, die sogenannte adjungierte Darstellung. Die Spur (J' von M; ist eine quadratische Form in den Parametern von a. Hat K die Charakteristik 0, so ist dann und nur dann (J' nichtentartet, wenn A helbeinfach ist (vergl. Nr. 17). Dann und nur dann ist A auflösbar, wenn für alle r aus A' die Spur von M; verschwindet. 31. Hinweis auf einige weitere Anwendungen hyperkomplexer Größen. Die Zahlentheorie in hyperkomplexen Systemen und ihre Anwendungen werdenin 22 (Hasse) behandelt. Auf dem Wege über die Darstel85) Vgl. etwa M. Zorn, Bull. Amer. Math. Soc. 43, 401, 1937. 86) S. 84 ). Vgl. ferner B. L. van der Waerden, Math. Z. 37, 446, 1933. 87) W. Landherr, Abh. Math. Sem. Hansische Univ. 11, 41, 1935; N. ]acobson, Proc. Nat. Acad. Sei. USA 23, 240, 1937; Ann. Math. (2) 38, 508, 1937, -Von weiteren Arbeiten über Lie-Algebren seien genannt: N. Jacobson, Ann. Math. (2) 36, 875, 1935; Trans. Amer. Math. Soc. 42, 206, 1937; E. Witt, J. Reine Angew. Math. 177, 152, 1937; G, Birkhoff, Ann. Math. (2) 38, 526, 1937,

188

THEORY OF ALGEBRAS

ALGEBREN

89

lungstheorie 14 (Deuring) erhält man durch Untersuchung der Gruppenringe (Nr. 4) wichtige Anwendungen in der Gruppentheorie (15 (Magnus)). Für eine Anwendung auf algebraische Gleichungen vergl. man 17 (Brauer). Die Theorie der Algebren spielt eine Rolle bei der Untersuchung von Riemannschen Matrizen88l_ Dies sind Matrizen Q, die von den Perioden der p linear: unabhängigen Integrale 1. Art auf einer Riemannschen Fläche vom Geschlecht p gebildet werden. Q hat p Zeilen und 2p Spalten, und es gibt eine schiefsymmetrische rationalzahlige Matrix C vom Grad 2p mit I Cl =I= 0, derart daß QC- 1Q1 = 0 gilt, und ferner Q(ic- 1)Q 1 die Matrix einer positiven definiten Hermiteschen Form ist. Man suche Matrizen M vom Grad 2p, für die eine Gleichunge MQ = QR besteht, wo R eine rationalzahlige Matrix p-ten Grades ist. Diese M bilden eine Algebra A über dem Körper der rationalen Zahlen, ebenso bilden die R eine isomorphe Algebra. Die Eigenschaften dieser Algebren A und die Konstruktion von Riemannschen Matrizen zu gegebenem A ist von A. A. Albert 89 ) eingehend untersucht worden. Eine andere Behandlung der Frage ist von H. Weyl 90 ) von etwas anderem Ausgangspunkt aus gegeben worden. . Von Bedeutung sind hyperkomplexe Größen ferner für Untersuchungen über die Grundlagen der Geometrie. So beweist D. Hilbert 91 ) die Unabhängigkeit des Pascalschen Satzes von den räumlichen Verknüpfungs-·und Anordnungsaxiomen, indem er mit Hilfe eines geordneten Schiefkörpers92 ) eine analytische Geometrie bildet. Mit Hilfe von Alternativsystemen kann R. Moufang 93 ) die ebenen Geometrien beschreiben, in denen der Satz vom vollständingen Vierseit gilt. 0. Veblen und ]. L. M Wedderburn haben nichtdistributive Systeme für Grundlagenfragen verwendet 94l_ Neuerdings hat J. von Neumann 95 ) mit Hilfe von Schiefkörpern kontinuierliche projektive Geometrien gebildet.

(Received August 14, 1978) 88) 89) geführt. 90) 91) 92) Angew. 93) 94) 95)

Vgl. dazu S. Lefschetz, Bull. Nat. Res. Counc. Washington 63, 310, 1928. Ann. Math. (2) 36, 886, 1935. Dort findet man Alberts vorangehende Arbeiten anAnn. Math. (2) 35, 714, 1934; 37, 709, 1936, Grundlagen der Geometrie, 7 Aufl. Leipzig 1930. Untersuchungen über geordnete Schiefkörper finden sich bei R. Moufang, J. Reine Math. 176, 203, 1937. Abh. Math. Sem. Hamburg Univ. 9, 207, 1933. 0. Veblen und ]. H. M. Wedderburn, Trans. Amer. Math. Soc. 8, 379, 1907. Proc. Nat. Acad. Sei. USA 22, 101, 1936.

189

REPRINT OF [22]

236

MA THEMA TICS: BRA UER A ND NESBITT

PROC. N. A. S.

abelian group of order p(m + 1)/2 which contains no operator of order p2 just as in the case when G involves no operator of order p2. In the present case the group may be extended arbitrarily by an operator of order p or by an operator of order p2 except that at least one of the (m - 1)/2 extending operators must be of order p2. Hence there result m - 2 distinct groups since the extending operators can always be so selected that they have different orders except in one case. When G involves an invariant operator of order p2, all the possible groups can be constructed by starting with the abelian subgroup of type 2, 1(m - 2)/2. Since the extending operator can be selected so as to have either of two different orders in each of the (m - 2)/2 possible cases, the number of the possible non-abelian G's in this case is again m - 2. As there is also one possible abelian group it results that, including direct products there are m - 1 groups of order pm which have the property that each of them involves invariant operators of order p2 and contains exactly m - 1 independent generators, m being even. When m is odd this number is m - 2. In the special case when p = 2 a necessary and sufficient condition that a group of order pm has m - 1 independent generators is that the squares of all its operators generate the subgroup of order 2. It is known that the number of these groups of order 2' is 3(m - 1)/2 when m is odd and (3m - 4)/2 when m is even.' Although these groups are somewhat more complex than those relating to the more general case when p is odd they seem to require no consideration here since their fundamental properties were determined in the article to which we referred at the close of the preceding sentence and in those referred to therein. 1 G. A. Miller, these PROCEEDINGS, 22, 112 (1936).

ON THE REGULAR REPRESENTATIONS OF ALGEBRAS By R. BRAUER AND C. NESBITr UNIVERSITY OF TORONTO

Communicated February 24, 1937

The regular representations play an important r6le in the work of Molien, Cartan and Frobenius' in the theory of hypercomplex numbers. More recently, the theory of groups of linear transformations has been extended and new concepts have been introduced. Our first aim was to study the regular representations of an algebra with regard to these new ideas.

VOL. 23, 1937

MA THEMA TICS: BRA UER AND NESBITT

237

We consider an associative algebra A over a field F, assuming that A has a unit element. Since we shall be concerned with the absolutely irreducible constituents of the regular representations of A, we may assume without restriction that F is algebraically closed. Let el, e2, . . ., Ex be a basis of A. For every a in A, we have equations

e,= ErK) 1. Since [S'.;J', f;) i-d = f;) ;, it follows that Q;-1 and Q; commute (mod f;);). If we interchange 0;-1 and O; in 01, 02, · · · , O., we obtain a set of residue systems belonging to the composition series ( 7) In a similar manner, we can prove a theorem which has the same relation to Schreier's extension of the Jordan-Holder theorem (Schreier [25], Zassenhaus [32]) as (1.1A) has to the Jordan-Holder theorem itself.

506

RICHARD BRAUER

~ = ~o :)

~1 :) · · · :) ~;-2 :) ~;-2 :) ~; :) · · · :) ~.

[May

= {1} ( 1

because 0;-10;~;= 0;0;-1~; and ~f-2= 0;-1Ü;+1Ü;+2 · · · 0,. We next interchange O; with 0;-2, etc., until O; finally stands at the first place. If (1.2A) is true for @1, as we may assume, it now follows for @. 2.

LOEWY SERIES

1. A group@ is completely reducible( 8 ), if it is the direct produd of simple groups %, '-132, · · · , '-13,. As indicated by this notation, @ = %'-132 ... '-13,,

@1 = '-132 ... '-13,, ... ' @,-1 = '-13,,

@, =

l 1}

is a composition series. Every normal ~subgroup ID1 of @ is completely reducible and is a direct factor, i.e., @= ID1 X \.n, where \.n is anormal subgroup of @. Because @Jm~m, the factor group @/\.ln is also completely reducible. If & is anormal subgroup of an arbitrary group @, we say that & is completely reducible with regard to @, if & is the direct product of minimal normal subgroups of @. More generally, if & and 5S are normal subgroups of@ and &::25S, we say that &/5S is completely reducible with regard to @, if &/5S is completely reducible with regard to @/,S. If we add the inner automorphism of @ to the operators of the groups considered (subgroups of @ and factor groups formed out of them), then complete reducibility of &/5S with regard to @ means the same as ordinary complete reducibility of &/5S. In the case of abelian groups @, the words "with regard to @" can always be omitted. For any group @, we prove easily: (2.1A) If 53 and m are normal subgroups of@which are completely reducible with regard to @, the same is truefor 53\.ln.

Proof. We add the set of all inner automorphisms of@ to the set of operators. If :D= [53, \.ln], we may set 53=531X:D, ID1=ID11X:D, where 531 and ID11 are normal subgroups of @. We then have 53ID1=531XID11X:D, since [531, ID11X:D] = { 1}. This shows that (2.1A) is true. (2.1B) If &, 5S and 3 are normal subgroups of @, where 5S~&, and &/5S is completely reducible with regard to @, then [3, &]/ [3, 5S] is completely reducible with regard to @, and isomorphic with a normal subgroup of &/5S.

Proof. We extend the domain of operators as in the proof of (2.1A). The statement is a consequence from the fact that [3, &]5S/5S~ [3, &]/ [3, &, 5S] = [-3, &]/ [3, 5S ], since [3, &]5S/5S is anormal subgroup of &/5S. (2.1C) If 5S and ~ are normal subgroups of @, where 5S~/5S and ,S~/~ are, both completely reducible with regard to @, so is 5S~/[5S, ~].

Proof. From (2.1B) it follows that [5S~, ~]/ [5S, ~] =~/ [5S, ~] is com(8) Cf. van der Waerden [28, vol. 1, p. 143 ].

210

THEORY OF ALGEBRAS

1941]

MATRICES OVER A DIVISION RING

507

pletely reducible with regard to @. The same is true for 58/(58, ~]. Then (2.1A) shows that 58~/ [58, ~] is completely reducible with regard to@/ [58, ~]. and hence with regard to@. 2. A Loewy series of@ is a series of normal subgroups of@: (5)

in which each factor group il.n,-i/9.n, is completely reducible with regard to@. Of special importance is the lower Loewy series (or lower cover series of@). Here iJ.n 1_1 is the normal cover ("Sockel")(9 ) of@, i.e., the union of all minimal normal subgroups of@. It follows from (2.1A) that iJ.n1_1 is completely reducible with regard to @. More generally, we take for iJ.n,_1 the group for which iJ.n,_iJiJ.n, is the normal cover of @/9.n, (r=t, t-1, ···).Then we actually obtain a Loewy series of @. Obviously, iJ.n,_1 is the largest group which can precede il.n, in any Loewy series of@. Let S;, be a second group, and (6)

be a Loewy series of Sj. We then state (2.2A) Let 0 be a homomorphic mapping of Sj upon a subgroup Sj* of@ (Sj*C@) which maps normal subgroups \n upon normal subgroups \n* of @( 10 ). If (5) is the lower Loewy series of@, and (6) any Loewy series of Sj, then

Proof. Let \)1 be a minimal normal subgroup of Sj. If its image \n* contains a normal subgroup~ of@ with { 1} C~C\n*, the elements of \)1 which are mapped upon elements of~ form a proper subgroup of \)1 which is normal in Sj. This is impossible, and hence \n* is a minimal normal subgroup of @, and belongs therefore to iJ.n 1_ 1, the normal cover of @. It now follows easily that Wu~1 Cil.n1-1. The mapping 0 induces a homomorphic mapping of S;,/Wu-1 upon a subgroup of @/ID11-1, which maps normal subgroups upon normal subgroups. Using the same argument, we obtain (Wu-dWu-1)*C(il.n1-dil.n1-1), and hence Wu~2Cil.n1-2, etc. 3. The dual of the lower Loewy series is the upper Loewy series or upper cover series. Here, il.n, is the upper cover of iJ.n,_1(11), i.e., the intersection of all maximal normal subgroups of il.n,-1, T = 1, 2, · · · . We see successively that il.n1, il.n2, · · · are normal in @. Then il.n, can also be defined as the intersection of the normal subgroups of @ which are maximal in il.n,-1. From Remak [24], Cf. also Ore [22]. This assumption is necessary whereas in the dual theorem (2.3A) it is sufficient to assume that .p* is normal in @. ( 11) Ore [22 ]. ( 9)

( 10 )

508

RICHARD BRAUER

[May

(2.lC) it follows easily that ffi1,-1Lffi1, is completely reducible with regard to@, so that we actually have a Loewy series. Obviously, 9)1, is the smallest group which can follow ffic,-1 in any Loewy series of @. We no.w show (2.3A) Let 0 be a homomorphic mapping of S;i upon a normal subgroup S;i* of@ (S;i*C@). If (6) is the upper Loewy series of S;i, and (5) any Loewy series of@, then \np*CfficP (p = 1, 2, · · · ) where \np* again denotes the image of \np.

Proof. Without restriction, we may assume that to every inner automorphism of S;i there corresponds an operator in r which produces this automorphism. Form (5')

The distinct groups in (5') form a Loewy series as follows from (2.1B), and 0 maps S;i upon®· We replace@ by@, and (5) by this Loewy series. If we can prove (2.3A) ~n this case, it also will be true in the original case. It is, therefore, sufficient to prove (2.3A) in the case where@= S;i*. Here, \np* is a normal subgroup of @. The totality of elements of S;i whose images lie in ffi11 form a normal subgroup~ of S;i. We map S;i/~ upon ®/ffi11 by H~-H*ffic1 (Hin S;i). Since H*ffi11 = ffi11 only if H is in ~. this mapping is an isomorphism. With ®/ffi11, then S;i/~ also is completely reducible, and hence~ contains the upper cover \)1 1 of S;i. This implies \ni*Cffi1 1. If for ffi1 1, \n1, and the mapping induced by 0 the statement has been proved, as we may assume, it now follows for@, S;i and the mapping 0. 4. We now consider the case that @ = S;i, and 0 is the identical isomorphism. From (2.2A) it follows that any Loewy series (6) of @ has at least the same length as the lower Loewy series (5), since for u.

in K,

form a complete residue system '13; of m;-1 (mod m;). All the vectors E~0 • arranged according to increasing i form a basis of m, and with regard to this basis, & has the form

(11)

where &; is an irreducible set of square matrices of degree a;. These &; are called the irreducible constituents of &. From Jordan-Holder's theorem, we obtain at once(16 ) (4.4A) The irreducible constituents of a set & of square matrices are uniquely determined apart from their arrangement, if similar sets are considered as equal.

When we replace the Ei1l by another basis of m;_ 1 (mod m;), then &; is replaced by a similar set. We obtain this new form of & by a similarity transformation of type (3.3C). If a formula (11) holds where each &; is a reducible or irreducible set of square matrices of some degree a;, we say that each &; is a constituent of &. In particular, we call &1 a top constituent and &, a bottom constituent. Let & and 5S again be two related intertwined sets, &P=P5S and P~O. We consider again the mapping of 5IB upon a certain admissible subgroup \E of mwhich is defined by P. The vectors of 5IB which are mapped upon O form an admissible subgroup ® of 5IB, and we have ~~5IB/®. If we use these subgroups in order to split & and 5S, we have with regard to suitable coordinate systems

&=(* u0) ' *

This gives Schur's lemma(17 ). ( 16)

This simple proof for the uniqueness of the irreducible constituents is due to W. Krull

[11 ]. ( 17 ) I. Schur [26 ]. Schur's proof is extremely simple. By means of (7), similarity transformations of l!l and 5B are performed such that P assumes the desired form, and then 21 and 5B must have the form given here.

515

MATRICES OVER A DIVISION RING

1941]

(4.4B) If two related sets 2( and .SS are intertwined by a matrix P-;6-0, then there exists a bottom constituent of 2( which appears as a top constituent of .SS. If 2( and .SS are irreducible, then Pis nonsingular, and 2( and .SS are similar. 5. We now apply the results of Section 1. We choose the residue systems of .SSi-i/.SS; always as in (10), consisting of all linear combinations of some r-independent vectors. From (1.lB) we see that the results of Section 1 remain valid, if we restrict the choice of residue systems by this condition. Any change of the residue system '-lJi as used in Section 1 can be accomplished by a succession of changes of the following kind: The elements of 'l3i are multiplied by elements of some$;, with i> j. This now corresponds to replacing E!1l by E~1>+s;1> where each s!0 is of the form (10). This basis transformation corresponds to the linear transformation X,*=X, for K-;6-i, X;*=X;-QXi where the vector X is broken up according to the scheme (a1, · · ·, akJ 1) and the matrix Q of type (ai, ai) is formed by the components Z}o., (10), of the vectors S~1l. This is a transformation T;i(-Q) = Ti/Q)- 1 (cf. §3.3), and 2( is there replaced by Ti/Q)- 12(Ti/Q). According to (3.3A), we have to add the ith column in (11), r-multiplied by Q, to thejth column, and thejth row, l-multiplied by -Q, to the ith row. Because of i>j, the triangular form (11) of 2( is not disturbed:

*

--2(1

l --2(ii *

21:;

I - I

.*.

Only the sets 21:;}o. with A ~j and 21:iii with µ'i!;i will be changed. We denote such a special similarity transformation of 2( as an elementary similarity transformation of 2(. All the 21:. remain unchanged. Consider again two related sets of square matrices 2( and .SS, operating in the vector spaces mand 5ffi respectively. We assume that both split into irreducible constituents

5.81

2!:1 (12)

2(

=

*

2!:2

*

*

5.8= ... 21:, J

*

.SS2

*

*

.. • 5.8s

where 2(P has the degree ap and 5.8, has the degree b,. If Pis an intertwining matrix, we break up P according to the scheme (a1, · · · , ar/ b1, · · · , bs); say

516

RICHARD BRAUER

[May

P = (P.x). Then the products ~p and P'ilj can be obtained in the ordinary manner (§3.3). We say, therefore, that the intertwining matrix P has been broken up in accordance with the splitting of~ and 'ilj in (12). Application of (1.lA) to the homomorphic mapping of 5ffi upon a subgroup of mthen yields

(4.SA) Let ~ and 'iS be two related sets of square matrices which split into irreducible constituents (12), and let P be an intertwining matrix. We can apply to ~ and 'ilj a succession of elementary similarity trans!ormations such that the matrix P* which afterwards takes the place of P (cf. §4.2) contains in each row and each column at most one term not equal to 0, if broken up in accordance with the sp_litting of~ and 'iS.

If P*=(P:x), then ~.P:x=P:x'iSx because of this form of P*. If P:x~O. then P.x is nonsingular, according to (4.4B). Since for a given X this may occur for at most one value of K, after a succession of similarity transformations of type (3.3C), each P:x is either O or a unit matrix. Assume now that Pis nonsingular so that~ and 'ilj are similar. Then every row of P* must contain one P:x~O, say for instance Pf1 ~0. We denote the sets similar to ~ and 'iS, which we have obtained by ~ and 'ilj again, and use the notation (12). Then it easily follows from ~P*=P*'ilj by forming the first rows of the products that

We replace 'ilj by the similar set Zj·:__\ 1'iSZi-I,i (cf. (3.3B)). Because 'ilji.i_1=0, the triangular form (12) of 'ilj is not disturbed, *

~-'ilji-1

_t_o *

'iS·

1-1

I

*

The irreducible constituents of 'ilj remain the same, only 'iS i-l and 'ilj i are interchanged. Such a similarity transformation of 'ilj is an admissible permutation of rows and columns which can always be applied, if 'iSi,i-1=0. According to §4.2, P* must be replaced by P*Zi-I,i, i.e., the columnsj-1 andj are to be interchanged (3.3B); but the essential properties of P* are not destroyed. Similarly, we can interchange 'ilji with 'ilji-2, 'iSi-3, · · · , 'iS1. The matrix P** which takes the place of P will have the first row (I, 0, · · · , 0). We now work with the second row of P**. The element P;: = I in it will not stand in the first column. After a number of further admissible permutations of rows and columns, we may bring it into the second column. Continuing in this manner, we will finally replace P by I. This gives (cf. (1.2A))

(4.SB) If~ and 'iS, (12), are two similar sets of square matrices which break

1941]

517

MATRICES OVER A DIVISION RING

up into irreducible constituents, then it is possible to carry 5B into~ by a succession of similarity transformations of types (3.3A), (3.3B), and (3.3C) (18 ). 5.

THE LOEWY CONSTITUENTS

1. A set ~ of square matrices of degree n is completely reducible, if the corresponding vector space m (with ~1 and K as sets of operators) is completely reducible. If we choose the composition series of mand the~. as in §2.1, then the formula (11) takes the form ~1

~(P

irreducible,

~. with zeros above and below the main diagonal. Conversely, if such a formula holds, then~ is completely reducible. In the general case, let be a Loewy series for m. If we choose the basis of mby first taking a maximal set of vectors of 9)10 which are r-independent (mod 9)11), then a maximal set of vectors of ID11 which are r-independent (mod 9)12), etc., then ~ has the form ~1

(13)

*

~2

*

*

. . . ~t

and each ~>. is completely reducible, since ID1>.-i/ID1x is completely reducible. We say that~ here appears in a Loewy form; every Loewy form of~ is obtained from a Loewy series of m. Two Loewy forms are of special importance, the lower and the upper Loewy form( 19 ), corresponding to the lower and upper Loewy series of 5B, both having the same length (cf. (2.4A)) which will be denoted by L=L(~). We write them:

*

(14)

* ( 18) ( 19)

~L-1(~)

*

*

... ~1(~)

*

* • . • ~L(~)

These transformations are to be applied to the form (12) of ~ and !B. Cf. A. Loewy (14, 15], W. Krull (12], B. L. van der Waerden [29].

518

[May

RICHARD BRAUER

where the first is the lower and the second is the upper Loewy form. The lower Loewy constituents 21(~). 22 (~), • • • are numerated starting from the bottom, and the upper Loewy constituents ~1 (~), ~ 2 (~). • • • starting from the top. The constituent 21(~) is the maximal completely reducible set which can appear as bottom constituent of ~- If ~ splits into 58 and 21(~). then 2i+1(~),..._,2i(58).

Similarly, fi1(~) is the maximal completely reducible set which can appear as a top constituent of ~; and if ~ splits into ~1 (~) and Q3, then ~i+1 t - X. (ft) If 58 is in its upp61' Loewy normal form, then P .x = 0 for IC t, the first s - t rows in P consist of zeros. In the case (ft), P has the form

1941]

MATRICES OVER A DIVISION RING

519

( 16/j) ; for s < t, the last t - s columns consist of zeros:

(16a)

(1M)

P=

P=

· P,-2.1-2 0

0

· P,-1.1-2

P,-1,1-1 0

· P,.1-2

Ps,1-1

Pu

0

0

P21

P22

0

P31

P32

Paa·

P.,1

:J 3. When a set ij is given in the form (11), splitting into irreducible constituents, we can use the method of §2.5 in order to determine the Loewy constituents \l(ij), We consider one constituent iji in ij, *

(17)

where the rows and columns i+l, i+z, · · ·, r of (11) are grouped together in '.t). If (11) belongs to the composition series m, m1, m2, ... ' mr and $, is a complete residue system of m,_1 (mod m,), then the question is whether we can change$; so that it forms an admissible subgroup. The only freedom which we have is that we can add arbitrary vectors of mi to the basis elements of$;. This amounts to an elementary similarity transformation of (17), involving the second and third row and column (cf. §4.5). If after the change $i is an admissible subgroup, then ~ must become 0, since the modified $; are invariant under ij, But an elementary similarity transformation replaces ~ by ~+'.t)Q-Qiji; so that the residue system $; will be of the lowest kind, if and only if this is O.for a suitable Q, and iji will belong to L1(ij). Hence (5.3A) The first Loewy constituent 21(ij) consists of those irreducible constituents m;, (15), for which a matrix Q can be determined such that in (17) ~ = Qm; - '.t)Q. After similarity transformations, we may assume that all mi of this type stand in columns in which otherwise only zeros appear. In order to find 22(m) · we have to remove the rows and columns of the mi "of lowest kind" from m, and treat the remaining set 5B in the same manner; we have 2v+1(~l) =2.(58). Moving all the constituents~ of lowest kind to the bottom by admissible permutations §4.5, 2 1(~() will appear at the bottom of m. After removing its rows and columns from m and treating the remainder in the same fashion, we

520

RICHARD BRAUER

[May

finally arrive at the lower Loewy form of ~- It is remarkable in this connection that the criterion (5.3A) only depends on the solution of linear equations for the coefficients of the matrix Q. 4. The dualism between the upper and lower Loewy form can be realized in the following manner. We replace every matrix A of~ by its transposed A', §3.5. If ~ is in its lower normal form, (14), the new set 21:' formed by all A' will have the following form *

~'= If we arrange the rows and columns in reverse order, ~, splits into the constituents 21(~)', · · · , 2L(~)'. In this manner, we easily see that

2.(~)'=t(~') (v=l, 2, · · ·, L; L=L(~)=L(~')).

(5.4A)

Using this method, we can derive results concerning the upper Loewy form from those concerning the lower Loewy form in §5.3.

6.

ADDITIONAL REMARKS

1. We consider two related sets~ and 53 of matrices which split completely into irreducible constituents, i.e., ~1

~= 53. If Pis an intertwining matrix, ~P=P53, we break up P according to this splitting, P= (P,>.) (cf. §4.5). The condition for P,>. becomes ~.P,>.=P,>.53>.. Using Schur's lemma, we obtain

(6.lA) Let~ and 53 be two related sets of matrices which split completely into irreducible constituents ~!i. ~2, · · · , ~' and 531, 532, · · · , 53, respectively. If P = (P ,>.) is an intertwining matrix broken up in accordance with the splitting of ~ and 53, then either P,>. = 0, or ~.,..__,53>. and P,>. is nonsingular and intertwines ~. and 53>.. Conversely, if these conditions are satisfied P = (P,>.) intertwines ~ and 53. 2. The matrices P which intertwine a set~ of square matrices with itself, ~p = P~, form a ring, the commuting ring(£(~) of~- If Pin(£(~) is a nonsingu-

lar matrix, then p- 1 also belongs to(£(~). From Schur's lemma, we find that (6.2A) The commuting ring of an irreducible set is a division ring.

521

MATRICES OVER A DIVISION RING

1941]

Denote by kX~ the set which splits completely into k equal constituents ~. and by [~h the set of all matrices (A.>.) of degree kin which the A.» are arbitrary elements of ~- I.e., Au···

kX~=

(k times),

[~h: all

A1k

· · · · ·

with A.» in ~-

Ak1 • • • Akk

We then state (6.2B) (a) C[(['.t].)=sX([('.t):)sX5ill=([(@), which is impossible. Hence 5IB and '.t both are irreducible division rings consisting of matrices of degree t, and each is the commuting set of the other. We now apply theorem (9.1B) to '.t instead of @. If ho is the h-number of ([('.t) = 5IB, and z the l-rank of '.t, then ho =z/t. But (34) shows that the h-number of ([(@) is h = sh 0 , and hence

k/n = h = sh0 = sz/t

(35)

which implies k=s 2z since n=st. Consequently,@ and ['.t]. have the same l-rank, and therefore 9)1(@) =ID1(['.t].). Thus we have (9.3A) Any irreducible semigroup@ of degree n is, after a similarity transformation, contained in a set ['.t]. where '.t is an irreducible set of matrices of degree n/s=tforming a division ring, and@ and ['.t]. have the same l-rank and hence the same enveloping module, ID1(@)=ID1(['.t].). Further, 5ill=([('.t) is the only irreducible constituent of. where k is the l-rank of@ and >. the degree of the first Loewy constituent ~1 (ffi) of the regular representation ffiof@.

Proof. Let Mi, · · · , Mk be an l-basis of ID1(@) with regard to which the regular representation ffi appears in its lower Loewy normal form. lf Gis an arbitrary element of @, we have

M.G = I:r.,;.M-,. where R= (r.-,.) is the matrix of ffi, associated with G. If 5B is one of the ij., and M. corresponds to U. in ID'l(.58) and G corresponds to B, we have

U,B = I:r.-,.U-,.. We now apply (7.8A) setting U.= (h~). Then P,,= (h. rows of P„ contain coefficients not equal to 0. Hence for a ;;i k -

242

THEORY OF ALGEBRAS

>..

1941]

MATRICES OVER A DIVISION RING

539

This shows U.=O for K~k-';... Hence M 1, ···,Mk->. are represented by O in each wmµ) and, therefore, belong to 91. (10.4B) If the semigroup @ splits into irreducible constituents and the radical of ID?(@) vanishes, then@ is completely reducible.

Proof. We have here k =A; i.e., ffi is completely reducible, L(ffi) = 1. We denote by @ the set obtained from @ by replacing everything below the main diagonal in (39) by O's, and omitting all constituents 0. According to (8.2G), a suitable linear combination of the elements of@ is equal to the unit matrix. A corresponding linear combination of the elements of @ gives a matrix J of ID?(@) which in (39) has a unit matrix in the place of every ID1(i5\) ~O and, of course, 0 in the place of every ID?(l)p) =0. The product JG of J with an element G of @ has the same main diagonal as G. Then G - JG lies in the radical of ID?(@), and hence JG =G. Now (8.2F) can be applied. We obtain L(@) =L(ffi) = 1, i.e., @ is completely reducible. If K is noncommutative, the converse of assertion (10.4B) need not be true. 5. Repeated application of (10.2A) now gives {10.5A) If a set O of square matrices is a sum of sets 01, 02, · · · , Or, if O;Oi=O for i 0 is an integer and ~ an irreducible ring. Then mis isomorphic to the ring~ whose structure is described by (11.4A).

Proof. The radical of l8 must vanish. Therefore, l8 is completely reducible. From (11.4E) it follows that l8 has only one irreducible constituent. 5. We consider an arbitrary ring~ of matrices which contains n XK and a representation mof~- Let A1, · · · , A k be an l-basis of~ and A an arbitrary element of~- The regular representation mis defined by (42)

A.A

=

L;,. r.;,.A;,.,

If A.-B., A-Bare the associated elements in

r.x in K.

m, we find

(43) We may assume that for a certain t the elements B1 = · · · =Bt = 0 and that B 1+1, • · · , Bk are l-independent. On comparing (42) and (43), we see that the regular representation of l8 appears as an end constituent of m. Using (8.2A) we now find: (11.5A) Let~ be a ring of matrices containing n XK. Every representation lB of ~ of degree m appears as an end constituent of m X where is the regular representation of~- Further, lB appears as a constituent of mnX~.

m

m

As corollaries, we obtain : (11.5B) If lB is a representation of~. then L(lB)CL(~). (11.5C) Every irreducible representation of~ appears as a constituent of~The following theorems are sometimes useful.

544

RICHARD BRAUER

[May

(11.SD) If 581 and 582 are two representations of~ which have no irreducible constituent in common, then we can find an element Q of ~ which is represented by the unit -matrix in 581 and by O in 582.

Proof. According to (11.4B), we can find an element A of ~ which is represented by the unit matrix in every irreducible constituent of 58 1 and by O in every irreducible constituent of 582 • Then A corresponds to a radical element B2 of 582. If we replace A by a power of A, we may assume B 2= 0. If A is represented by B1 in 581, then B1- I lies in the radical of 581; we have (B1 - 1) 1 = 0 for some integer t>O. Hence we may write I as a polynomial f(B 1 ) without constant term of B 1 . Then Q = f(A) will satisfy the required conditions. (11.SE) If 581 and 582 are two representations of~ which have no irreducible constituent in common, and if B1 and B2 are arbitrary elements of 581 and 582 respectively, then we may find an element A of~ which is represented by B1 in 581 and by B2 in 58;.

Proof. Let A ,.. From (11.SF) and (11.4C) it follows that 4),. has the radical (0) and hence is completely reducible. Conversely, if 4>,. is completely reducible, it represents 91 by 0. Then ~,.mc~,.-1, as was stated. The complete reducibility of 4>,. is, of course, equivalent to the complete reducibility of~,./~i,-1 considered as an additive group with the elements of K as l-operators and the elements of ij as r-operators. In order to obtain the lower Loewy normal form of m, we must choose ~-1 as small as possible such that ~mmc~m-1; then ~m-2 as small as possible such that ~m-191C~m-2· Thus (12.3B) Let ij be a ring containing n XK whose radical is 91. The lower Loewy normal form of the regular representation mof ij corresponds to the chain of r-ideals (O) = 9(LC9(L-1cmL- 2c · · • cmcm 0 =ij(38). The exponent L here is equal to the number L(ij) =L(ffi) of Loewy constituents of ij and m (cf. (8.2B)). Similarly, we obtain from (12.3A) the theorem that (12.3C) Under the assumptions of (12.3B), the upper Loewy normal form of ffi corresponds to the series of ideals (O) = OoC01C · · · COL=ij, where 0; consists of the l-annihilators of 9(i in ij.

If we consider! as an additive group with the elements of K as l-operators and the elements of ij as r-operators, we may say that 9(LC9(L-tC · · · Cij and 0 0 C01C ; · · COL are the upper and the lower Loewy series of!. 4. From (12.3B) we see that the degree X of ~1(ffi) is equal to k- P where k is the l-rank of ij and vis the l-rank of 91. In the notation of (11.4B) X = Lk,.. But the argument of §10.1 easily shows that every ij,. appears at least k,./f,. times in ~1(ffi). Therefore, the constituents ij,. occupy at least k,. ordinary rows of ~1(ffi). Because X=Lk,., we obtain: (12.4A) If ij,. is an irreducible representation of ! of l-rank k,. and of degree f ,., then ij,. appears exactly k,./f,. times in the first Loewy constituent ~1(ffi) of the regular representation mof !. ( 38)

Cf. Nesbitt [19].

1941]

MATRICES OVER A DIVISION RING

547

ßIBLIOGRAPHY

1. A. A. Albert, Modern Higher Algebra, Chicago, 1937. 2. - - - , Structure of Algebras, American Mathematical Society Colloquium Publications, vol. 24, 1939. 3. R. Brauer, Ueber Systeme hyperkomplexer Zahlen, Mathematische Zeitschrift, vol. 29 (1929), pp. 79-107. 4. R. Brauer and C. Nesbitt, On the regular representation of algebras, Proceedings of the National Academy of Sciences, vol. 23 (1937), pp. 236-240. 5. W. Burnside, Reducibility of any group of linear substitutions, Proceedings of the London Mathematical Society, (2), vol. 3 (1905), pp. 430-434. 6. A. H. Clifford, Representations induced in an invariant subgroup, Annals of Mathematics, (2), vol. 38 (1937), pp. 533-550. 7. M. Deuring, Algebren, Ergebnisse der Mathematik, vol. 4, 1935. 8. H. Fitting, Die Theorie der Automorphismenringe Abelscher Gruppen und ihr Analogon bei nichtkommutativen Gruppen, Mathematische Annalen, vol. 107 (1932), pp. 514-542. 9. G. Frobenius, Theorie der hyperkomplexen Grössen, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1903, pp. 504-537. 10. G. Frobenius and I. Schur, Ueber die Aequivalenz von Gruppen linearer Substitutionen, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1906, pp. 209-217. 11. W. Krull, Ueber verallgemeinerte endliche Abelsche Gruppen, Mathematische Zeitschrift, vol. 23 (1925), pp. 161-186. 12. - - - , Theorie und Anwendung der verallgemeinerten Abelschen Gruppen, Sitzungsberichte der Heidelberg Akademie der Wissenschaften, 1926. 13. A. Loewy, Ueber die Reduzibilität der Gruppen linearer homogener Substitutionen, these Transactions, vol. 4 (1903), pp. 44-64. 14. - - - , Ueber die vollständig reduciblen Gruppen, die zu einer Gruppe linearer homogener Substitutionen gehören, these Transactions, vol. 6 (1905), p. 504. 15. - - - , Ueber Matrizen und Differentialkomplexe, I, II, III, Mathematische Annalen, vol. 78 (1917), pp. 1-51, 343-368. 16. C. C. MacDuffee, On the independence of the jirst and second matrices of an algebra, Bulletin of the American Mathematical Society, vol. 35 (1929), pp. 344-349. 17. F. D. Murnaghan, The Theory of Group Representations, Baltimore, 19.38. 18. T. Nakayama, Some studies on regular representations, induced representations and modular representations, Annals of Mathematics, (2), vol. 39 (1938), pp. 361-369. 19. C. Nesbitt, On the regular representations of algebras, Annals of Mathematics, (2), vol. 39 (1938), pp. 634-658. 20. E. Noether, Hyperkomplexe Grössen und Darstellungstheorie, Mathematische Zeitschrift, vol. 30 (1939), pp. 641-692. 21. - - - , Nichtkommutative Algebra, Mathematische Zeitschrift, vol. 37 (1933), pp. 514541, 22. 0. Ore, Structures and group theory II, Duke Mathematical Journal, vol. 4 (1938), PP· 247-269. 23. L. Pontrjagin, Ueber den algebraischen Inhalt der topologischen Dualitätssätze, Mathematische Annalen, vol. 105 (1931), pp. 165-205. 24. R. Remak, Ueber minimale invariante Untergruppen in der Theorie der endlichen Gruppen,, Journal für die reine und angewandte Mathematik, vol. 162 (1930), pp. 1-16. 25. 0. Schreier, Ueber den Jordan-Hölderschen Satz, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, vol. 6, pp. 300-302. 26. I. Schur, Neue Begründung der Theorie der Gruppencharaktere, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1905, pp. 406-432.

251

REPRINT OF [36]

548

RICHARD BRAUER

27. - - - , Beitritge zur Theorie der Gruppen linearer Substitutionen, these Transactions, vol. 15 (1909), pp. 159-175. 28. B. L. Van der Waerden, Moderne Algebra, 2 vols., Berlin, 1931. 29. - - - , Gruppen von linearen Transformationen, Ergebnisse der Mathematik, vol. 4, 1935. 30. J. L. Wedderburn, Lectures on Matrices, American Mathematical Society Colloquium Publications, vol. 17, 1934. 31. H. Wey!, The Classical Groups, Princeton, 1939. 32. Zassenhaus, Lehrbuch der Gruppentheorie, Leipzig, 1937. THE UNIVERSITY OF TORONTO, TORONTO, CANADA.

ON THE NILPOTENCY OF THE RADICAL OF A RING RICHARD BRAUER 1

1. Introduction. A few years ago, it was shown by C. Hopkins 2

that the structure theory of noncommutative rings 3 can be based on , the assumption of only the minimum condition for left-ideals. Before Hopkins, a maximum condition for ideals had also been used in order to prove that the radical of the ring is nilpotent. Actually this last fact is a special cas.e of the maximum condition, for example, the existence of a maximal nilpotent (two-sided) ideal, and this makes Hopkins' result appear rather surprising. In this note, I give a short and simple proof for Hopkins' theorem. I also show that it is sufficient to assume only the minimum condition for sets of two-sided nil-ideals (that is, ideals consisting only of nilpotent elements) in order to prove the nilpotency of the radical. The later sections are concerned with the existence of idempotents and primitive left-ideals contained in a given regular left~ideal. Here the assumptions concerning the ring R are those on which Köthe 4 and Deuring5 based their treatment of noncommutative rings. As was shown by Kiöthe, these assumptions are equivalent to the validity of the structure theory, so that it is natural to work with them. Once the results of the later sections have been cstablished, there is no difficulty in dcvcloping thc theory with thc usual methods. 6 2. Preliminaries. A ring R is a set of elemcnts for which an addition and a multiplication are defined such that the elements form an abelian group under addition and that the associative law of multiplication and both distributive laws hold. We may also have a set K of operators. Then the product t (ixv) R (ßv) = 0. But R (ß1 ), ••• , R (ß,,.) are Jinearly independent, and we find h~~>(ixv) = 0 for all r, s, v. Then H,,(ixv) = 0 for all v which gives a contradiction. 1 ) The relations for the coefficients of the representations given in this and the following theorems have also been obtained by Nakayama and Nesbitt by a different method. Their proof has not been yet published. 2 ) This result has been obtained by Nakayama, loc. cit.

6

265

REPRINT OF [44]

The representation (17) can now be written in the form

(20)

and this shall be done from now on. We next take V= W == UK in (15). commuting with U„ has the form 1 )

The most general matrix P

(21)

when broken up in the same manner as U K in (20). If P is singular, we must have P 11 = 0, P 33 = 0. Applying the same method as above, we obtain:

Theorem 4: I f FK( ~)

= (t~;l( ~)) ,

then uruler the assumptions of theo-

rem l, we have }; U~i) (1Xv)

/~;) (ßv)

=

0,

V

prrovide,d that uil does not belang to the part C K in the lower left corner in (20).

u,,,

In order to deal with the case that uii belongs to C K in (20), we make the further assumption that FK is absolutely irreducible. Then in (21), we must have P 11 = P 33 = hl where h lies in K and J is the unit matrix. This yields (22) V

V

where we set OK(~)= (cWm), i, j = l, 2, ... , Dg(FK). Here h;r is an element of K, depending only on the values of j and r (for fixed K). When ( 3.11 It now followsthat (22)

K = K(e),

V = To,

S = Z,

n = q=.

As shownin J., Footnote 24, we may assume that (23)

c(ao, 3o, ao) = 1

forall ao, i8oin W. This togetherwith (1) yields easily (24)

T1; c(ao, ,)goyo) = c(go, ao ,yo)

c(ao , go, yo) = c(0o,0TO, aCO).

Choose a complete residue system (25)

Pi =

1, P2

...* *Pn

of 9t (mod 3). The elementsk(o, fo) in (5) may now be taken as (26)

k(ao, 3o) =

TIc(ao,,3o, pi)

11A. A. Albert (Bull. Amer. Math. Soc. 36, p. 649 (1930)) has shown that for a central division algebra A of index m splitting fields K of degree m with the symmetric group as Galois group exist, provided that Hilbert's irreducibility theorem holds in F.

ON SPLITTING

FIELDS

OF SlMPLE

ALGEBRAS

as followsfrom(lc) and (4).

From (23) and (24), the equation

(27)

flk(pi, 1) = 1

87

can be deduced. We may then assume that the P in (8) satisfythe additional condition

IIr(Pi)= 1.

(28) LEMMA:

PROOF:

? 1. The elementsof [!9] have thedeterminant Under our presentassumptions,we have [ti] =

(Ei3(K

X))

and it is clear that the determinantis 1. On the otherhand, det [M(a)I by (16),

=

?

HC0(1,K

K ranges over the system (15). -1 Ya-, K-1) =

C(l

C(1,

K,

a1 K1)

Since C lies in F a!

K )

=

C(Kx, a, 1)

and (11) yields C(1,

K

X ) =

,Y

C(Koao, ao , 1) (Koao)/l(Ko)'v(ao).

If K rangesoverthe system(15), thenKowillrangeovera completeresiduesystem of at (mod 3), and withoutrestrictionit can be assumed that this is just the system (25). Thus det [M(a)] =

i

IJc(pitao, ao , 1) nt(pia0)/j|P(pi)an(a0)n.

The elementpiao will also range over a completeresiduesystem (mod A). followsfrom (4) that

It

JjeC(piao, ao, 1) = J7c(pi, toa 1)

and from(9) that

= IlR(pi) H1r(piao) By (28) this productis 1 and so is its symbolicath power. In orderto obtain t(ao)n, use 1 = n and apply (8) and (26): (ao) n=

k(ao, 1) = IIc(ao , 1, Pi).

On account of these facts,we have det [M(a)] =

Jjjc(pi, ao, 1)/IIc(ao,

1, pi).

Now (24) shows that the determinantis i 1 and this proves the lemma.12 considerations up to this point can be given under the weaker assumption that n is an odd number, not necessarily a prime. 12 The

88

RICHARD BRAUER

If 0 and X are two elementsof Q, we write 0 Then (11) fory = 1 gives t (aoF

) t(o)

-,' ,

when

e/71

lies in N(K).

v(0o)

and when ao is replaced by aoBo, to considerthe n Because of (9), it is sufficient We thenobtainn - 1 relations (29)

(P)

fI

-

1 elementsD(pj) with i > 1.

,( )m(Pa)

wherep is an elementi#1of (25) and wherea on the rightranges over the same n - 1 elements. We have m(p, a) = 1 when S3p#= Ha; m(p, a) = -1 when , = He, and m(p, a) = 0 in all othercases. It is seen easily that the matrix T(f3) = (m(p, a)) furnishesa representationT of WI. If we interpretat as a permutationgroupof degreen, thenfi is the subgroup whichleaves the firstsymbolfixed. It is seen at once that if we omit this first of degreen - 1 is identipermutationrepresentation symbol,the corresponding cal with T(23). Under our present assumption,e3 is still doubly transitive. and an irreduciblerepresentaHence T(iB) will split into the 1-representation tion Q(Q3)of degreen - 2. This holds not onlyin the sense of representations but also in the modularsense (mod n), because the with complexcoefficients, order of e is prime to n.

Returningto (29) for arbitrary,3in WI,we remarkthat it is morelogical to considerT(,3) as a modularrepresentation(mod n), since the m(p, a) could be changed(mod n). This modularrepresentationT(W) still containsthe 1-representationas can be shown without difficulty.The remainingrepresentation Q(9) can be obtainedas follows. Using P(1) = 1 and (28), we can write (30)

3(P.)

=

(t(P2) *(P3) ...*

(Pn-1))

?

If we introducethis on the rightside of (29), we obtain formulae (31)

I

D(P)'

I

D(Pff)q

a)

Here, Q(,3) = (q(p, a)) can be seen wherep, a now rangeover p2, p3,X. *p*-i. to be therequiredrepresentation. Since Q(Q) has beenrecognizedas irreducible, Q(W) is, a fortiori,irreduciblein the modular sense. Consider now all systems of n -

n - 1) for which (32)

2 rational integers a(pi) (i = 2, 3,

IfR(P)a(P)

(%

...

1.

modular vectors (a(p2), a(p3), It followsfrom(31) that the corresponding a(pn..i)) forman invariantsubspace V of the space of the representationQ. Since Q is irreducible,we must have one of the followingtwo cases CASE I: V = 0. ...

ON SPLITrING

FIELDS

OF SlMPLE

ALGEBRAS

89

II: V is the full representationspace. In the case I, onlytrivialrelations(32) existand thisimpliesthat [Q:N(K)] = nn-2, because (30) implies that the degree cannot be larger than nn-2. Then Theorem6 showsthatCo/l has the ordern'-2, thatis, to has the ordern'l. It then followsthat to consistsof all unimodulardiagonal matricesof degreen in which the coefficients are nth roots of unity. It follows furtherthat ! consistsof all monomialunimodulartransformations such that the coefficients are nth roots of unityand that the corresponding permutationgroupis W. It willbe shownlater that Case II is impossiblefor A $ F. Then we shall have proved THEOREM 9: Assume thatF containsthenth rootsof unityfor an odd prime numbern and thatK is an extensionfieldof degreen whoseGalois group (interpretedas a permutation groupofdegreen) is at leastthreetimestransitive.Every centraldivisionalgebraA $ F withthesplitting fieldK is associatedwiththefull are nth rootsofunity,whichhas ?l as monomialgroup p, in whichthecoefficients its corresponding permutation group,and in whichthedeterminants are 4 1. In our presentcase, we have to investigateonly the one definitegroup .& in Theorems6, 7, and 8. K will be a splittingfieldof non-trivialA, if and onlyif the embeddingproblemof Theorem6 has a solutionQ forwhichthe embedding problemof Theorem8 has no solution. It remainsto discard the Case II. Here any systemof rationalintegersis possiblein (32), and as a consequence,all ~(p) lie in N(K). This implies CASE

(33)

Q = N(K),

(5 = W1.

The factorset c thenis associatedin N(K) witha factorset C whichconsistsof We want to show that C can be chosen in such a manner that (4) holds forC. If f3is an elementof fi, (11) shows that

nth roots of unity.

r (0:1),

=

(a)C(a

A1)-l

Replacing a by ad, we have a formula (a)d = (0) .X

(34)

whereX is an nth root of unity. Take a = p. and choose f3as an elementw of (35)

U =

e3

n

-1tp.

Then ad = prwhas the formSp. with T in Z. and (34) reads (36)

(;(Pn)

=

(37)

1; that is

(Pn)

;(pn)

t(Pn)X.

The order of w is certainlyprime to n.

readily obtain X =

Hence P(af) = r(,rp.) =

= (n)=

Applyingw repeatedly in (36), we

90

RICHARD BRAUER

If a, ( now are any two elementsof Wwhichdo not belongto the same residue class mod 0, we can findan elementa of 2[ such that 50k = 5B3,Opna = 58a. This is a simpleconsequenceof the factthat 2[ is doubly (even triply)transitive. If a' is anotherelementwiththe same property,we have a' = wAwithX in (35); and because of (37), the quantityt(pn)U dependsonly on a, (3and we may write k(a, A)

(38)

=

If a, #belong to the same residue class mod fi, set k(a, () = 1.

(38*)

Then k(a, () satisfiesthe conditions(3) and (6). = t(pn) and hence k(pn X1)n

In particular,we have Jc(p.,1)

= t(Pn)n = k(pnX 1).

Applying all a of 9 we derive

k(a,,3)n = k(a, ()

(39)

forall a, ( whichdo not belong to the same residueclass (mod 3). If a, ( lie in the same residue class, (39) followsfrom(38*) and (3). The factorset C*(a, (, 7) = c(a, (, Y)k(a, 7)/k(a, )k(3, -y) is associated to c and satisfiesthe condition(4). Because of (39) and (5) (with 1 = n), the nth power of c*(a, (, -y)is 1 and c* consistsof nth roots of unity. we may assume that c itselfconsistsof nth roots of unity, Withoutrestriction, and we may stillassumethat (23) holds. Then, by (24), forall a, 0, -yin W = 0 c(a, (3,BY)= c(3, a, Y)1-

(40)

residueclasses mod Q3. Since 2fis threetimestransiTake a, (,,y fromdifferent tive, thereexistsa a in 2[ such that Zacr = 2(3,B,

Since c(a,

(3, y)

A38

=

3a2,

Z37a

=

37y

lies in F, (lb) and (4) yield

(41)

c(a, (3,y) = c((, a1,7).

Comparing(40) and (41), we see that c(a, 3,y) = 41, and since c(a, (3,7) is an nthroot of unitywithn odd, we must have c(a, (, y) = 1. From (la) and (23), it followsthat the same equation is true,if a, (,2y do not lie in different residueclasses. It is then shown that c is the unit systemand hence A = F. Consequently,Case II cannot appear for non-trivialA, and Theorem 9 is proved completely. UNIVERSITY OF TORONTO.

Representations of Groups and Rings Richard Brauer ColloqUium Lectures: September, 1948

Most abstract concepts of mathematics arise from some concrete notion; we collect the most significant features and postulate them as axioms. Thus a study of the one-to-one transformations of a system on 1tself leads to the group concept. A study of the linear transformations of a vector space, or more generally of the endomorphisms of an abelian group, leads to the concept of a ring. Once the abstract concept has been obtained, we may sever all connections with the original notion and study the systems satisfying our axioms purely on the strength of these axioms. On the other hand, we can ask whether there exist concrete systems of the original kind which represent our abstract system faithfully, and then proceed to study this system by means of such representations. In these lectures, we shall be interested in the linear representations of rings and groups, primarily from an algebraic and arithmetical point of view.

1 Fundamental Definitions Let A be a system of elements for which an "addition" and a "multiplication" are defined within A. We have a representation M of A by linear transformations of a vector space V over a given field K if to every element a of A there corresponds a linear transformation M(a) of V such that M(a

+ ß) = M(a) + M(ß),

M(aß)

= M(a)M(ß),

(a,

ßE

A).

The dimension m of the representation space V is called the degree, m = deg M. If coordinates are introduced in V, every M(a) is described by a matrix M(a) of degree m. (We could have taken V more generally as an arbitrary abelian group and the M(a) as endomorphism of V. However, for the sake of simplicity, we shall take V as a vector space of a finite number of dimensions.) If the linear transformation M(a) carries v E V into v' = vM(a) we set v' = va. Then V becomes an A-module: (v 1

(v,

Vi,

+ v 2)a = v 1a + v 2 a;

v 2 E V, c E K, a,

= c(va); v(aß) = (va)ß (cv)a

v(a

+ ß) =

ß E A). Conversely, every A-module

sentation M, with M(a) defined as v - va.

285

[56]

va

+ vß;

V defines a repre-

Representation spaces are additive groups with operators. When we apply the group-theoretical terms subgroup or homomorphism they are to mean admissible subgroup = submodule, operator-homomorphism, etc. (a) Similarity If two representation spaces V and V0 of A are (operator)-isomorphic, the corresponding representations M and M 0 are similar; M - M 0 • If suitable coordinate systems are introduced, we have for the matrices M(a) = M0 (a) for all a. Similar representations are considered as not essentially different. If arbitrary coordinate systems are used, M(a) = p- 1 M 0 (a)P with a fixed nonsingular matrix P. (b) Reducibility An invariant subspace V* of V is an admissible subgroup. Then both V* and the residue dass space V/V* are representation spaces of A. If a suitable coordinate system is used, the matrices M(a) have the form

(1. 1)

M(a)

M*(a) = (

*

where M* is the representation in V* and M** that in V/V*. Then M is reducible and M*, M** are constituents of M (provided that V* =/= (0), V). The representations M*, M** may themselves be reducible. If we break them up in a corresponding manner and continue, we finally obtain a splitting of M into irreducible constituents Mi, Mz, ... , Mr, This corresponds to the forming of a composition series V = Vr ::J Vr-l ::J · · · ::J V0 = (0) of V with M; belonging to V;/V;_ 1, The Jordan-Hölder theorem yields at once the uniqueness of the irreducible constituents M; of a given representation. (c) Decomposition If V is the direct sum of two invariant subspaces V = V* + V**, we say that M is the direct sum of the corresponding representations M* and M**: M = M* --i-- M**. If suitable coordinate systems are used,

(1.2)

M(a)

=

(OM*(a)

0 ) M**(a) .

In this case, M is decomposable (provided V*, V** =/= (0)). If M*, M** are decomposable, we treat them in the same fashion, and finally arrive at the decomposition of M into indecomposable parts. This corresponds to a decomposition of V into a direct sum of indecomposable invariant subspaces. The corresponding group-theoretical uniqueness theorem shows the uniqueness of the indecomposable parts of M.

286

THEORY OF ALGEBRAS

(d) Schur's Lemma If V and W are two representation spaces for A, the operator-homomorphisms P of V into W form a K-module Hom(V, W). The kernel V0 of P is an invariant subspace of V, the image VP = W* of V is an invariant subspace of W, and V/V 0 = W*. For P -4= 0, it follows that the representations M in V and N in W have a common constituent. In particular, if M and N are irreducible and Horn (V, W) -4= (0), then M - N, and every P E Hom(V, W) is either O or nonsingular. In general, we denote the rank of Hom(V, W)as the intertwining number I(M, N) of M and N. (e) The Commuting Algebra of a Representation In the case V= W, the Kmodule Hom(V, V) becomes an algebra over K, the commuting algebra C(M) of the representation 'M in V. If M is irreducible, Schur's lemma shows that C(M) is a division algebra. If M is indecomposable, Fitting's lemma shows that every element P of C(M) is either nilpotent or nonsingular.

2 The Regular Representation

So far, the question has remained open whether a given system A possesses nontrivial representations. If we have a faithful (i.e., one-to-one) representation M of A, the elements M(a) together with the identity 1 generate an algebra of finite rank n over K and it will not mean an essential restriction to assume that A is itself an algebra over K with a 1-element 1. The original system A can be imbedded in such an algebra and the original representation can be extended to one of the algebra. The same is true if we have only a multiplication but no addition in the original A (with a corresponding change in the definition of representation). We assume then that A is an algebra with a 1-element of rank n over K, so that K may be considered as contained in the center of A. Without essential restriction, we may assume that M(c) = c · 1 if M is a representation of A, c E K. The algebra A is a vector space over K and, moreover, A is an A-module. Hence A itself defines a representation, the regular representation R of A. If g denotes a variable element of A, then R(a) is the transformation g - ta, the right multiplication by a. Clearly, this is a linear transformation in A (taken as a K-space). In the case of the regular representation, the invariant subspaces Z are the right ideals of A. Of particular importance are the components of A, i.e., the right ideals Z for which A is the direct sum of Z and another right ideal Z*, A = Z + Z. * If we write 1 accordingly as 1 = e + e*, then e is an idempotent, e 2 = e, and Z = eA. Conversely, if e is an idempotent, eA is a component of A.

287

[56]

In the following lemmas, eA is a component of A, and V, W are arbitrary representation spaces. (2A)

The set Hom(eA, V) consists of all mappings a -

va (a E eA), where

v is an arbitrary fixed vector of V.

(2B) If P E Hom(eA, W), if Q E Hom(V, W) and Q maps Von W, there exists a T E Hom(eA, V) such that P = TQ. (2C) If P E Hom(V, eA) and P maps V on eA, then V is the direct sum V 0 -i- V* of two invariant subspaces. Here, V 0 is the kernel of P and V* is mapped isomorphically on eA by P. Lemma (2A) together with Schur's lemma shows that every irreducible representation M appears as a constituent in R. A reducible representation M need not be a constituent of R but is a constituent of a representation R -i- R -i- · · · -i- R. Another corollary of (2A) is the equation I(R, M)

(2.1)

= degM.

For eA = V = A, lemma (2A) shows that the commuting algebra C(R) of R consists of the left multiplications L(a): f - af These L(a) form a skewrepresentation of A: L(a ±

ß) = L(a) ±

L(ß);

L(aß)

= L(ß)L(a).

The transposes of the matrices of L(a) define a representation S(a) of A, the second regular representation.

3 The Main Properties of the Representations The indecomposable parts of the regular representation will be called the principal indecomposable representations. (3A)

The number k of distinct principal indecomposable representations U i, U k of A is equal to the number of distinct irreducible representations of A. The space eµ4 of U 1 has a unique maximal invariant subspace N 1 C eµ4. If F 1 U 2,

••• ,

is the irreducible representation belonging to eµ4/Ni, then Fi, F2, ... , Fk are the distinct irreducible representations of A.

288

THEORY OF ALGEBRAS

(3B) The representation U; appears m R with the multiplicity q; = f;/r;, wheref; = degF; and r; = rankC(F;): (3.1) (3C)

If M (with the space V) is an arbitrary representation, then I(U;,M)

= rankHom(e;A, V)= h;r;,

where h; is the multiplicity of F; as irreducible constitutent of M. The proof of (3A) and (3C) can be based on the lemmas in Section 2 and (3B) is a corollary of (3C). (3D) If n; is the maximum number of linearly independent matrices F;(a), a E A, then q; = f;/r; = n;{f; (generalized Burnside theorem). (3E) If ß 1, ß 2, ... , ßk are k arbitrary elements of A, there exists an element a E A such that F 1 (a) = F 1(ß1), F2(a) = F2(ß2), ... , Fk(a) = Fk(ßk). 4 Connection with the Structure Theory of Algebras and Rings

The radical N of an algebra A can be defined as the set of all a E A represented by O by all irreducible representations F;. Then N is an ideal of A. An alternative definition is as follows. Write A as a direct sum of indecomposable components e;A and determine the maximal right ideal N; C e;A; then N = LN;. lt is easy to see that N is the maximal nilpotent right ideal of A. The algebra A is semisimple if N = 0. In the general case, A/N is semisimple. If A is semisimple, all N; = 0 and (3A) shows that the principal indecomposable representation U; coincides with the irrreducible representation F;. Now (2C) yields (4A) Every representation of a semisimple algebra is completely reducible, i.e., a direct sum of irreducible representations. Conversely, if A has a faithful completely reducible representation, then A is semisimple. We can use (3.1) in the semisimple case to find the commuting ring C(R). lt follows that C(R) is a direct sum of k algebras B; where B; = C(q; X F;). Then B; consists of all matrices of degree q; with coefficients in C(F;), B; = [C(F;)]q;, and by Schur's lemma C(F;) is a division ring. The algebra C(R) is anti-isomorphic to A. We thus obtain Wedderburn's theorems:

289

[56]

(4B) Every semisimple algebra is a direct sum of simple algebtas. Every simple algebra is a complete matric algebra over a division algebra. This shows the true significance of the quantities introduced in Section 3. (4C) If Ai is the algebra of all Fi(a) (a E A, i fixed), then Ai is a simple algebra homomorphic to A. (This A 1 is a complete matric algebra of degree qi over a division algebra D 1 which is anti-isomorphic to C(Fi), The rank of Ai is ni; the rank of Di is ri. We treated here only algebras. lt should be mentioned that rings with minimum conditions for right ideals can be treated by the same method, if the notion of a representation is generalized suitably (see Section 1). On the other hand, one may first build up the theory of rings (for instance, following Jacobson) and then derive the results in Section 3 (introducing the necessary restrictive assumptions for each statement).

5 Representation in Extension Fields If A is an algebra over the field K, and if O is an extension field of K, we can consider representations M of A in 0, i.e., in vector spaces over O_;, Al'\ remarked above, M can then be extended to a representation of an algebra over 0, and the previous results can be applied. We remark that if two representations of A in K are not similar, they remain nonsimilar in any ext~nsion field; if two representations of A have no common irreducible constituent, the same holds after an extension of the field. An irreducible representation of A in K may become reducible in extension fields O; the irreducible representations of A in 0 are obtained by breaking up the irreducible representations of A in K. If a representation M of A remains irreducible in every extension field of the underlying field, then M is absolutely irreducible. (3D) yields

(5A) A representation F of degree f is absolutely irreducible if and only if there exist.f2 linearly independent matrices F(a). An absolute splitting field is an extension field O such that every irreducible representation of A in O is absolutely irreducible. (5B) If 0 0 is an extension field of K, there exist absolute splitting fields of A of finite degree over 0 0 •

290

THEORY OF ALGEBRAS

The behavior of irreducible representations in extension fields is dosely related to the theory of simple algebras (see A. A. Albert, Structure of Algebras). 6 The Commutative Case A number of simplifications occur if A is commutative. We note the following: (6A) All irreducible constituents of an indecomposable representation of a commutative A are equal. (6B) The numbers q; are equal to 1, i.e., n; = f; = r;. All absolutely irreducible representations are of degree 1. If !1 is an absolute splitting field, an absolutely irreducible representation F of A is a ring homomorphism of the algebra A into !1 which leaves the elements of K fixed. Let B be a subalgebra of A. If R is the regular representation of A, the elements R(ß), ß E B, form a representation of B which contains the regular representation of B as constituent. This leads at once to (6C) If !1 is an absolute splitting field of the commutative algebra A, then !1 is an absolute splitting field for every subalgebra B and every irreducible representation of B in !1 can be extended to a representation of A. This is a basic result in commutative algebra. 7 Applications In order to illustrate the preceding results, we discuss briefly some well-known theories from our point of view. (a) Normal Form of a Matrix (See MacDuffee's book, Vectors and Matrices.) Given a linear transformation M 0 in the vector space V over the field K, the problem is to find a coordinate system such that the matrix M0 of M 0 has a specially simple form. Let g(x) = 0 be the minimal equation of M 0 • The polynomials of M 0 form an algebra A isomorphic to the residue dass algebra K[x]/(g(x)). If a 0 denotes the residue dass of x, then h(a 0 ) - h(M 0 ) (h(x) E K[x]) defines a faithful representation M of A. Without essential restriction, it can be assumed that M O is indecomposable. Then g(x) is a power of an irreducible polynomial. If W is the space of a faithful representation Z of A, it follows easily that W contains an

291

[56]

invariant subspace W* such that the representation m W* is similar to the regular representation R of A. Using the commutativity of A and duality we derive that W also contains an invariant subspace W** such that the representation in W/W** is similar to R. Then (2C) shows that R is a component of Z. In particular, if Z = M, we obtain M - R because of the indecomposability of M. Hence M 0 - R(a 0 ). In order to obtain a specially simple form of the matrix M0 , we only have to choose a specially simple basis of A and write down the matrix of R(a0 ) with regard to this basis. (b) Structure of Fields If A is taken as an extension field of degree n of K, the general theorems yield easily the basic facts of field theory. We illustrate this by some remarks. If O is an absolute splitting field, every irreducible representation F; of Ais a K-isomorphism of A into 0. The number k of distinct F;, then, is the number of conjugates of A, i.e., the degree of separability of A over k. In the field A itself, we have the identical representation a - a of A. lf U is the corresponding principal indecomposable representation, the principal indecomposable representation U; of A in O is obtained from U by application of the isomorphism F; to the coefficients. All U; have the same degree and this degree is the degree of inseparability of A over K. In particular, A is separable over K if and only if deg U = I, i.e., if the extension AA of A is semisimple. If, in the inseparable case, 0 U(a) - ( :

d(a)

then d(a) is a derivation of A with regard to K, and every such derivation can appear here. The minimal absolute splitting fields are the Galois fields of K; a separable field A is normal if it is its own absolute splitting field. (c) Galois Theory Assume that the field A is normal and separable over K. Then A possesses n irreducible representations F; in A. Each F; is a K-automorphism; the n automorphisms F; form the Galois group G of A. Considering A as a vector space over K, we have two types of linear transformations: the

right multiplications R(a), g - (a (a E A); and the automorphisms F, g - F(fl (F E G). The transformations generate a simple algebra B of rank n 2 • If H is a subgroup of G, the F EH and the R(a) generate a simple subalgebra B(H). The commuting algebra C(B(H)) consists of all R(ß) with ß in the subfieldJ(H) of A left invariant by the F EH. Application of theorems of ring theory easily yields the fundamental theorem of Galois theory. Actually, it is sufficient to

292

THEORY OF ALGEBRAS

apply Burnside's theorem (3D), which shows that the degree of J(H) over K equals the index of H in G. (See also (6C).) These considerations will show that the modern "noncommutative" algebra embraces the older algebraic theories. This indicates the possibility of generalizations of the classical theories, for instance, of an extension of Galois theory to noncommutative domains, as given recently by several writers. On the other hand, an analysis of the proofs leads to important new concepts, for instance, types of algebras such as Frobenius algebras and uniserial algebras. To mention another example: the F in Section 7(c) form a linear representation of the Galois group G. If instead we use a representation of G by collineations, i.e., replace F by a linear transformation P(F) such that P(F)P(F') and P(FF') may differ by a scalar factor, we are led to the idea of a crossed product, qf fundamental importance for the theory of simple algebras. 8 Group Algehras Belonging to Groups of Finite Order

A group algebra f is an algebra whose n basis elements gi, g 2, ••• , gn form a group G under multiplication. To every given group G of finite order, and to every given field K, there exists a unique such group algebra consisting of all L a;g;, a; E K. lt is the same problem to find the representations of G and those of f. The representations of groups can therefore be treated as application of the theory of representation of algebras. (8A)

If K has characteristic 0, f is semisimple.

We shall need the following facts concerning the center A of f: (8B) IfCi, C2, ... , C 1 are the classes ofconjugate elements in G, a basis ofthe center A consists of the l elements (C;) where (C;) is the sum of the elements in the dass C;. The constants of multiplication aum of A are rational integers

~

0, defined by

(8.1) m

They are completely determined by the structure of G. 9 Representations of Groups in the Field of Complex Numbers

The classical theory of group representations deals with the representations of a group G of finite order n in the field K of complex numbers, more

293

[56]

generally, in an algebraically closed field of characteristic 0. We only state the basic facts: (9A)

The representations of G are completely reducible.

(9B) The number of distinct irreducible representations of G is equal to the number l of classes of conjugate elements in G. The character x(g) of a representation M(g) of G is the trace of the matrix M(g), g ranging over G. The character x(g) is a class function, i.e., its value depends only on the dass Ci to which g belongs. Since x(g) is the sum of the characteristic roots of M(g), it is a sum of tth roots of unity, t being the order of g. Also, x(g- 1 ) = x(g). (9C) Two representations of G are similar if and only if they have the same character. The product of two characters x(g) and x*(g) is again a character. We shall denote the l irreducible representations of G by Xi, X2, ... , X 1 and the corresponding irreducible characters by Xi, X2 , ••• , Xi· We have the orthogonality relations

o(g)a~~(g- 1 )

= (g/xA)BAµ,8;,ßJm

o•G

Every irreducible representation XA of G corresponds to a representation wA of degree one of the center A of r with (9.2)

where g; is an arbitrary element of C;. In the notation (8.1), (9.3)

wA(C;)wA(C;)

=

L

aumWA(Cm)

m

294

THEORY OF ALGEBRAS

As a consequence, the w»(C;) and the X>.(g;) are completely determined by the aum· The w»(C;) are algebraic integers. (9E)

The degree x» of the irreducible representation X» divides n.

We took K as an algebraically closed field. However, we can now replace K by an absolute splitting field of the group algebra of G over the field of rational numbers. In particular, we may choose K as an algebraic number field.

10 Arithmetical Questions If an arbitrary subring o of a field K has been chosen as the ring of "integers" of K, the group ring Ü of a group G with regard to o can be formed. lt consists of the elements 2.aigi with ai E o. We shall assume that 1 E o and that K is the quotient field of o. lt will be sufficient for our purposes to consider the case of a principal ideal ring o. If A is an arbitrary algebra of rank n over K, an order or domain of integers Ü of A with regard to o is a subring () of A which contains o and is a finite o-module. lt is no essential restriction to assume that Ü generates the algebra A. Then () has a o-basis consisting of n elements. If we write the regular representation R in matrix form, choosing an arbitrary coordinate system, the elements a represented by integral matrices R(a) will form an order; and every order can be obtained in this manner. Here, an integral matrix is a matrix with coefficients in o. We shall now think of representations M as given in matrix form assuming that in each case a definite coordinate system has been chosen. Let o be a fixed order of A. A representation M of A is integral (for Ü) if M(a) is an integral matrix for all a E Ü. (IOA)

Every representation of A is similar to an integral representation.

Two similar representations M 1 and M 2 are integrally equivalent if there exists an integral matrix P, whose determinant is a unit of o, such that p- 1M 1 (a)P M 2 (a) for all a.

r

(IOB) If o is the ring of rational integers, and is a group algebra, every dass of similar integral representations breaks up into a finite number of subclasses of integrally equivalent representations. Generalizations of this theorem of C. Jordan have been given by Zassenhaus. One of the main arithmetical problems is that of the behavior of a prime Po

295

[56]

of O in 0. This question is dosely connected with a study of the residue dass ring 0/p 0 0 = A *. If a* denotes the residue dass (modp 0 0) of a E Ü, the c* with c in o form a field K which may be identified with o/p 0 o. Then A * is an algebra of rank n over K*. For the investigation of A *, the representations of A * in K* can be used. Every integral repre_sentation M of O yields a representation M* of A * in K* if every coefficient a is replaced by the corresponding a*. In general, not every representation of A * can be obtained in this manner. The connection between the arithmetical questions and the K*-representations is given by (lOC) If A * has k distinct irreducible representations F 1, F2, ... , F k in K*, then Po has k prime ideal divisors ~1, ~ 2 , ••• , ~k in Ü. The residue dass algebra 0/~i is isomorphic to the algebra of all Ft(a*), a E 0, and ~i consists of the a E O with Fi(a*) = 0. If M 1 and M 2 are two similar integral representations which are not integrally equivalent, the corresponding representations M 1 and M 2 need not be similar. However:

( 1OD) If two integral representations M 1 and M 2 are similar, the irreducible constituents of Mf and M; are the same. This result enables us to speak of the irreducible K*-constituents of a Krepresentation of A.

11 The Case of a Complete Field We shall assume now that the field K in Section 10 is complete with regard to the valuation defined by the prime p0 • For the sake of simplicity, it will be assumed that A is semisimple. Then the irreducible representations Xi, X2, ... , X1 of A in K coincide with the principal indecomposable representations of A. On the other hand, the residue dass algebra A * need not be semisimple. Let U 1, U 2, ... , U k be the principal indecomposable representations and let F 1, F 2, ... , F k be the irreducible representations of A * in K*. U sing the completeness of K, we can show (llA) There exist k integral representations (U 1 ), ..• , (Uk) of A in K such that (U;)* = U;. If R* contains U; with the multiplicity q;, then R 1s integrally equivalent to the direct sum of the k representations q; X (U;).

(llB)

If M is an integral representation of A, then J((U;),M) = J(U;,M*).

296

THEORY OF ALGEBRAS

Applying this to M = X>.., we find (l lC)

If X>.. appears in (U;) with the multiplicity d>..;, and if F; appears in X{

with the multiplicity d>..;, then

where w;

=

rank C(X;), ri

=

rank C(Fi).

Set cu = I(Ui, U;). Then Fi appears in U; with the multiplicity cu/ri. These cu are the Cartan invariants of A *: (11.1)

cu =

L >..

d>..;w>..d>..i = cii·

12 Modular Group Characters

We are now in a position to build up the theory of group representations in an algebraically closed field of characteristic p. Let K 0 be an algebraic number field which is an absolute splitting field for the group algebra of the group G over the field of rational numbers. Let +> be a prime ideal dividing p and let K be the corresponging +J-adic extension field. Then the field K* in Sections 10 and 11 is a finite field of characteristic p, and the representations M* become modular representations of G in a field of characteristic p. If K 0 is extended sufficiently, it can be assumed that K* is an absolute splitting field for the modular representations of G. This implies that not only the w>.. but also the ri in ( 11 C) are equal to 1. Every element g of G can be written uniquely in the form g = bs, where b and s commute and where b is a p-regular element of G (i.e., an element whose order is prime top) while the order of s is a power of p. If T is any representation of G in a field of characteristic p, the traces of T(g) and of T(b) are the same. lt will therefore be sufficient to define the modular group characters for the pregular classes C; of G, i.e., the classes consisting of Psregular elements b. The characteristic roots z; of T(g) are roots of unity, actually in a field of characteristic p. .If now g is p-regular, the exponent of z; is prime top, and we may identify z; with a root of unity (z;) in the complex field. We define the character cp(g) of T as the sum of these complex numbers (z;). The character will not be defined for p-singular elements g. We then have the results: ( 12A) The group algebra of the group G over an arbitrary field of characteristic p is semisimple if and only if p does not divide the order n of G.

297

[56]

(12B) The number k of distinct (absolutely) irreducible modular representations F 1, F 2, ••• , F k of G is equal to the number of classes of p-regular elements in G. (12C) Two modular representations have the same irreducible constituents if and only if they have the same character. Let cf,; be the character of the irreducible representation F; and let ; be the character of the corresponding principal indecomposable representation U;. The orthogonality relations are (l2D)

L

(a)

m(g)cf>r(g')

= ni>mr,

g

where g ranges over all p-regular elements of G, k

(b)

~1

{on/h;

_

>..(g)cp>..(g') =

The connection between ; and

(g, g' in different classes) (g, g' E C;),

'Pi is obtained from

(11.1):

(12.1)

where the cii are the Cartan invariants of the modular group algebra. We call the rational integers d>..; ~ 0 in (llC) the decomposition numbers of G (for p). The ordinary and the modular characters of G are connected by (12E) X>..(g) =

L d>,.;cp;(g)

(g p-regular),

i

(12.2)

;(g)

(12.3)

(12F)

=

L d>..iX>..(g), >..

Ld>..;d>,.j >..

If n

=f=.

= Cij,

0 (mod p), then the ordinary and the mc:>dular characters of G

coincide; dii = 8ii (Kronecker delta). lt remains to give a substitute for (12E) for the case that g is p-singular. Choose a maximal system of elements of G, (12.4)

So

298

= 1, Si,

S2, ... , Sm,

THEORY OF ALGEBRAS

such that the order of S; is a power pa, of p and such that s; and si are not conjugate in G for i fj. Let G; be the centralizer of s; and denote the modular characters of G; by ?>, j = 1, 2, ... , k;. The, total number '2:.k; of sµch characters is equal to the full number l of classes of conjugate elements of G. If b is a pregular element of G;, we have (12.5)

X1..(bs;) =

L

d'!J?(b ),

j

where the generalized decomposition numbers d k.

(2)

Dann kann man Grössen x 1 von

r*

nichttrivial aus

l

l:x1z,(1) = 0,

(3)

(v

= 1, 2, ... , k)

1=1

bestimmen. Da jede Gleichung algebraische Koeffizienten in bezug auf r hat und mit jeder Gleichung auch alle konjugierten vorkommen, so kann man dabei die Zahlen Xi. sogar aus r wählen, das heisst also als ganze rationale nichtnegative Zahlen, die nicht sämtlich (mod p) verschwinden. Dann stellt

x. (R) =

l

l:xi.z(1) (R) 1=1

einen Charakter von G dar, der wegen (3) die im Hilfssatz genannte Voraussetzung erfüllt. Also folgt für alle R x.(R)

=

0.

Dies ist aber nach dem Satz von Frobenius und Schur (1 ) nur möglich, wenn alle x 1 als Elemente von r* verschwind.en, d. h. durch p teilbar sind. Man erhält also einen Widerspruch. Die Anzahl der wesentlich verschiedenen irreduzi• blen Darstellungen kann daher k nicht übertreffen. Wir zeigen jetzt umgekehrt, dass es k wesentlich verschiedene irreduzible Darstellungen gibt. Angenommen, es seien (1) alle zu irreduziblen Darstellungen gehörigen Charaktere und es. sei l < k.

(4)

s.

Sitzungsberichte der Preussischen 209.

(1 )

326

Akademie der Wissenschaft, 1906,

FINITE GROUPS

DARSTELLUNG VON GRUPPEN IN GALOISSCHEN FELDERN

r*

Dann kann man Grössen y, aus so bestimmen, dass k

!P,z.{A)

(5)

7

auf nichttriviale Weise

=0

(i.

= 1, 2, ... , l)

K=i

gilt. Da jeder Charakter von G aus den Charakteren (1) durch lineare Kombination mit ganzen rationalen Koeffizienten entsteht, folgt aus (5), dass für alle Charaktere x von G.gelten muss k

!i,YxZ.x =

(5')

0,

x=i

wo wieder X• den Wert von X für die Elemente der Klasse C. bedeutet. Wir wollen (5') als unmöglich nachweisen und nehmen dazu an, dass die Unmöglichkeit der entsprechenden. Aussag_en für Gruppen kleinerer Ordnung bereits nachgewiesen sei. Es sei Q ein Element von G, dessen Ordnung q zu p teilerfremd ist. Wir betrachten die Gruppe der mit Q vertauschbaren Elemente von G und suchen in ihr eine zu p gehörige Sylowgruppe P, die Ordnung heisse p~. Dann erzeugen Q und P eine Untergruppe H der Ordnung qp.,. von G. Ist U die durch Q erzeugte Untergruppe, so stimmt H mit dem direkten Produkt U X P überein. Es sei

... , ein vollständiges rechtsseitiges Restsystem von G nach H. Ist ~ (H) ein beliebiger Charakter von H, und definiert man für nicht in H enthaltene Elemente S von G

t!i (S)

= 0,

so stellt nach Frobenius (1 ) (6)

.

l:it!i (P.,RP,- = 1)

z. (R),

(R . and these quantities dP„ is given by the formula

which plays a central röle in our work (§4). As an application we obtain formulas which correspond to the fundamental relations for the characters in the ordinary theory (§5). At the same time, we obtain new proofs for some known results, namely, a theorem of Dickson3 concerning the multiplicities of the F>. in the regular representation, a theorem of R. Brauer4 which gives the number of distinct irreducible representations F>., and finally results of Dickson and Speiser6 for the non-singular case g =I"' 0 (mod p). 'The last three sections·of this paper deal with the decomposition of r into a direct sum of invariant subalgebras in connection with the properties of the representations. The basis for our work is formed by the general theory of algebras and their representations. 6 The ordinary theory of group characters is assumed. A number of our results could be generalized for the case of the modular representations of an algebra over an algebraic number field. L. E. Dick_son, Transactions Amer. Math. Soc. 8 (1907), p. 389. R. Brauer, Actualites scientifiques et industr. No. 195 (1935). 6L. E. Dickson, Transactions Amer. Math. Soc. 3 (1902), p. 285; A. Speiser, loc. cit. Cf. also van der Waerden, loc. cit., p. 74. 6 Cf. L. E. Dicksan, Algebras and their arithmetics, Chicago, 1923; M. Deuring, Algebren, Berlin, 1935. Further, see R. Brauer and C. Nesbitt, r,roceedings National Academy of Sciences 23 (1937), p. 236. We refer to this paper as B.N. 3

4

337

REPRINT OF [27]

MODULAR REPRESENTATIONS

5

2. Let K be an algebraically closed field. We consider representations of groups G by linear transformations with coefficients in K. 7 LEMMA : Let A and B be two representations of a group G which associate the matrices AQ and BQ with the element Q of G. lf both AQ and BQ have the same characteristic roots for every Q in G, then A and B have the same irreducible constituents. Prooj: lt follows from our assumption that (1) ..... .

trace (AQ) = trace (BQ)

and that AQ and BQ are of the same degree n. If K has characteristic 0, the lemma follows from a theorem of Frobenius and Schur. 8 We assume, therefore, that K is of characteristic p:j::O. Let F 1 , A, .. , F, be the non-equivalent irreducible constituents of A and B. We denote the multiplicity of Fx in A by ax, the multiplicity of Fx in B by bx. The equation (1) implies, according to the argument of Frobenius and Schur, that

(2) ..... .

ax=bx (mod p)

(but not necessarily a>- =bx). We assume that the lemma is true for representations of smaller degree than n. We replace A and B by the completely reducible representations Ao and Bo with the same irreducible constituents. The assumption of our lemma then is true for the two representations Ao, B 0 and it is sufficient to prove the lemma in this case. We may hence assume without loss of generality that A and B themselves are completely reducible. If A and B have a constituent Fx in common, we leave Fx in A and B away. Thus we obtain two representations A1 and 1?1, of degree smaller than n which also satisfy the assumption of the lemma. Since the lemma then is true for A1 and B1 it also holds for A and B. 7For the sake of simplicity, we exclude representations which contain the 0-representation as constituent (i.e., the representation which associates the rtumber O with every Q of G). The following lemma would hold without this restriction. lt also holds for representations of pseudo-groups G in which the elements do not necessarily possess inverses. 8G. Frobenius-1. Schur, Sitzungsberichte d. Preuss. Akad. d. Wiss. 1906, p. 209.

338

FINITE GROUPS

6

MODULAR REPRESENTATIONS

We have, therefore, only to deal with the case that either a-,.. or We may, of course, assume that not all a-,.. and b-,.. are equal to zero. From (2) it follows that b-,.. vanishes for every A.

a-,..=O, b-,..=O (mod p). Denote by A 2 the completely reducible representation which contains F-,.. with the multiplicity a>./P, A= 1, 2, ... , r. Similarly, denote by B2 the completely reducible representation which contains F-,.. with the multiplicity b-,../p. Then A 2 and B 2 again satisfy the assumption of the lemma and their degree is smaller than n. Since the lemma then is true for A 2 and B 2 , these two representations have the same constituents. Hence a>- =b-,.. and, because either a-,.. =0 or b-,.. =0 for every A, we have a>- =b>- =0, which gives a contradiction. The converse of the lemma is trivial. 3. Let G now be a group of finite order g and p any rational prime. We denote by A, B, ... the representations of G by linear transformations with coefficients in the field of complex numbers. We are going to derive modular representations Ä, B, ... from A,B, .... We may replace A by an equivalent representation whose coefficients lie in an algebraic number field !.1 and are all integers for p. 9 We then take a prime ideal 1,) in Q which divides p. For every integer for p, the residue dass (mod l,l) is defined, and all these residue dasses form a Galois field of characteristic p. We replace now every coefficient in A by its residue dass (mod 1,J). Obviously, we obtain a modular representation Ä of G. By the same process, we may derive a modular representation .B from B, provided that the coefficients in B lie in an algebraic number field !.1 1 and are all integers for p. The coefficients in .B are residue dasses modulo a prime ideal 1,) 1 of !.1' which divides p. We assume now that A and B are equivalent. lt can happen that Ä and B are not equivalent. We shall, however, show that Ä and .B have the same absolutely irreducible constituents if the Prime ideal \.) 1 of p in Q' is properly chosen. 9 By an integer for p we mean a number which can be written as a quotient a/ß of two algebraic integers such that ß is prime top.

339

REPRINT OF [27]

MODULAR REPRESENTATIONS

7

Let 0 0 be a common algebraic extension field of O and O'. We denote by lJo a prime divisor of lJ in 0 0 and we choose the prime ideal ll' in O' so that lJo also divides lJ'. Obviously, we obtain the same modular representations, Ä and B, as before, if we consider A and B as representations with coefficients in 0 0 and take these coefficients (mod lJ 0). We take 0 0 such that it contains the characteristic roots of all elements of A and B. As in the introduction, we denote the Galois field with p elements by II, and its minimal algebraically closed extension field by K. The field of residue classes (mod lJo) is a subfield of K. Two corresponding elements of A and B always have the same characteristic roots, since A and B are equivalent. lt follows at once that corresponding elements of Ä and B have the same characteristic roots. Then the Iemma of §2 shows that the two modular representations Ä and B have the same absolutely irreducible consti tuen ts. Let k* be the number of classes of conjugate elements in G. We denote the ordinary absolutely irreducible representations of G by F1*, F2*, ... , FZ* and their degrees by Ji*,f2*, ... ,h*· We choose the F-1,. * so that they have coefficients which lie in some algebraic number field O and are integers for p. We may assume that O contains the g-th roots of unity. For lJ we take a fixed prime divisor of p in 0. The modular representations F'1*, F2*, ... , 'PZ* break up into irreducible constituents. Let Pi, F2, ... , Fk be the absolutely irreducible modular representations of G, and d.1,. be the multiplicity of Fx in F.*. We call these d.1,. the decomposition numbers of G for p. Then (3) ...... . D= (d.x) is a matrix with k* rows and k columns. The coefficients are nonnegative rational integers. We may express the splitting of F. * in the form k

F.*+-+ ~ d.1,.Fx. 10

(4) ..... .

X=l

The relation A+-+B is to express that two representations A and B have the same absolutely irreducible constituents. The sum on the right-hand side of (4) denotes a representation which contains Fx exactly d.x times as irreducible constituent. 10

340

FINITE GROUPS

8

MODULAR REPRESENTATIONS

We form the ordinary representation Roof G which contains F. * exactly J. * times as irreducible constituent. Then R 0 is equivalent to the regular representation R of G. We now consider R. 0 and R.. The representation R is the regular representation for the modular theory. Let u-,.. be the multiplicity of F-,.. in R.. The multiplicity of k*

F">- in R..o is, evidently, ~ f.*d•">-· Thus we obtain k*

u">- = ~ J. *d•">-·

(5) ..... .

•=l

In particular, there exists at least one K for a given X such that d.">- =l=O. Every irreducible modular representation can be obtained from splitting the irreducible ordinary representations after taking them (mod µ). On comparing the degrees in (4), we find k

J. * =

(6) ..... .

~

">..=1

d.">.A,

when A denotes the degree of F">- . We had to choose a prime factor µ of p for our construction of the modular representation Ä by means of A. If we take another prime factor \)1 of p, then in general we obtain another modular representation Ä 1, instead of Ä, which is not necessarily equivalent to Ä. However, it can be easily shown that we may replace A by a conjugate representation A .

with

4>.» = rr (ui•) +vl))

(12) ..... .

a, {J

where 11

u~-t•d,,.x. K,

X=>l

Comparing this with (11), we find 12To be exact, the uf'l, vf;"l do not lie in Q but in some extension field TI: Similarly, the ff> lie in some extension field K of K. The domain of integers for p in Q is mapped homomorphically on K. We may extend this homomorphism to a homomorphism of a domain oi._ integrity I in Q upon a part of K. Here I consists of those quantities of Q which satisfy algebraic equations

u~), v

where the A,,. are polynomials in Xi, • • • Xg, Y1, •.• , Yg with coefficients which are integers for p in n. The uf'>, vf;") lie in 11 We may, therefore, speak of the (},) (>.). element of K to wh1ch Ua or VfJ 1s "congruent".

343

REPRINT OF [27]

MODULAR REPRESENTATIONS

11

k* k

II

1: dµ,,. ,. i,=l

•• }..=l

Since the 4,,>,. are distinct irreducible polynomials, we obtain (9). The formula (10) shows at once that Cis symmetric. Further, the quadratic form belonging to C,

..

~ C,>,.X/C>,. }..

is not negative. Since det Cis =l=O, as we shall see in §5, the form is positive definite. 5. We denote the k* classes of conjugate elements in G by K 1, K2 .•. , Kk* in such a way that the orders of the elements in the first h classes are prime top, whereas they are divisible by p in the last k*-h classes. We shall prove later that h is equal to the number k of distinct irreducible modular representations. We set g=Pa·g', (g', p) =l. Let Q be an element in one of the last k*-h classes. We may write it as a product of two commutative elements P and M,

Q=P·M where the order of P is a power of p, and the order of M prime to p. Let A be any modular representation of G. We have Aa=Ap·AM.

The characteristic roots of Ap are pa-th roots of unity and since our field is of characteristic p, they are all equal to 1. The characteristic roots of A 0 are obtained by multiplying each characteristic root of AM by one of the characteristic roots of Ap. Accordingly, we see that A 0 and AM have the same characteristic roots. An immediate consequence is (15) ..... . Here, of course, M lies in one of the first h classes of conjugate elements. Combining these facts with the results of §2, we obtain LEMMA: If two modular representations A and B of G represent an element M of an order prime to p by matrices AM and A 0 , both of

344

FINITE GROUPS

12

MODULAR REPRESENTATIONS

which have the same characteristic roots (Jor every such M in G), then A and B have the same irreducible constituents. Jf both A and B are irreducible and tr (AM) =tr (BM) f or every M of an order Prime to p, then A and B are equivalent. In the table of modular characters of the group we may omit the last k*-h classes, since the values of every character, according to (15) is known, if we know the value for the first h classes. The values of a character are, of course, defined as numbers of the field K of characteristic p. lt is, however, advisable to change the point of view. Every value of a character tr(AM), M of an order m prime top, is a sum of m-th roots of unity. Set g' =g/pa. The g'-th roots of unity in the algebraic number field forma cyclic group Hoforder g'. No two of them are congruent mod p, since the difference of any two distinct ones is a divisor of g'. If we replace each of these roots by its residue dass (mod p), we-obtain an isomorphic mapping of H upon the group of g'-th roots of unity in K. We replace now every modular root of unityin tr(AM) by the corresponding complex root of unity. Then tr(AM) becomes a complex number. If this number is known for every M of an order prime top, we know tr(A1), that is, the sum of the r-th powers of the characteristic roots of AM. Since we are now in the field of complex numbers, we may find the characteristic roots of AM, In what is to follow, we always shall understand the modular characters in this sense. We do not here define the values of such a character for elements whose order is divisible by p. We have now

Theorem II: Jf two modular representations A and B have the same character (Jor elements of an order prime to p) then they contain the same irreducible constituents. We denote the character of the irreducible modular representation F>. by x-l, and the value of x.) for the dass KP of conjugate elements by x!>-l (p = 1, 2, ... , h). Similarly, let x*.. elements in the dass Kx of conjugate elements in G. These Z>.. constitute a basis of the centre Z of r. The algebra Z is the direct sum of its radical and a semisimple algebra Z 0 • According to the general theory of algebras, Zo 15 Cf.

the Jiterature given in note 5.

349

REPRINT OF [27]

MODULAR REPRESENTATIONS

17

is the direct sum of t invariant subalgebras which are fields. Let 7/1, 7/2, ••• , 7/t be the unit elements of these fields. We then have for T=l, 2, ... , t. =11.,.r11.,. The number t and the elements 7/.,. can be found if the multiplication quantities of the z.,. are known. These depend on the multiplication of the classes of conjugate elements in G. We discuss the representation of r now from the point of view of the theory of representations. We say that two indecomposable parts UK and U..,.. of R belong to the same block if there is a sequence of up of the form ~. .

(32)°. .....

UK,

u.,, ..... ' U.,., u..,.

such that any two neighbouring UP in (32) have at lea-st one irreducible constituent in common. The k representations U1, U2, ... , Uk are then distributed into a number of blocks B1, B2, ... , Bn such that each BK consists of a number r. of U.,. We write R so that it is decomposed into its indecomposable parts and that each indecomposable part is split into irreducible constituents. In W..,.. we gather all the indecomposable parts which belong to the block B..,...

(33) ..... .

We take one of the elements 7/ = 7/o and denote the matrices representing 7/ in F«, U..,.., W, by F';, ~. w?. F~ is commutative with all the elements of F«. According to Schur's lemma, it is a multiple j«E of the unit matrix. Since 7/ is idempotent, we have either i«=l or jK=O. Further, V~ is commutative with every element of U..,... Since U..,.. is indecomposable, U~ has only one characteristic root. 16 If U..,.. contains a F. withj. = 1, this root must be 1. In the other case, it is equal to 0. Each U..,.. contains either only F.'s withj.=l, or only F.'s withj«=O. In the first case, all the UP belonging to the same block also contain only F/s with j. = 1.R. Brauer-[. Schur, Sitzungsberichte d. Preuss.Akad. d. Wiss., 1930, p. 209, §8.

16

350

FINITE GROUPS

18

MODULAR REPRESF;NTATIONS

We see that 'W? has O above its main diagonal, and either only 1 or only O in its main diagonal. This is true for 'IJ = 111, 112, ••• , 11,. But no linear combination of these 11T can have O in the whole main diagonal in (33) and above the main diagonal also. This shows that t is at most equal to the number n of the W•.

t ~ n.

(34) ..... .

Let H be a matrix commutative with all the elements of R. The matrix H' then belongs to the second regular representation. Since WP and W„ have no irreducible constituent in common, Schur's lemma shows that H breaks up in the same manner as R in (33)

(35) ..... .

If we take for H, a multiple h,E. of the unit matrix E, of the degree in question, then this particular H is commutative with all the elements of Rand all the elements of S'. This latter fact shows that our H belongs to R and, therefore, to the centre of R. The element 'Y of r, which is represented by the matrix Hof R, then lies in Z. All the elements 'Y form a semi-simple subalgebra of order n of Z. This gives n ~ t, and in connection with (34) requires n=t.

(36) ..... .

We may take for Z 0 the algebra of the quantities 'Y· The elements 11T then are obtained by choosing hT=l and hP=O for p:j:::r. From (33) and (35) it follows that the elements of 11Tr11T are represented by O in Wp, p:j:::1. We then have be the totality of elements of r which are represented by the 0-matrix in every UP which does not belang to the block BT. Then ~T is an invariant subalgebra of r which cannot be represented as a direct sum oj invariant subalgebras. M oreover r is equal to the direct Si[m of ~1, ~2, ••• , ~t· We take the F„ for which the U„ belong to a block BT. We also say that these F„ belong to the block BT. We numerate the F„ in

Theorem VI: Let

~T

351

REPRINT OF [27]

MODULAR REPRESENTATIONS

19

such a manner that we first take the r 1 representations Fµ of B 1, then the r2 representations Fµ of B2, etc. Because of the definition of the blocks and (7), C has the form

C

(37) ..... .

where CT is a square rnatrix of degree rn corresponding to the block BT. The matrix CT does not split further into two parts. Two representations F-i:. and Fµ belong to the same block, if, and only if, they represent the centre in the same manner: Denote by w?> E the matrix representing ZP in F}... Then the condition becomes (p=l, 2, ... 'k*). 17

(38) ..... .

r formed with regard to the field of all complex numbers. Here ZP is represented in F: by a multiple w;.. contains elements of an order prime to p, and the second, elements of an order divisible by P, then the (ordinary) characters of G satisfy the relations (47) ..... . p

where p runs over all values f or which F; belongs to a given block B;. In particular, .t* *(p) = O. ~ JpX>.. p

354

FINITE GROUPS

290

MA THEMA TICS: R. BRA UER

PROC.N. A. S.

theorem cannot be proved in the present situation, for if II1and 12 are admissible sets of ideals, II U IH2and II1 II2are not in general admissible. 2. We shall assume in what follows that K k is a normal field whose group is solvable. Then K = Kg has a chain of subfields over k: Kt -2 = . . . D K :D k, which has the property that K,IKI,Kt D Kt-I is abelian and that Ki is a maximal abelian subfield of Kt]Ki-1. We shall say that H is immersed in an ideal group H in k if II is a subset of II(H), the set of prime ideals occurring in H. By means of the abelian

class field theory the following theorem can be proved: THEOREM (Existence Theorem). Let II be an admissible set of ideals in k. Then II determinesa solvableclass field Klk if and only if II is immersiblein an ideal group H in k which has no proper ideal subgroup. An analogue can also be obtained of the Artin law of reciprocity in alge-

braic number fields. This generalized law yields a one-one multiplicative correspondence between the sets of conjugate automorphisms of Klk and certain unordered vectors r whose components are ordered vectors of elements of the class groups defining the Ki Ki-1. Incidentally, a non-abelian definition is given for multiplication of ideals in algebraic number fields.

ON THE REPRESENTATION

OF GROUPS OF FINITE ORDER

BY RICHARD BRAUER UNIVERSITY

OF TORONTO

Communicated April 17, 1939

1. Introduction.-The modular representations of a group ( of finite order have been studied by C. Nesbitt and the author in a joint paper' (cf. ?2). These investigations will be continued in ?3 of this note. The results enable us to derive a number of new properties of the ordinary group characters of ? in the case that g is of the form g = pg' where p is an odd prime which does not divide g' (?4). I mention here some applications of this theory. I. Let Z be an irreducible group of linear transformations of degree n which has no normal subgroup of order p. If the order of T is divisible by the prime p to the first power only, then p < 2n + 1. This improves for the case g 9 0 (mod p2) a theorem of H. F. Blichfeldt2 who proved p < (2n + l)(n - 1). For p = 2n + 1 we can show that Z, considered as collineation group, is isomorphic with LF(2, p).

II. If a group of order g = p. q. rm, (p, q, r distinct primes) is simple, then g = 60 or g = 168. Using different methods, Burnside has treated

such groups for m = 4 and odd g, and W. K. Turkin3 for any odd g.

MA THEMATICS:

VOL. 25, 1939

R. BRAUER

291

The order of a transitive permutation group @ of degree p is of the form g = qp(l + np) where q divides p - 1. These groups are the subject of a number of papers by Mathieu, Jordan, Sylow, Frobenius, Burnside, Miller and others.4 It is possible to restrict oneself to the case that ? is simple. The elements of order p commute only with their own powers. We can prove III. Let ? be a simple group of order g = qp(l + np) where qlp - 1, in which the elements of order p commute only with their own powers. If n = (2p + 7)/3 then either (1) ? is cyclic, or (2) ? - LF(2, p), or (3) p is a prime of theform 2h + 1, and ? - LF(2, 2h). The degrees of the irreducible representations for which f 0 0 (mod p) can be given explicitly for many larger values of n.

The group LF (3, 3),

p = 13 is an example for the case n = (2p + 7)/3. For the proof we need not assume that ? is a permutation group of degree p. There are numerous groups for which our method allows us to find the degrees of the irreducible representations. 2.5 Let ? be a group of order g = pagI where the prime p does not divide g'.

1i, Z2, * . ., 2I be the distinct (= non-similar) irreducible

Let

representations of ? with coefficients in an algebraic number field K. If p is a prime ideal dividing p then the co6rdinates in the underlying spaces can be chosen so that all the group elements are represented by matrices with p-integral coefficients. We replace every coefficient by its residue class (mod p). This operation will generally be indicated by a bar. Every representation Zx goes over into a modular representation 2x of ( with coefficients in some Galois field K of characteristic p. Each ~x can therefore be split into the irreducible modular representations i1, i2, ..., k of @ in K, k

Sp

E >

(1)

d,^S.

If K is suitably chosen, the Zp and H. are absolutely irreducible. The Cartan invariants of the modular group ring of ( then are given by c

=

E ddpd

p=-

(K, X = 1, 2 ...,

k).

(2)

We may assume that K contains the g'th roots of unity. Let G be a regular element of ?, i.e., an element whose order is prime to p. Every modular representation i of ? in K represents G by a matrix i(G) whose characteristic roots are the residue classes of g'th roots of unity mod p. There is, however, a (1-1)-correspondence between the g'th roots of unity and their residue classes. We may, therefore, define the character co(G) of i as the sum of the roots of unity which correspond to the characteristic roots of 0(G). The character w(G) is a complex number; it is defined

MA THEMA TICS: R. BRA UER

292

PROC.N. A. S.

for regular elements only. Two modular representations split into the same irreducible constituents, if and only if they have the same character (for all regular G). Let

1, 2, . . ., Gk be the classes of conjugate elements which consist of

regular elements, let gx be the number of elements in Gx. The inverse elements of the elements of Gx form again a class 6x*, (X = 1, 2, .. , k). We arrange the values o) of the character w( of k for the elements of Gx

in form of a matrix, X = (wK)), where K is the row-index and X the column-index. The place of the orthogonality relations of the ordinary group characters is taken in the modular theory by X'CX=

(- ,x.*)

(K row-index,

X column-index)

(3)

where 5p.. = 0 for p == . and =pp= 1, and where C = (CK,) is the matrix of

the Cartan invariants. The ordinary and at the same time the modular representations of ? are distributed into "blocks" Bi, B2, . ..6 The ordinary representations of Bt contain only modular representations of B, in (1).

The results of the remaining part of ?2 were obtained jointly by C. Nesbitt and the author, and the proofs will be published in a separate paper. Let 3K be an irreducible ordinary representation whose degree is divisible by the highest possible power pa of p. Then the corresponding modular

representation X3 is also irreducible; it is an indecomposable constituent of the modular regular representation. The representation , for itself forms a block B, (blocks of "highest" kind). On the other hand, we consider blocks of the lowest kind, i.e., blocks which contain at least one ordinary character whose degree is prime to p.

It can be shown that the number of such blocks is equal to the number of those Ex for which (gx, p) = 1. That is to say, the order g/gx of the normalizor of an element of (x is divisible by pa.

3.

We say that two ordinary irreducible characters are p-conjugate if

they are obtained from each other by a change in the choice of the primi-

tive path root of unity. Such characters have the same value for regular elements and consist therefore of the same modular constituents. This shows that they belong to the same block B,. degree tp of an irreducible ordinary representation

Assume now that the ip is divisible by pa-1,

that on the other hand the number rp of distinct p-conjugate representations for Xp is not divisible by p. Then it can be shown that 1p contains each of its modular irreducible constituents

only with the multiplicity

1,

dp, = 0 or 1 in (1). If a second 3X of degree t, is not p-conjugate to Sp, and if again the number r, of p-conjugates is prime to p, then 3, and 3, can only have a modular constituent a in common, if rct, + r,,t, -0

(mod pa).

(4)

MATHEMATICS: R. BRAUER

VoL. 25, 1939

293

=

lt follows that t,,. 0 (mod pa- 1). Further, in the case of an odd p, it follows easily from (4) and §2 that '1:P and '1:,,. and their p-conjugates are the only representations which contain \J as a modular constituent., and for which the number of p-conjugates is ~ 0 (mod p). If a = l, then rP I p - l, and hence always (rp, p) = l. Our results can here be applied to any representation '1:P of a block of lowest kind, since the assumptions concerning tp, rP are always satisfied. Using these facts, we can set up all the linear relations which are satisfied by the ordinary group characters, when we restrict ourselves to regular elements. This is the starting point for the considerations of §4 in which new properties of the group characters in the case a = l will be given. For the actual determination of the numbers rP of p-conjugate characters we need another fact which holds for any a and also for p = 2. The determinant of C in (3) is always a power of p, more exactly the highest power of p which divides l /(g1g2 ... gk). This can be shown by proving that the other factors of the determinant on the right side in (3) are absorbed by the determinants \XI, \X'I = !XI on the left side. lt follows that the elementary divisors of C are the powers of p which divide g/g 1, g/g2, ... , g/gk. If a block BT contains at least one representation '1'.P of a degree ~ 0 (mod pa), then at least one elementary divisor of the part of C, which corresponds to BT, is ~ pa-a. We thus obtain new conditions for the blocks of different types. I mention some further results which hold for representations '1'.P of a degree tP 0 (mod pa- 1) in the case where rP = l" Here, the arrangement of the modular constituents fö.. of [P is uniquely determined apart from a cyclic permutation. If we have h such modular constituents in '1\, then we can find h representations '1:?), '1:~2), ••• , '1:?) which represent @ by matrices with l)-integral coefficients such that all the '1:~") are similar to '1'.P but no two of them can be transformed into each other by means of a similarity transformation with lJ-integral coefficients with a determinant prime to lJ. They represent all the subclasses into which the dass of all representations similar to '1'.P splits, if ordinary similarity is replaced by similarity in this narrower sense. 4. From now on, we restrict ourselves to the case a = l, p odd. THEOREM: Let @ be a group of order g = pg' where p is an odd prime and g' ~ 0 (mod p). Denote by SR the normalizor of a Sylow-subgroup 1,13 of order p. If the degree t of an ordinary irreducible character x of @ is divisible by p, then x vanishes for all elements of an order _ 0 (mod p). For the characters x of a degree t ~ 0 (mod p), we can set up a (1 -1)-correspondence with the irreducible characters if; of SR such that if x corresponds to if;; we have

=

e(x)·x(G) = if;(G), e(x) =

" i. - 1c,; I.A.S(;!d.

prov'1JQ.J

0.

1

5...,;+o..b JQ !';?li+1-·,~o

L'R· is,·-:} 358

:1:

FINITE GROUPS

(5)

MA THEMA TICS: R. BRA UER

294

PROC.N. A. S.

for elements G of an orderdivisible by p. The sign E(x) is independent of G. In particular, the number of charactersx with t 0 0 (mod p) is equal to the number of classes of conjugateelements of 1. The normalizor 9' of an element P of order p of is contained in 9 On the other hand, 9' = 941 X $3, where 9) has as a normal subgroup. an order prime to p. If the irreducible representations of 9 are known, then the characters of 9 can be obtained without difficulty, and we can therefore write down the values of the characters of ? for all elements of an order - 0 (mod p), only the sign of the characters remaining undetermined. For an element P of order p, we have 9

t

x(P) (mod p).

(6)

If therefore the degree of the character of ? is known, this sign can also be found. On the other hand, if the character 4 of 91 has the degree h, it follows from (5) and (6) that the degree t of the corresponding representation 3 of ? satisfies t _ =~h (mod p). (7) Combining this with the known properties of the degrees and with the formulas (9) below, we can actually find the degrees t for numerous groups even of very high orders. The structure of 9) is very simple in many important cases. Often, it is of order 1. We now study the distribution of the ordinary characters of a degree t 0 0 (mod p) into blocks. Suppose that B1, B2, . .., Bs are the blocks consisting of such representations. All the following blocks consist of exactly one character of a degree t - 0 (mod p) (?2). There are exactly such that E, consists of s classes of conjugate elements, say 1,, (2, . ., *(, regular elements and the number g, of its elements is ? 0 (mod p) (?2). In each G,. (- = 1, 2, ...,

s) we can find an element V. which commutes

with the element P of order p, we may take V1 = 1. Let z, be the number of classes of conjugate elements which contain an element VPa" with a 0O (mod p) for a fixed ar. If n(G) denotes the order of the normalizor of the group element G, then z, can also be characterized by the conditions z,lp -

1,

n(V,)z,.

+ n(V,P)

= 0

(mod p2).

We can show that the blocks B1, B2, . . , Bs can be arranged so that Bx

p-1 consists (1) of --zx

characters,which lie in the field of g'th rootsof unity,

and (2) of zx further characters which are all p-conjugate to each other. denote one of the latter onles by X(X,0)and the first oices by x( '").,

We 1 2,

MATHEMA 'ICS: R. BRA UER

VOL.25, 1939 -

295

1

p . In the case zx = 1, any character of Bx can be taken as ..., ---zx (x,'?) For u 4: O, (X')(P) is either positive or negative, we set e(X' = +1 or e(X') = -1 E(X,O)as the sign of

accordingly.

In the case ,i = 0, we define the sign

x(X') (Pa).

Using the results of ?3 we can show that

p-1

a=l1

the characters of the blocks Bx satisfy the relation p-1/zx

E

E(

X(X')(G) = 0 for regular elements G, (e(X,) = =1l).

(8)

A=0

In particular, the degrees t('") = x("') (1) of the x(x)" satisfy p- 1/zX

E

=

(X')t(X')

JA=0

o.

(9)

Finally, we mention a property of the characters x(X') for nonregular elements. It can be shown that the expression e,)

x(,)

(Vap ) =

has the same value for all , > 0 and all a formula must be replaced by e(X,0)

EX(X,0)(

VaP,)

0 (mod p). =

(10a)

aX)

For Au= 0, the

(X)

(

where the sum on the left side is extended over all the zx distinct p-conjugates of x(X'). The expression (x) on the right: side is the same as in (10a).

There exists a character

(X) of 9)1, for which ) (V( ) -==

(11)

and #x is the sum of zx/zl irreducible associated characters of the subgroup 9 of 1. Each character of 51 appears in exactly one a(x). There is a close connection beginning of ?4.

between these last facts and the theorem

at the

1 University of Toronto Studies, Mathematical Series No. 4 (referred to under M. R. (1937)). 2 Finite Collineation Groups, University of Chicago Press, 1917. 3 W. Burnside, Proc. Lond. Math. Soc., 33, 266 (1901); W. K. Turkin, Rec. Moscow, 40, 229 (1933). 4 For references see E. Pascal, Repertorium der hoheren Mathematik, Vol. I, part 1, pp. 211-214, Leipzig, 1910. 5 For the results of ?2, cf. M. R. For alternative definitions and proofs, cf. T. Nakayama, Ann. Math. (2) 39, 361 (1938), and R. Brauer, these PROCEEDINGS, 25, 252-258 (1939). 6 Cf. M. R., pp. 17-21, and the last paper in 5

ANNAL OF MATHEMATICS Vol. 42, No. 1, January, 1941

ON THE CARTAN INVARIANTS OF GROUPS OF FINITE ORDER* BY RICHARD BRAUER (Received April 18, 1940) 1. INTRODUCTION

E. Cartan, in his fundamentalpaper on hypercomplexnumbers,'introduced an importantset of invariants CKix, (K, X = 1, 2, -.. , k) of an algebra A witha are principalunit 1. Here, k is the numberof primeideals $3Kof A. The c,,KX non-negativeintegerswhichalso play an importantrole in the decomposition of the regularrepresentationof A. Let us considernowa semisimplealgebra F ofrankn overan algebraicnumber fieldK, and let J be an integraldomain2of F. Every primeideal p of K generates an ideal pJof J. The natureof thisideal is determinedby the structure of the residueclass ringJ/pJ. This ringcan be consideredas an algebra over the residue class ring o/p, where o denotes the domain of all integersof K. We have in this Hence, we may form the Cartan invariants ca(p) of J/pi. case cKa(p) = cx.(p), and the cKx)(p)are the coefficientsof a non-negative quadratic

formX,6= E CK.(p3)XKXx. In particular,these notionscan be used in the case of the group ring r of a we choose an algebraicnumber group 3 of finiteorderg. As fieldof reference, fieldK, such that all the absolutely irreduciblerepresentationsof (5 can be in K.3 The linearcombinationsof the groupelements writtenwith coefficients in K forman integraldomain J of r. Hence, we may withintegralcoefficients formthe Cartan invariantsc,,),(p)foreveryprimeideal p of K. It appears that they actually depend only on the rational prime p which is divisible by p. we denotethemby CKX(P).4 If p is not a divisorof the grouporder Accordingly, g, then the matrix C(p)

=

(Cx),(p))

is the unit matrix 1, i.e. cKx(p)

=- bA .

We

thereforerestrictour attentionto the case wherep divides g, in which case p is a divisorofthe discriminantof J. To everysuch primep, we obtain in cGx(p) * Presented to the American Mathematical Society on April 26, 1940. Cf. furtherR. Brauer, Proc. Nat. Acad. Sci. 25, (1939), p. 252. 2 By an integral domain (Ordnung) J of r, we understand a subring J of r with the following properties: (a) J contains all the integers of K; (b) The rank of J is n; (c) The 1 E. Cartan, Annales de Toulouse, 12 B, (1898), p. 1.

elements of J, when expressed by a basis el,

. ,2

en

of J, have the forma E

where the as are integers of K and w is a fixedinteger of K which is independent of a. 3 Cf., for instance, A. Speiser, Theorie der Gruppen von endlicher Ordnung, 3 rd ed., Berlin 1937, theorem 181, p. 204. 4For the properties of the cx(p), in this case, cf. R. Brauer and C. Nesbitt, University of Toronto Studies, Math. Series No. 4, 1937, and a paper forthcomingin the Ann. of Math.

53

54

BRAUER

RICHARD

a set of invariantsof the group6, whichare of greatimportanceforthe theory of group characters. The aim of this paper is the determinationof the discriminantI C(p) I of Af. We prove 1: The determinant THEOREM I cKX(p)I of thematrixof Cartan invariantsof a group65offiniteorderis a powerof p. The exact exponentof p in I Cx(P) I is given below in theorem1*. 2.

FOR THE

PREPARATIONS

PROOF

We denote by (i1, (2 , ***, Gk, the classes of conjugate elementsof the group 5, which consist of p-regularelements.i Let g = g/nxbe the number of elementsin Gxi,so that nxis the orderof the normalizorof an elementof Al containingthe reciproTo each class (E, , therecorrespondsa reciprocalclass Grin irreduciblemodular k absolutely then, cals of the elementsof (Ex. We have, arrange the values ?(K) of ,k) of 65 (mod p); we ,. may characters,so(1), (P(2) matrix of a (P forthe class Chiin the form (s(K)) #X]>=_

1

(ECUX =

2,

.

,k).

The numberk here also gives the degree of the Cartan matrixC - (cx(p)) and the matrices1 and C are connectedby the formula f =(1) F'Cb4 )

(ic) k are takes in a suitable arrangement. On form2, ., providedthat &p', (2) in we obtain (1), ing the determinant

(2)

12IC[

=

Jn1n2 *..

nk.

This shows that l-C l 0 0. Hence, the inon-negativequadratic formif is positivedefinite,i.e. I C I > 0. Further,the determinantj i I is primeto p;7 Therefore,theorem1 will be provedwhen we can show , 6,k be the classes of p-regular, conjugateele2: Let (S, 2- 2 THEOREM mentsof 65,and letgo,= g/nxbe thenumberofelementsof d,. If 4 is thematrix of themodulargroupcharactersof 5(mod p), thenthesquare of thedeterminant is givenby [

(3)(3)~~~~~~~

(

4FI2 12 =

i

n2

...

pa

nk.

wherepa is thehighestpowerofp dividingn1n2... n/ At the same time,we obtain of theCartan matrix,(ckx(p)),is equal to pa, 1*: The determinant THEOREM 2. as in theorem wherepa has thesame significance For primesp whichdo not divide the orderof 5, theorem2 is trivial. Here, 5

By a p-regular element of (5, we understand an element whose order is prime to p.

6Cf. the formulae (29) and (15), respectively, of the papers mentioned in 4.

Cf. the formulae (26), (17), respectively, of the papers mentioned in

4.

ON CARTAN INVARIANTS OF GROUP~ OF FINI1'EI ORDER

55

k is the number of all the classes of conjugate elements of @, and p" = l. The relation (3) is obtained by multiplying ' and tl>, using the orthogonality relations

for group characters. In the same manner, the analogous formula for ordinary group characters (instead of modular characters) can be obtained at once. In order to prove theorem 2, it is sufficient to show that if q ~ p is a rational prime, and if r/ divides the right hand side of (2), then r/ divides I tl> [2 • We shall prove that by proving a similar statement for certain minors of . We first have to give some simple group theoretical considerations. Let A be an element of@ such that the order of A is prime top and q. We shall say that an clbment G of @ contains A a,s its q-regular factor, if G is of the form G = AQ, where the order of Q is a power q' ~ I of q, and where AQ = QA. Of course, A and Q are uniquely determined by G; both can be written as powers of G. If G1 is conjugate to AQ, then the q-regular factor of G1 is conjugate to A. Let A1, A2, · · · , Ambe a maximal system of elements of @, such that A;, A; are not conjugate for i ~ j and the order of each A; is prime to p and q. With each A;, we associate those classes of conjugate elements, ([fi>, ([i'l, . , . , ([t>, which contain elements with A, as their q-regular factor. Each of the classes, ([1 , ([2 , · · · , ([k , then appears exactly once in the form ([;il. By expanding I tl> 1, we see that I tl> 1is a sum of terms T1T2 · · · T,..,

(4)

where T;, is a minor of degree h, of tl>, containing only the colm'nnswhich belong rc-(i) «Ci) • • • rc-Ci) to ~1 , ~2 1 , ~h; • We now state THEOREM 3: Let A be an element of an order not divisible by the two primes p and q, and assume that the h classes, ([P , ([.. , , · · , ([, , are all the classes of conjugate elements of the group @, which contain elements G with A as their q- 11?.Jl 1 is divisi~le by the same number. lt is therefore sufficient to prove theorem 3 in order to prove theorems 1 and 2. Changing the not.at.ion, if necessary, we may assume without restriction that p = ], (j = 2, • • • , T = h, r = 1, s = 2, , .. , t = h, 8 An analogous theorem holds for ordinary group characters. Here, the assumption that the order of Ais prime top is not necessary. The proof is the same as for theorem 3.

363

REPRINT OF [33]

56

RICHARD

BRAUER

and then (5) assumes the form (6)

..I

(1) I P1,

=.0

.

. 'I

Ph

...

(h)

'P1 ,

,

(1)

_

(mod (qlq2

(h) '*Ph

...

qh).

The proofof (6) will be given in ?3. First, we must formulateand prove a forthe , AQh be representatives grouptheoreticallemma. Let AQ1, AQ2, h classes CS, C2, * * *, Sh, whereA is the q-regularfactorof AQi, and where Q,= 1. Let 91 be the normalizorof A in (, and let e be a Sylow subgroup of 9 belongingto the primeq. Then n1is the orderof 91 and q1 the orderof Z&. Each AQi will commutewith every elementof a certainsubgroupCi of order qi of (M. Since A and Qi both are powers of AQi, each of them commutes with every elementof Hi. In particular,we have shy 9W. Replacing Qi by an element Nt1QiNi, withNi in 91,we may assume that *

.

C C

(7)

-as followseasily fromSylow's theorem. Since Qj must belong to Hi, the elementQ. itselfwill belong to S. If Q is any elementof Z?, then AQ will be conjugate in 5 to some AQi, i.e., G-'AQG = AQi. Raising this equation to suitable exponents,we obtain G-'AG = A, G-1QG = Qi. Therefore,Q , Qh forma completesystem and Qi are conjugate in 9%,and hence Q1, Q2, forthose classes of conjugate elementsin 9, in whichthe orof representatives ders of the elementsare powers of q. , Qh need not forma completesystemofrepreIn ?7 the elementsQ,, Q2, sentativesforthe classes of conjugate elements. However, we may construct , Qh. Each such a systemby adding furtherelementsQ to the set Q1, Q2, Q will,in 91,be conjugate to a certainQi wherei is uniquely determined,i = 1, 2, ... , h. We denote the elementsQ belongingto Qi by Qi = Q( ), QIl), , (I > 0). Let q(A) be the highestpowerof q dividingthe order Q2), of the normalizorof Qi() in SD. Accordingto (7), we have .

.

..

(8)

q~o)

_=qj

We now prove the LEMMA: The numbersquX),(X = 0, 1, 2, di of themare equal to qi, then (9)

...

, 1s)are divisorsofqi . If exactly

di 4 0

(mod q).

PROOF: Let Vi denote the class of conjugate elementsof 9t, which contains the elementAQi. The iumber Mi of elementsin ti is equal to the orderof 9 divided by the orderof the normalizorof AQi . Hence (10)

(Msq Ms. Mi = qil

w ith

q) = I

57

ON CARTAN INVARIANTS OF GROUPS OF FINITE ORDER

The class Sri can be broken up into partial classes .sr~"), where each Sr~"l consists of elements which are conjugate by means of transformations by elements of O s;;; 91. The elements AQ?>, AQ~ 1>,. ... , AQ}l•) will each determine such a partial class, but there may be further partial classes which do not contain elements AQ with Q in 0. In any case, if AT\") is an element of Sr~"\ and if T~"> commutes with exactly w~"l elements of D, then the number M~") of elements of Sri"l is given by

(11) We have, of course, (12) µ

In @, the elements AQi and AT~µ) are conjugate. Hence, the order of the normalizor of AT}") in@ is divisible by qi but not by a higher power of q. On considering the subgroup generated by AT~µ) and the w~µJ commuting elements of 0, we readily see that w~µ) ~ qi and that the equality sign can hold only if T~µJ belongs to 0. When Ti") = Qf'\ w~") = qf'\ and we thu:hbtain the first part of the lemma. If w~µ) = qi, then the partial class Sr~") contains exactly one element/Qf'\ (;\ = 0, 1, 2, · · · , li), and we have qf') = qi. According to our assumption, there are exactly d; such partial classes. Therefore, d, of the numbers M~>'>, (cf. (11)), are equal tp qifq;, the remaining ones being divisible by a higher power power of q. Then (12) gives Mi= d; IQ qi

(mod ~

1 )

and, on comparing this with (10), we obtain (9). 3.

PROOF OF THE THEOREMS

As we have seen, it is sufficient to prove (6). If ~ is any representation of @, we may assume that the matrix ~A representing the fixed element A appears in canonical form, i.e. a1T, 1

A

-+~A

=

(

)

ad:2 ...

where a1 , a2 , · · · are distinct roots of unity and v1 , v2 , , .. are positive integers. The matrices ~ Q , representing elements Q of 0, then break up in the form

365

REPRINT OF [33]

58

BRAUER

RICHARD

whereV is of degreev . For a fixedj, the matricesV forma representation , a~m) be the distinct,irreduciblecharacters'of Z. Let z(1', 12 Z iof ?. Then m is the numberof classes of conjugate elementsin Z, i.e., the number of elementsQV?. If x denotesthe characterof j, we readilyobtain m

x(AQ) -E

zz

(Q)

A=1

where the z, are algebraic integers which are independentof Q. We set, accordingly, (13)

zaps'(Q).

0, and withthe same firstindex. Then, accordingto (14), ZOl on the righthand herebetweenthe ordinaryand 9 Since the orderofi is primeto p, thereis no difference the modularcharacters(mod p) of A. '? i.e., a matrixwithm rowsand h columns.

59

ON CARTAN INVARIANTS OF GROUPS OF FINITE ORDER

side will be replaced by 0. If 0i is obtained from 0 1 by subtracting from each column (i, A), with X > 0, the column (i, O) of 0o, then, by taking the determinant, we find 1

Here,

1

.z 11 e 1 =

Z I is a.n algebraic integer, and

1 t:.. II 10 1

ue: 1.

has the value

-+1·t-G1 E.. -·..) _-y-J .. On account of (8), we obtain

(15)

1Ä 11

U0i 1

= o, (mod (q1q2 ... q1.)½ CIL IL>o q:f-l)i). t

Let us assume now that the formula (6) does not hold. powers of q, it follows that 1 U0i 1

(16)

Since all the q~"J are

= 0 (mod (q IL IL.>o q~>.)) 1),

for any choice of U. Taking a sliitable U, we see that any minor of degree m - h of 0i can be obtained in the form I U0i" I- But the determinant 1 (0i)'0i I is equal to the sum of the squares of all these minors. Hence, (17)

1

(0i)'0i !

= 0 (mod q IL Il»>o qtl).

Any row in (0i)', the transpose of 0i , is characterized by a pair of indices, i, µ, (i = 1, 2, ·. · , h; µ = l, 2, · · · , li), and any column is characteri!'led by an index K, (K = 1, 2, ... , m). ·The rows of (0i)'0i are given in the same manner, and each column is characterized by a pair of indices j, v, with j = 1, 2, · · · , h; v = l, 2, ... , l;. For the element y(i, µ; j, v) at the place (i, µ), (j, v) in (0i)'0i , we obtain easily m

y(i, µ;j, v)

=

E

(t1 in D has the order qi">, we find y(i, µ;j, v) (18)

= r,(Qi"\ Q}"))qf"l - r,(Qi' Q}"))qi - , In each case we have i fixed, and we denote the row-index by µ, the column index by v.

Xi3J, X~ 4J.

(25)

As the lemma in §2 shows, we have d; - 1 values for µ ~ 1, for which q~µJ = q,. W e may assume that these are the values µ = 1, 2, ... , d; - 1. 13 For µ ~ d, , we have (mod q).

(26)

Three cases must be considered separately. CASE I: i ~ i*. We may then assume that Q~µJ-i = Q!~' for all µ. Then X} 1' is the,unit matrix, X1 2J = Xf 3' = 0, whereas in Xf 4 > each coefficient in the µ-th row is equal to q,;/q1µJ, Hence, (mod q), the matrix X?' has d; - l rows consisting of 1, and the other rows are all 0. Then (mod q). 12 13

If l; = 0, we must set I O; 1 = 1. Ford; = 1, the corresponding kinds of rows of Xfl, do not occur.

368

FINITE GROUPS

61

ON CARTAN INVARIANTS OF GROUPS OF F1NITE ORDER

(

This shows that (mod q) one characteristic root of /( has the value di - l, and that the others have the value 0. Then the characteristic roots of Xi= I + Xi 4> are given by di , 1, 1, · . · , 1 (mod q). Hence (mod q). OASE II. i = i*, but Qi and Q,1 are not conjugate in 0. We may assume here that Q,1 = QPl. Then X} 4l = 0. In xF>, only the first column contains elements different from 0, and the coefficients in this column are given by

- qi / qi(1) ' - qi / q;,(2) ' ... ' - qi. / q;,Cl,) .

In Xf 3l, only the first row contains elements different from 0. All the coefficients in the first row are equal to -1. In xpi, the first row and column are 0. Each of the other rows contains exactly one coeffi.cient 1, and all the other coeffi.cjents are 0. The same is true for the second, third, ... , last column. On adding all the other rows to the first row in Xi, we obtain easily

IXd =

±(~!l + qi~~l +· ... + q;,Zz\i + 1). qi

Hence, since di - l of the fractions are 1, and the other ones 1

xi

1

= ±di

=

0 (m0d q),

(mod q).

III. i = i*, and Qi and Q, 1 are conjugate in O. Here Xi 2l = 0, Xi = 0. As in case I, the first di - l rows in x?i contain only coefficients = 1 (mod q), and the latter rows contain only coefficients = 0 (mod q). The matrix xpi can be changed into the unit matrix, if the columns are taken in another order; the value of X! 4l (mod q) is not altered hereby. The argmnent used in the first case then gives OASE 3>

± 1 xi 1 = di

(27a)

(mod q).

The three formulae (27) show, in connection with the lemma in §2, +,hat in any case I Xi 1 ~ 0 (mod q). Then (24) is impossible. Thus, the assumption that (6) is not true leads to a contradiction, and the theorems 1, 2 and 3 11re oroved. UNIVERSITY OF TORONTO

1 i.; ~

[R.8.]

369

REPRINT OF [33]

ANNALS

OF MATHEMATICS

Vol. 42, No. 2, April, 1941

ON THE MODULAR BY R.

CHARACTERS

BRAUER

OF GROUPS

AND C. NESBITT*

(Received, December 15, 1939)

PART I. Introduction ?1. Ordinary representations. Group ring. ?2. Arithmetical questions. ?3. Modular representations. ?4. Decomposition numbers. ?5. Cartan invariants. ?6. Characters. ?7. The character relations. ?8. Corollaries. ?9. Blocks. ?10. Decomposition of P. ?11. Summary of the results. PART II.

Blocks of highest kind

?12. Conditionforthe reducibilityof Zi . ?13. Blocks of highestkind. ?14. Vanishing of the character for p-singular elements of (X. ?15. Example. PART III. The elementarydivisors of the Cartan matrix ?16. Computation of the elementarydivisors of C. ?17. Blocks of type a. ?18. Ordinary characters which are linearly independent (mod P). ?19. Application of the lemma of ?18. ?20. Blocks of the lowest kind. ?21. Alternative proof of theorem 2. PART IV. On the multiplicationof characters ?22. Relations between the problems of determiningthe ordinary and the modular characters of (5. ?23. The multiplicationof characters. ?24. Upper and lower bounds for the degrees of the indecomposable constituentsUK. PART V. Relations between the charactersof a group (5 and those of a subgroup 'S of 05 ?25. The induced character. ?26. The formulas of Nakayama. ?27. On the converseof theorem 1. ?28. An upper and a lower bound forcil . PART VI. Special cases and examples

?29. Special cases. ?30. The groups GLH(2, pa), SLH(2, pa), and LF(2, pa). ?31. The Cartan invariants and decompositionnumbers (for p) of LF(2, p). *

Presented to the American Mathematical 556

Society, October 30, 1937.

ON THE

MODULAR

CHARACTERS

OF GROUPS

557

I. INTRODUCTION'

of a group 1. OrdinaryRepresentations. Group ring. The representations 6 of finiteorderg were firsttreatedby FrobeniUs2in his theoryof group charare taken as complex of the linear transformations acters. The coefficients if we take themas the elements numbers,but it does not make any difference of an algebraicallyclosed fieldK of characteristic0. The theory has been extendedby I. Schur3to the case whereK is any fieldof characteristic0. It ifwe take K as an algebraicnumberfield. does not mean an essentialrestriction, of 65,we may considerrepresentations Instead of consideringrepresentations ofthe group ring4r of 0 with regardto K. This r is an associative algebra consistingof all symbols (1)

a = aG, + a2G2+ -.- +agGg

whereG,, G2, ** , G;,are the elementsof 6, and a1, a2, ..., a, are arbitrary elementsof K. The equality of two such elements,theiraddition,and their multiplicationare definedin a natural manner. The study of the representationsof r is closelytied up withthe investigationof the algebra r. 2. Arithmetical questions. We may also study r froman arithmeticalpoint ofview. TakingK as an algebraicnumberfield,we obtaina domainofintegrity a ifwe take the as in (1) fromthe domaino of the integersofK. The question arisesin what mannerdoes a primeideal p behave when consideredas an ideal of a. The behaviorof p in a is characterizedby the structureof the residue class ring 3/p. This ringcan be consideredas an algebra F over the residue class fieldK = o/p of the integersof K taken (mod p). Obviously,P is the group ring of 6 with regardto the finitegroundfieldK. The study of the structureof P then amountsessentiallyto the same thingas the study of the in the finitefieldK. of P or of 6 by matriceswith coefficients representations We thus are led to the problemof extendingFrobenius'theoryto the case ofa modularfieldof reference(i.e. a fieldof a characteristicp - 0). 3. Modular representations. Modular representationsof a group 6 (i.e. in a modular field)were first of 65by matriceswith coefficients representations 1 In ??4-10 of the introduction, we give a short account of the theory of modular representations of a group as developed in our paper: On the modular representations of groups of finite order, University of Toronto Studies, Math. Series No. 4, 1937 (we refer to this paper as M.R.). We tried to make it unnecessary for a reader, who is familiar with the theory of representations in general, to read our former paper. An exception is formed perhaps by the proof of formula (5) below, but literature for other proofs of this formula are mentioned in footnote 10. 2 For Frobenius' theory, see the accounts in L. E. Dickson, Modern Algebraic Theories, Chicago, 1926, chapter XIV; G. A. Miller, H. F. Blichfeldt, L. E. Dickson, Theory and Application of Finite Groups, New York 1916, chapter XIII, H. F. Blichfeldt, Finite Collineation Groups, Chicago 1917, chapter VI. 3 I. Schur, Sitzungsber. Preuss. Akad., 1906, p. 164. 4 Cf., for instance, H. Weyl, The Classical Groups, Princeton 1939, Chapter III.

558

]R. BRAUER

AND

C. NESBITT

studied by Dickson.! He proved that Frobenius' theory remains valid, if the characteristicp of the fieldis primeto the orderg of 65. Since the discriminant of F is a power of 9, this correspondsto the case that the prime ideal P in ?2 is not a discriminantdivisor. If, however,p divides g, then we must expect results which differfromthose of Frobenius. This was shown firstby a theorem of Dicison6 concerningthe splittingof the regularrepresentation(cf. ?8 below). A coherenttheoryof the modular representationswas given by the authors in a previous paper.7 In the following??4-9, we shall discuss brieflyour former results. We prefer,in most of what follows,to use the language of the theory of representations(instead of that of the theoryof algebras or of the theory of ideals). 4. Decomposition numbers. We choose the algebraic number field K such that the absolutely irreduciblerepresentationsof (5 in the sense of Frobenius , Z2n be the essentially can be writtenwith coefficientsin K. Let Z1, Z2, ones among these representations,and let zi denote the degree of Zi. different Then n is the numberof classesof conjugateelements(i, , (L in R. 22, . Let p be a fixedrational prime number,and p be a fixedprime ideal divisor of all the Zi are p-integers(i.e. of p in K. We may assume that the coefficients a of K, and A is prime to p). integers numbersof the forma/f where and d are Let opbe the ringof the p-integersofK, and K the residue class fieldofop(mod P) whichis identicalwith the fieldo/p in ?2. We denote generallythe residue class of an element z of K (mod p) by 2. Similarly,replacing every coefficientz in in opby its residue class 2, we obtain a a representationZ of ? with coefficients in K. In this manner we may form modular representationZ with coefficients Z,, Z2 , * * * , Zn . These modular representationswill, in general,be reducible and will then split into irreduciblemodular representationsF, with coefficients in K. We indicate by !

* *

(2)

2

Z

dikF,

that F. will appear in Zi with some multiplicitydi, These rational integers d,,,>0_ and are called the "decompositionnumbers" of 6). In the sense of ?2, they describe a connectionbetween the simple invariant subalgebras of r, and the prime ideal divisors of p in S. 5. Cartan invariants. Of special importanceis the regular representationR of (M (or F) formedwith regard to K as ground field. Since the group ring is no longer semisimplein the modular case, the theorem of the full reducibility of R does not hold any more. Let U1, (2, *. * , b',, be the distinct indecomE. Dickson, Transact. Am. Math. Soc. 8, 1907,p. 389. L. E. Dickson, Bull. Amer.Math. Soc. 13, 1907,p. 477. 7 For the following,cf. M.R., and also R. Brauer, Nat. Ac. of Sciences 25, 1939,p. 252. We referto this last paper as R.A. 6 L. 6

ON THE

CHARACTERS

MODULAR

559

OF GROUPS

posable constituentsof R. Each UKcan still be brokenup into its irreducible constituentsin K. This splittingis of the form

FK (3)

UK

F

(FK

*F, if the notation is chosen suitably.8 The representationsF1, F2, ***, Fk are of 65in K. Furall distinct,and thereare no otherirreduciblerepresentations ther,the FK are absolutelyirreducible. We denote the degree of F, by f,,,that of U, by UK. Then U, appears f, timesas indecomposableconstituentof R and F, appears u, timesas irreducible constituentof R. of Fx as irreducibleconstituentof UK Let CKXbe the multiplicity (4)

UKE+

Fx.

Here, the C,,a are rational integers > 0, the Cartan invariants9of 65 (forp). They also can be characterizedby means of structuralpropertiesof P; theyexprime ideal divisors of p in 3. press mutual relationsbetween the different Between the decompositionnumbersand the Cartan invariants,we have the followingequations'0 nI

(5)

= CKXA i=1

or in matrixform (6)

C

dgdx

(K,

X =

1,2, *..

,

k)

= D'D

whereC = (cK,,),D = (di,,)and D' is the transposeof D. There existsa representation(UK) of 6 in K whichif taken (mod P) becomes similarto UK, (UK) = UK. We then have"

(7)

( UK) ul > gq/(g- m).

(79)

The multiplicationof charactersis used to obtain new charactersif some charactershave already been found. It is often convenientto determinethe (K) at the same timewiththe (K) Here formula(76) can be used. Formulas (77) and (79) can sometimesbe used, if we want to show that a character , whichwe have obtained,is an 77(K) and not a sum of severalsuch n(K). V. RELATIONS BETWEEN

THE CHARACTERS OF A GROUP OF A SUBGROUP '

5

AND THOSE

25. The induced character. The second importantmethodof Frobeniusfor the constructionof charactersassumes that the characterx of a representation "induces" a representaV of a subgroupIDof 5 is known. This representation tion V* of 5 whose characterx* can be obtained. The methodof formingV* and so does the remainsvalid in case we start with a modularrepresentation, proofhere, formulaforx*, but this last formularequiresa somewhatdifferent due to the modifieddefinitionof the characterof a representation. Let h be the orderof &, and let Q. (A = 1, 2, ** ,m; m = g/h) be a complete residuesystemof (5 (mod .S) 0 = DQ1 +

DQ2+ .+

tQm.

581

ON THE MODULAR CHARACTERS OF GROUPS

We set V(G) = 0 if G does not belong to ments of @, and define V*(G)

(80)

~

so that Y(G) is defined for all ele-

= (V(Q.GQ~1))

(K row index, A column index).

lt is easily seen that this is a representation V* of@ of degree tm = tg/h where t is the degree of V. We shall determine the character of V*. Let G be an arbitrary element of @. The element QµG belongs to some residue class ~QPa 0. The determinant to determinethe exact value of the determinantof D. It is not difficult

,

,

BN 1, p. 15.-BN 2, (22). It should be noted that the d,, are not the ordinary decomposition numbers of 91(P1) though they satisfy exactly the same relations. I Cf. R. Brauer, On the Cartan invariants of groups of finiteorder, Annals of Math. 42, p. 53, 1941. 10 BN 2, ?8.-In BN 1, it is shown that the determinant I X I of the matrix X of the modular group characters of a group is prime to a fixedprime-ideal divisor p of p, cf. BN 1, (26). The proof given in BN 1, p. 14 for the fact that I X I is prime to p, is not correct. However, this result follows fromBN 1, (26), since I X I1is a rational integer, cf. BN 1, (29). 8

ORDINARY

BETWEEN

CONNECTION

AND

MODULAR

CHARACTERS

931

4. p-conjugatecharacters If we replace a primitivegth root of unity E0by anotherone E , (X, g) = 1, into a conjugate charactert,. Choose then every characterX, is transformed Examountsto an interchange now X -1 (mod g') so that the substitutionci of the path roots of unity,such that the geth roots of unity remain unaltered. In this case, we say that the two conjugate charactersi, and G are p-conjugate. All the charactersare distributedintofamiliesof p-conjugatecharacters. If i, and Paare p-conjugate,then i,(G) = Ra(G)forp-regularelementsG. It followsthat G and Pahave the same modularconstituents(forp). Hence ?, and ? havethesamemodularconstit6: Two p-conjugatecharacters THEORFEM uents;theylie in thesame blockof characters(for p). 5. The decompositionnumberscorrespondingto a block of characters Let B be a block" of charactersof (M(forp). We considera sum analogous to (6) but whereA' rangesonly over the values forwhichi;, belongsto B. We shall say forshortthat these are the indicesin B. Similar to (5) we have

EZ d,,wd~= A,' A,' V

MuinB

W

1 4(V')4i(W) 1 n(Pi) n(PI)

E

;,-(PiV)4,(PiW).

uinB

As the d,., the whole sum is an integerof the fieldof the path roots of unity. But with every I,, all its p-conjugatesappear, and the expressionon the right hand side shows that the sum is a rational integer. Now collect the termsforwhich V lies in a fixedclass of 91(Pi) and W in a fixed class of 9(P3). Since these classes consist of n(Pi)/n(PiV) and of n(PD)/n(PjW) elementsrespectively,and all the correspondingtermshave the same value, we find Ed',,d'p (7)

=

E

V

Z W

[ -'(V)/n(PiV)][4)'(W)/n(PjW)]

E

in B

A,(PiV)t,,(PiW)

where V and W range over certainelementsof %(Pi) and 9(Pi) respectively. The numbersin the square brackets are p-integers.'2 If P is a prime ideal divisorof p in the fieldgeneratedby the charactersI,,, then i;,(PV)=- ,,(V) (mod P). On the otherhand, ifj > 0 and, therefore, PiW p-singular,we have X

inin B

~-U(V)r-y(P1W)

=

0.13

1, ??6, 7.-BN 2, ?9. BN 1, theorem V.-This theorem is a consequence of BN 2, (16) and (17). Is BN 1, theorem VIII.-This can also be seen from BN 2, (28), and the formulae (1) and (6) of the present paper. 11 BN 12

932

RICHARD

BRAUER

Hence forj > 0 the sum (7) is divisible by p, and since it is rational,it is divisibleby p. THEOREM 7: If Pi 5 1, i.e. ifj > 0, thenforeveryblockB Z

(8)

p in B

O (mod I);

d,>,p

theleftside is a rationalinteger. Assume that the blocks B = Bx consistsof the ordinarycharactersP1, t2, ... , = and the modularcharacters (p2, ... , as (so = p?) and formthe = 1, 2,-.. , y). The matrixD matrixDx = (dp) = (do.),(p = 1, 2, . ..,x; blocks.'4 breaks up completelyinto D1, D2, ... correspondingto the different From (6), it followsthat

EI

(8*)

Auin BAAP

do

if i = 0, and p,, cppboth belongto B. in all othercases.

= c

=

02

This supplements(8). familiesof p-conjugatecharacters,then If Pi, 2, *... , As belongto w different , t all belongto different families. Assume we arrangethe Aso that 1, P2, that the familyof tv consistsof r, charactersand set

(p = 122 Al w;

Do=(dpa,)

(9)

1 2y.. *y I ). ly

If i, and Dpare p-conjugate,then dsA= dp,by theorem6. Therefore,DA has From (8*) the same rowsas A, but the A' row of DA appears r, timesin A. and (1), it followsthat

Z dA, vA(S)

AL=1

= 0

(forall p-singularS of (M).

Here, p-conjugatecharactershave the same coefficients.If we denote by A the sum of all characterswhich are p-conjugateto i;, (includingC), the equation can be writtenin the form IV

(10)

E

dsy(S)

= O

(forall p-singularS of (5).

;A=1

We set dA =

E

dvw,forany fixednumberswi, .. ., xv .

(l0a)

E

d,,J,,(S)= 0

Then

(forall p-singularS of (5).

;A=1

Every characterC,(G) of (Mmay be consideredas a characterof the p-Sylowsubgroup$ of (M,when we take G as an elementP of $. If, in this sense, I, containsthe 1-character[1] of 3 q. times,the same will holdforthe p-conjugate times. The expression charactersand, therefore, I, will contain [1] exactlyrpqA d X(P) = d,,,(P) is a linearcombinationof the charactersof$, and [1] appears 14

BN 1, theorem VII.-BN

2, (28).

CONNECTION

BETWEEN

ORDINARY

in it withthe coefficient E drq, z. is the degreeof i, , then x(1) ordinarygroup charactersgive and hence pa

1j

CHARACTERS

933

But, from(i0a), x(P) = 0 forP # 1. If . The orthogonalityrelationsfor r,,d,,z,, of [1] in x, (1/pa) r,,dz,, as the coefficient =E

w

(11)

AND MODULAR

w

rdeq,

=

E rdz,

If only one of the numbersd,, is different from0, (11) becomes paqA = z,, and then the characterG is of the highestkind.'5 If we exclude this case, it so that only one of the followsthat it is impossibleto determinew,, W2, *.. WY, d, does not vanish. This impliesthat the rank of A is smallerthan w. But A has the same rank as Dx, and this rank is y.1'6 Hence ,

THEOREM 8: Every block B which is not of the highest kind contains more families of ordinary characters than it contains modular characters: w > y.17 Each relation(10) must containat least two ,. This gives THEOREM 9: If the block B is not of the highest kind, then each of its modular constituents appears in at least two characters i of B which are not p-conjugate.

From (10) it also followsthat

(12)

, C(R)f(S)

=

0

forany p-regularR and any p-regularS. The blocks of highestkind consistof exactlyone ?, whose degreeis divisible by pa, and each such C,formsa block of highestkind.'8 Since such a I, vanishes for all p-singularelements,(2) gives THEOREM

10: If G forms a block of highest kind, then d>, =

0 for all i >

0

and all v.

6. The permutation lemma

We now derive a simple lemma which we shall need. Consider a matrix

M = (mij) withu rowsand v columns. Every permutationA of the rowsof M

can be effected ofM witha suitable "permutationmatrix" by left-multiplication 1 and PA of degreeu whichin everyrowand in everycolumnhas one coefficient m - 1 coefficients 0. Similarly,every permutationB of the columns of M can be effectedby right-multiplication of M witha suitablepermutationmatrix QB of degreev. We prove

1: Let M be a non-singular matrix. If there exists a permutation A of LEMMA the rows of M and a permutation B of the columns of M such that both carry M into the same matrix, then the cycles of the permutation A have the same lengths as In particular, A and B have the same number of cycles. those of B. 15

BN 2, theorem 1.

16 BN 1, p. 21.-BN 17 18

2, (29), (15) and (14). This improves the inequality x > y given in BN 2.

Cf. footnote15.

934 PROOF:

(13)

RICHARD BRAUER

Accordingto the assumption,we have PAM

= MQB .

Since M is non-singular,PA and QB have the same characteristicroots. To each cycle of lengthr of A, therecorrespondthe r r 1h roots of unityas characteristicrootsof PA .19 On comparingthe rootsof PA and QB, startingwiththe maximalr, we readilyobtain lemma 1. Similarly,we prove matrixwhosecolumnsare linearlyindependent, LEMMA 2: Let M bea rectangular B of A of therowsand a permutation and assume thatthereexistsa permutation thecolumnsof M whichbothcarryM into thesame matrix. If B has a cycleof lengthr thenA has a cyclewhoselengthis divisiblebyr. PROOF: Again, (13) holds. Schur's lemma shows that QBis a constituentof On comrootsof QBappear among those of PA. PA so that the characteristic paringthese roots,we obtain lemma 2. We also have the result matrix 'fdegreem, and let 21and e3 be two LEMMA 3: Let M be a non-singular to thesame groupZ. groupsof degreem whichare bothhomomorphic permutation elementA T of X, applied to therowsof M, If for everyT in Z, thecorresponding element BT ofA3,applied tothecolumnsof M, bothcarryM and thecorresponding is thesamefor 2I into thesame matrix,thenthenumberof systemsof transitivity and for S0. All A = AT and B = BT. PROOF: Again (13) will hold forcorresponding is an invariant, we have to showis that the numberra of systemsof transitivity and similaritytransif 21 is interpretedas a group of linear transformations, formationsof a are performed. But this followsfromthe fact that rw is the numberof 1-constituents of the lineargroup W. 7. Families of characters,classes, and d-columns The resultsof ?6 can be applied when M is the matrixof group characters of 5. We may constructcorresponding permutationsA and B in the following manner. Let e, be a primitivegth root of unity,and let X be a rationalinteger EqX carriesevery characterx whichis primeto g. The substitutionTx: E of (Minto a conjugate characterxW, we have

(14)

x (G) = x(GX).

On the other hand, the substitutionG -+ GXcarriesevery class of conjugate elementsQ into a new class ? Then (14) shows that the value of x forG(? is the same as the value of x(? forG. Hence the permutationA: x -f x( of the rowsof M, and the permutationB: (E-- G_( of the columnsboth carryM into the same matrix. We are interestedin the case that X 1 (mod g'). Then x and x(? belongto 19A modificationis necessary, if the underlying field is modular, but the lemma remains valid. The same is true for lemma 2 and lemma 3. We shall use the lemmas only in the case of a non-modular field.

CONNECTION

BETWEEN

ORDINARY

AND MODULAR

935

CHARACTERS

the same familyof p-conjugateclasses (?4). We shall also say that the classes E and G(? belongto the same familyof classes (X 1 (mod 9')). If (E contains the elementPYV (V in 91(Pi), p-regular),then C(? containsP"V in this case. Before formulatingthe results,we also considerthe matrixD, (3). With everycolumnd', all the algebraicallyconjugatecolumnswill appear. Indeed, the substitutionTxn,(X _ 1 (mod g')) resultsin the replacingof Pi in (2) by P., and on account of Theorem 2, this new columncan again be expressedin the formd>. The d-columnsthus appear distributedinto familiesof algebraically conjugated-columns. The effectof the substitutionT, on D then consistsof a permutationB* of the columns;the membersof each familyare interchanged among themselves. On the otherhand,theeffectof Txon D can also be describedby the permutation A: x -+ x( of the rowsof D as followsfrom(2). Hence the assumptions of lemma 1 are also satisfiedforM = D and the permutationsA and B*. Let Z be the group of all substitutionsTx with X-- 1 (mod 9'). We then have homomorphicgroups Xf,Q3,A3*consistingof the A, the B, and the B* respectively. In each of the three cases, a systemof transitivitycorresponds exactlyto a family(of characters,classes,or d-columns). Hence lemma3 gives 11: The numberof distinct THEOREM familiesis thesamefor each of thethree kinds of families: Families of p-conjugatecharacters,families of classes, and familiesofconjugated-columns.20 Next let p be an odd prime. Thus ? is cyclic,and a primitiveelementis obtainedby takingforXa primitiveroot (mod pa), (1 (mod g')). For this Tx, the lengthsof the cycles of the permutationA are the numbersof members familiesof characters. A similarstatementholds for belongingto the different B and B*. Lemma 1 now yields 12: Let p $ 2. If the different THEOREM families of characterscontain , if the different families of classes contain rf membersrespectively, ri , r2, ... and if thedifferent , Sf respectively, familiesof d-columnscontainti, t2, 81, 82, ... respectively, thenthethreesets (ri, r2 , ... , rf), (S, , S2 ... ,S tfmembers , (t, , *2 tf) are identicalapartfromthearrangement.2' Remark: The kop-regularclasses formeach a familyof its own,s" = 1. Similarly,the kod-columnsof Do = D formeach a familyof its own, t, = 1. There is anothercase in whichlemma 1 and lemma3 can be applied. Every automorphismof (Mpermutesthe characters,the classes,and the d-columns,and again the assumptionsof the lemmas are satisfied. It seems unnecessaryto formulatethe resultsexplicitly. *

UNIVERSITY OF TORONTO, TORONTO, CANADA.

30For the first two kinds of families, compare the similar statement and proof in W. Burnside, Theory of groups of finite order, 2nd. ed., Cambridge 1911, p. 315, theorem VI. 21 For p = 2 this will hold, if G does not contain elements of order 8.

ANNALS

OF MATHEMATICS

Vol. 42, No. 4, October, 1941

INVESTIGATIONS ON GROUP CHARACTERS* BY RICHARD

BRAUER

(Received November 19, 1940)

Introduction Let 5 be a finitegroup of orderg. It is well knownthat the distinctirreducible representations&31,.32, * **, &k of ( in the field of complex numbers can be so chosen that theircoefficients belong to an algebraic numberfieldQ of finitedegree. Furthermore, if p is a fixedrational primenumber,we may assumethat the coefficients are p-integers, i.e. are of the forma/d wherea and fi are integersofQ and fiis primeto p. Let 93be a fixedprimeideal divisorof p in U. If every coefficientof 38,is replaced by its residue class (mod $3),then we in a fieldof characteristicp. obtaina modularrepresentation3 withcoefficients It was shown by Dickson and Speiser' that the ordinarytheoryof group charactersremainsvalid formodulargroup characters,if the primep does not divide the group order g. Every modular representationthen is completely reducible;all the distinct,absolutely irreduciblemodular representationsare givenby 1, ,2, * *8kI In a recentpaper,2C. Nesbittand the authorobtained resultswhichmay be consideredas generalizationsof these theorems. Let p be any rationalprime, and assume that pa is the highestpowerof p whichdividesg, say =paIy

(p,

g')

1.

It was We then consideredrepresentations & whosedegreeis divisibleby pa shownthat 8 is absolutelyirreducibleas a modularrepresentation. Whenever .3gappears as a constituentof a modular representation0, then 0 breaks up completelyinto 8, and anotherconstituent2[ (reducibleor irreducible) 0

S1

None of the representations S8xforX 5 K contains 3 as a constituent. Since everyirreduciblemodularrepresentation appears as a constituentof at least one of the representations .231, .21 ... I, 8k, this, (in the case a = 0, i.e. g 4 0 (mod p)), actually yields the theoremof Dickson and Speiser. * Presented to the American Mathematical Society on September 5, 1941.

1 L. E. Dickson, Trans. Am. Math. Soc. 3, p. 285, 1902. A. Speiser, Theorie der Gruppen

von endlicherOrdnung 3rd ed. Berlin 1937,?71. 2 On the modular characters of groups, Ann. of Math., 42, p. 556, 1941. I refer to this paper as BN.

936

INVESTIGATIONS

ON GROUP

CHARACTERS

937

In this paper, I study representations whose degreezKis divisibleby pa-l but not by pa. If the;orderg of (Mis divisibleby p to the firstpoweronly,then 0 everyrepresentation23,is eitherof this type,or of the "highest"type, z(mod pa), whichwas studiedin the formerpaper. Our resultswill enable us to derive a great numberof propertiesof the charactersof groups (Mof such an orderg = pg'; these propertiesforma powerfulweapon in the investigationof these groups.3 Our firstresultconcerningrepresentations . of a degreez-= 0 (mod pal) is that the degreeof all thoserepresentations S3xwhichbelongto the same block4 B as 8X, is also divisibleby pa1 I mentionhere the followingapplicationof thistheorem:The onlysimplegroupof an order4paqb, (p, q primes)witha < 2 is the simplegroup of order60; the only simple groupsof an order3paqb (pi q primes)witha ? 2 are the simplegroupsof order60 and 168. (?9.) With the block B containinga representation Sk of degreez, _ 0 (mod pal), z$ p 0 (mod pa), we associate a lineargraph. Every vertexVxcorrespondsto a ", familyof p-conjugatecharacters5 ,, ... of B, everyedge S,, to a modular if is a modularconstituentof I; characterp,.ofB, and the edge S, containsVxA, (and so of all its p-conjugates). It will be shown that every So containsonly two vertices,furtherthat the complex actually is a tree T. Since we also have the result that p,,never appears with highermultiplicitythan 1 in an ordinaryirreduciblecharacter,the tree describesthe completestructureof the block B, if at every vertex V, the numberr, of charactersin the familyof go is indicated. If T and these numbersr, are given,the decompositionnumbers and hence the Cartan invariantscorrespondingto the block can be easily obtained. Of course,there exist two verticesof T which lie on only one edge. Consequently,theblockB containsat least two characterswhichare not p-conjugate, and whichremainirreducible,when consideredas modular characters. If 3,xis any irreduciblerepresentation of 5, ifz, is its degreeand r, the number of its p-conjugates,then we can show that rxzx 4 0 (mod pa+l)i If rxz), 0 (mod pa), thenz-= 0 (mod pa), i.e. S3xis of the highestkind. In the case zx =0 (mod pa-), zx # 0 (mod pa), it followsthat rxdividesp - 1. Besides, we have the relation St

r r2 *

rw

-+-+*

(r

r2

+

rw

P

forthe numbersrx belongingto the different verticesVxof the tree T. In the last threesections,the arrangementof the modularconstituentsof an 3 Cf. R. Brauer, On groups whose ordercontains a primefactor to thefirstorder, to appear soon. 4BN, ?9. Cf. also R. Brauer and C. Nesbitt, On the modular representationsof groups of finiteorder,University of Toronto Studies, Math. Series No. 4, 1937. 6 For the definition of p-conjugate characters, cf. the list of notations at the end of this introduction. 6 This improves the theorem that the degree of an irreducible representation divides the order of the group.

938

RICHARD BRAUER

0 (mod pa-) is discussed. We irreduciblerepresentationS3xof a degreeZA= assumeherethat the splittingfield2 is normalover the fieldof rationalnumbers e ofp in Q is equal to the numberrxofp-conjuand that the orderof ramification gates; thereexistfieldsQ satisfyingthese conditions,but e can never be smaller than rx. It turnsout that the arrangementof the modularconstituentsof 3x is uniquely determinedapart froma cyclic permutationof the constituents. M whose If we restrictourselvesto the applicationof similaritytransformations coefficients are $-integersand whosedeterminantis primeto $, thenthe cyclic permutationmust be the identity. The class of all representations3, with in Q. which are similarto 3x, splits into j subclasses of 13-integral coefficients whichare similarin thisnarrowersense. Here, j is the number representations of modularconstituentsof 3x , and each of the subclassescorrespondsto one of the j cyclicpermutationsof the modularconstituents. NOTATION: 6 is a group of order g = pg' where p is a fixedprime and (g', p) = 1. The words "representation"and "character" always referto in the fieldof all complexnumbersand theircharacters,unless representations in a field the word"modular" is added, in whichcase we mean a representation of characteristicp. The distinctirreduciblerepresentationsof 5 are denoted by 31, 32, ... I 3, , theircharactersby j1, 2 , * * *, rk and theirdegreesby Two charactersi, and , are p-conjugateif OK can be carried Z1, Z2, *... , Zk. into hx by a change of the primitivepath rootsof unitywhich leaves the g'th , I, rootsof unityunaltered. rhen 1, 2... k are distributedinto "families" of p-conjugatecharacters;the numberof membersof the familyof Xis usually denotedby rx. If G is an elementof 6, then Dx(G)is the value of tx forthis elementG. The distinct,absolutely irreduciblemodular charactersof G are denotedby spl,(P2, * * - . An elementG of 65is p-regularifits orderis primeto p. If its orderis divisibleby p, thenG is p-singular. 1. Constructionof a suitable splittingfield' Let 65be a groupof finiteorderg, and consideran irreduciblerepresentation

3 of the group65in the fieldof all complexnumbers. We want to constructa splittingfieldA for3 such that A is normalover the fieldP of rationalnumbers,

e ofa givenprimep in A is as small as possible. and that the orderof ramification Of course,A mustcontainthe fieldZ = P(r) generatedby the characterr of 3, and this fact imposesa conditionon e. The followinglemma shows that, for suitable A, the numbere has the smallestvalue whichis compatiblewith this condition. LEMMA 1: Thereexistsplitting fieldsof theformZ(r) wherer is a rootof unity, Ir = 1, of an orders whichis primeto p. primeideal of PROOF: Let q be a primeideal of Z and e be a corresponding I For the concepts of the theory of algebras used in ?1, cf. M. Deuring, Algebren, Berlin 1935, and A. A. Albert, Structureof algebras, New York 1939.

INVESTIGATIONS

ON GROUP

CHARACTERS

939

Z(r). Accoriing to a theoremof Hasse,8 the fieldZ(T) will be a splittingfield, ifforeveryq the q-indexmqof Z dividesthe s-degree nq of Z(r). Let q be one of the prime ideals for which mq 1, and let r' be the highestpower of the , rational prime r which divides mq. We shall show below that we can finda rootof unitya = 4(q, r') of an orderj = j(q, r') primeto p, such that forevery primedivisorCo of q in Z(4), the C-degree is divisibleby r'. If we thenadjoin the numberst9(q,r') for all discriminantdivisors q, and all prime powers r' dividingmq, a fieldZ(r) with the desiredpropertyis obtained. The constructionof t is immediatefor infiniteprimeideals; any imaginary root of unityof an orderprimeto p can be taken. Let q be finiteand let q be to finda positiverational the rationalprimedivisibleby q. It will be sufficient integerj whichis primeto p, such that in a fieldof thejth rootsof unityeither the orderof ramificatione, or the degreefq of the primeideal divisorsof q is divisibleby a preassignedprimepower r'.9 We distinguishseveral cases: (a) if r # p, r $ q then assume that q belongs to the exponentp (mod r). We set j = rXwhereA is a positiverationalinteger,and have q"1 (mod j). The degreefq equals the exponentto which q belongs (mod j). Hence p I fq, but f0I pj and consequently,f, is of the formpr'. For sufficiently large X,we shall have a 2 t) i.e. r' fq . (b) If r = q, q $ p, take j = r'+'. Then e, = r'(r-1). (c) Ifr = p, q 4 1 (mod p), take j = qt 1. Then (j, p) = 1,fq = r . 1 (mod p), we may writeq_ 1 = cxrbx (d) If r =p, q where(cx, r) = 1, bN> 0."0 Raising this equation to the rth power we easily obtain coil < cxo. For j = c\, X > t, we have fq = rx r'. In all these cases j is prime to p. This finishesthe proofof the lemma. Withoutrestriction, we may assume that all g'throotsof unitybelongto Z(r). We may furtherassume that T is so chosen that it can be used simultaneously forall irreduciblerepresentations of 63. If we set K = P(r), we have LEMMA 1*: Thereexists a field K overthefield of rational numberswiththe followingproperties: (a) ThefieldK containsthegtth rootsofunity. (,) The primep is notramifiedin K: (p) = pm,wherep is a primeideal of K, and (p, m) = (1). (,y) Every irreduciblerepresentation 3 of 6 can be writtenin thefield K(r) obtainedfromK byadjoiningthecharactert of S3. Two characterst and t' are algebraicallyconjugate with regardto K, if and onlyiftheyare p-conjugate. The degreer of K(r) withregardto K is, therefore equal to the numberof charactersin thefamilyof A. Let e be a path rootof unitysuch that r lies in K(e) = Q; we may always take -

_

I

Cf. the books mentionedin footnote7 or the originalpaper of Hasse, Math. Ann. 107, p. 731 (1933). 9Since the fieldZ is fixed,the fieldZ(a) willsatisfytheconditionabove,iftis takenlarge enough. 10 If r = 2, choose X 2 2.

RICHARD

940

BRAUER

The representation3 1 ? a < a. Let $3be the primeideal divisorof p in a." coefficients belongingto U. When we replace can be writtenwith 15-integral by its residueclass (mod $), we obtaina modularrepresentation everycoefficient belongto the fieldf2of residueclasses of integersofQ (mod , whosecoefficients $). Afterreplacing3 by a similar representation,we may assume that 3 splits into irreducibleconstituentsin Q2. of(Min the algebraicallyclosed extension If a is any irreduciblerepresentation fieldof Q, then the traces of all the matricesare sums of modulargth roots of unity. Since Q has the characteristicp, they are sums of gith roots of unity, and hence theybelong to Q. It followsthat a can be writtenwithcoefficients in 1.12 Any modular representationwhich is irreduciblein P is absolutely irreducible. We may set 8=

(1)

!921 1S2[21

W22

an.

..

.\ '

where21,x 0 (mod $) forK < X,and wherethe A (mod A) are the absolutely irreduciblemodular constituentsof 3. 2. On the numberof p-conjugatecharacters The degreeofQ over K is m = (p(p') = p'-(p - 1); we denote the conjugates notationfor ,C(M-i) and use the corresponding of a numberw of Q by w, cot * matricesand representations. Then 3.() and 3(a) are either non-similaror identical. Let 3(G) be the matrixrepresentingthe group element G, and denote by in the upper rightcornerof 3(G) and by 7y(G)the coefficient ,(G) the coefficient in the lower left cornerof 3(G). The fundamentalrelationsof I. Schur"3for of a representationgive the coefficients (2)

E

Gin't

t3(G)(P)y(G-01) f = 77prg/Z,

wherez is the degreeof 3, and sp, = 0 forr(p) $ r(,), and sp, = 1 fort(p) = Let w be any integerof Q, and set mS-1 mt-1 = tr (y(G)), 42(G) = t1(G) = Eco(P)f3(G)(P) = tr (wt3(G)); y(G)(6f) (3) y wheretr (...) yields

p-0

a-O

denotesthe trace of an elementofQ withregardto K. E t1(G)%2(G-)= (g/z). a

We thenhave p = $0, e =

X:

Pwf

Then (2)

'P'

pa-l(p - 1). Cf. R. Brauer, Math. Zeitschr. 29, p. 79 (1929), p. 101. The fact used is equivalent to Wedderburn's theorem on finite division rings. The argument shows that Wcan be writtenin the fieldof the modular glth roots of unity and, therefore,in the fieldK of residue classes of integers of K (mod p). 12 I. Schur, Sitzungsber. Preuss. Akad. d. Wiss. 1905 p. 406, theorem IV. 12

941

INVESTIGATIONS ON GROUP CHARACTERS

If r is thedegreeof K(v) over K, we have m/rcharactersr(p)whichare equal to a fixed . Hence

t1 i(G)%2(Gr')=

(4)

G

'm tr (c). zr

The trace of any 15-integer is divisible by pa-1.4 for41(G)and ti(G). For co = 1, we thusfind

This holds, in particular,

gm= 0 (mod P2a-2) rz

Since the leftside is rational,p may be replacedby p (cf. lemma1*, (3)). Since m = (p(p), this gives THEOREM 1: If r is an irreducible character ofdegreez, and if r is thenumberof theng/(rz) is a p-integer p-conjugatecharacters, forany primep. We nextassume that 3 is reducible(mod 0). Then (1) showsthat #(G) 0 (mod $5). Set (5)

v=

p ,-

(3=(

/1-e

e

=1+e+..+f-1

Since et is a primitivepth rootof unity, is divisibleby A13'1 If a 3-integer is divisibleby Ad,its trace is divisibleby p.i5 For w = 0, we have cowB(G) 0 (mod $5V) and, therefore, 41(G) 0 (mod Id). As before,42(G) 0 (mod p From (5) it followsthat tr (E-) = m, and (4) now yields gm2/rz0_ (mod p2-i). Hence

LEMMA 2: If r in theorem 1 is reducible as a modularcharacter, thenthep-integer g/rzis still divisibleby p.

3. Characterswitha commonmodularconstituent We use a similarargumentin orderto study two irreduciblerepresentations 31 and 32 whichhave a modularconstituentin common. We exclude the case when the charactersri and t2 of 31 and 32 are p-conjugate. The path root of unity e may be so chosen that Q = K(e) containsboth to and t ; let I be the Galois group of Q(with regardto K. Each 3, can be writtenin the form(1), say 3,, = (ad). For at least one pair of indicesi, j, we musthave (6)

K1i =I

(mod A).

14Any $-integer X can be written in the formX =

ups, where u = 0,1, *

,

m - 1 and

the u,,are p-integersof K. We have tr (ei) = m ifp 0 (modpa), tr (et) = -m/(p - 1), if p - 0 (mod pa-i), p a0 (mod pa), and tr (it) = 0 in all othercases. This provesthe statement. 16 It is sufficient to provethisfornumbersoftheform(1 - et)em,cf.footnote14,and here it is evident.

942

RICHARD

BRAUER

in the upper left cornerof 21,, and 7Y2(G)the coLet 'y(G) be the coefficient so that in the upperleftcornerof 22, and setfl(G) = -yi(G)-2(G) efficient

0 (mod I).

8(G) = 7yi(G) -Y2(G)

(7)

Schur's relations'3give here Ej i(G)'P'y(G-'?)

(8)

=

G

Ej G

(G)jP)

92

I(G)(P)Y2(G-')

-

(Y-2(-)

a

9

Z2

0,

(7) =

or P= wherezMis thedegreeof 3,and e4 = 0 or 1 accordingas The same relationshold, when 31 = .32, and .31containsone of.its modular constituentsmorethan once. We thenhave a formula(6) withi $ j, and (7) and (8) are also true. From (8) it followsthat d

Z

(9)

G

3(G)(P)(Gl)()

9

-

Z

+

Z2

7 ,,

We set (cf. (3)) 41(G) = tr (w#(G)),

(10)

42(G) = tr (IPO(G)),

wherew and it are two integersofQ. Then (9) gives Ei (1(G)42(G1) G z0

-

Eg 9

Z

lk +

d

77P, W

p,

g

E

Z2 pa

7t

(P)i/i

When K(t,) correspondsto the subgroupV, of the Galois group I, this can be writtenin the form (11)

(12)

~

jE a t(G)%2(G') u=

=

U1 9 + Us 9 Z2 zI

= XimX2

(mod 2'.)

wherein (12) the sum is to be extendedover all pairs of elementsX1, X2 of I, forwhichX1X2- belongsto ?, . We firstchoose w = e, , = 1, where is definedin (5). Then wB(G) 0 Furthermore, (mod $3), and as in ?2, the trace41(G)ofw#(G)is divisibleby pa. we may even state 4(G)= 1, a If a trace. is it since by 42(G)is divisible pa-l, = = us by J. Finally, tr (O)m/r, divisible is 0 (mod p), because ,8(G) = Then (11) yields m. (() tr and since Q, has the orderm/r,

e

z1 + gm2/r2z2 0 (mod gm2/r

1).

If (r1, p) = 1 and (r2, p) For a = 1,thiscongruenceeven holds (mod p2). we may choosea = 1. As before,pcan be replacedby p. Since m = pa-(p we have

= -

1,

1),

943

INVESTIGATIONS ON GROUP CHARACTERS

LEMMA 3: Let ri and r2 be twocharacters of @ whichare not p-conjugatebut in common. If J' has r,.p-conjugates, and if z,, whichhavea modularconstituent is thedegreeof D, then

(13)

gl/rlz+ g/rAz2x0

(13*) g/rlzl+ g/r.z2 0 (mod p2),

if (r1, p)

(mod p), =

1

and (r2, p) = 1.

0 (mod Ate). Then t1(G) such thatw O.1' Secondly,we choosew, ^,6 0, and we obtain 0 (mod pa), 6(G) LEMMA 4: Suppose thatri and r2 satisfythe assumptionsof lemma 3. Let X and 4tbetwointegers ofK(e) whichare divisibleby$", v = pa-l. Then

(14)

+ u2g/z2- 0 (mod p2") umg/zi

whereupis definedby (12). Accordingto a remarkabove, the same argumentwillhold,if 81 = 32 and 3 containsone of its modularconstituentsmorethan once. whichcontainsone ofits modularconstituents LEMMA 5: If r is a character more thanonce,thenfor odd p, (15)

g/(rz)

0 (mod p2)

wherez is thedegreeofr, and wherethenumberr ofp-conjugates ofr is assumedto be primetop. 4. On representationsforwhichrz=

0 (mod pa)

We now prove 2: Let r be an irreduciblecharacterof degreez whichhas r distinct p-conjugates. Thenrz can be divisiblebypa onlyif z is divisiblebypa, i.e. whenr is ofthehighest kind. PROOF: Assumethatrz =0 (mod pa), z * 0 (modpa). Since thecorresponding representation8 is not of the highestkind,the blockB of r = r cannot consist of the familyof r, only."6 Hence we may finda characterr2 in B whichis not p-conjugateto ri, but has a modular constituentin commonwith ri. Then (13) shows that also r2z2 0 (mod pa). On the otherhand, z2 4 0 (mod pa), becauseB is not ofthehighestkind. Consequently,r satisfiesthesame assumptions as ri. On consideringchains of charactersof B such that any two consecutive termshave a modular characterin common,we conclude that every characterr, of B satisfiesthe conditionsraz, _0 (mod pa), z, $ 0 (mod pa), wherez, again is the degreeof r, and r, the numberof p-conjugates. It now followsfromlemma 2 that all the t, of B are modular-irreducible, and this implies that they all are equal when consideredas modular characters. To theblockB = Bxtherecorrespondsa partDx7 ofthematrixD ofthedecompoTHEOREM

16 R. Brauer, On the connectionbetweenthe ordinary and the modular charactersof groups of finiteorder, Ann. of Math., 42, pp. 926-935, 1941, theorem 8. 17 Cf. BN, ?9, (28).

944

RICHARD

BRAUER

sitionnumbersof 5. Our resultsshowthatthe matrixDx consistsofone column only,and all the coefficients are 1. All the R, have the same degree z, and if z = p~z' with (z', p) = 1, then all the r, are divisibleby paP. The numberof rowsin DA is equal to the sum of all r, belongingto the different familiesof B, and hencethisnumberis largerthanpa-. There correspondsto B a part CA = DAD\of the matrixC of the Cartan invariants.'7 This CAis of degree 1; its only coefficient is equal to the numberof rowsof DA, and therefore is largerthan paP. On the otherhand, the block BA is of type p, and then CAhas p` as an elementarydivisor,'8whichmeans that and hence the theoremis proved. CA= (paP). This gives a contradiction, 6. Evaluation of the numbersu in lemma 4 = v rational integersPi Assume that a > 2. Let e be a systemof pa which forma complete residue system (mod pa-), a > 2, and let 4: P2, ... X. *. . be a second systemwiththe same properties. We set ?2 l

Ca) =,&=;E,e r (mod v), wherep rangesover the values of Q, and a over those of d. If p It followsthat w and i1 both are then ep_ { (mod 2$V)when again v = pa-. congruentto thenumbere in (5) and hencebothare divisibleby X'v- as assumed in lemma4. Now (12) becomes

(16)

U,,=

E

X1I=X2 (mod 2")

EE p

a

CPXI-OX2

(X1 , X2 in

If eX appears, all its conjugates appear with the same multiplicity. Consequently,ifA' termsof (16) are equal to 1, and B., termsare primitivepthrootsof unity,we have u, = AB/(p - 1). The termon the right-handside of (16) is 1, ifpX1 =X2 (mod pa), i.e. ifpXiX1_ or(mod pa). Similarly,the termis a primitivepth rootofunity,ifthiscongruenceholds (mod pa-l) but not (mod pa). Now X1X2' is an elementof V, Let A,, denote the numberof pairs (p, L), p in e, L in V,, forwhichpL is congruentto one of the numbersa (mod pa), and (a in d) holds (mod pa-l) B, the numberof pairs,forwhicha congruencepL = and not A, B,' are obtained from Then but (mod pa). A, and B,,by multiplying the latterby the ordersp(pa) = v~p - 1) ofA, and hence

18 In BN, ?17, it was proved that at least one elementary divisor corresponding to a Actually, exactly one elementary divisor is equal to block of type a is divisible hy pa-a. This can be proved pa-a while the other elementary divisors are smaller than pa-.a by a generalization of the method of BN, ?21; cf. below section 8 of this paper where the same method is used (for p = 2). 19 We may understand pX, and OX2 as rational integers (mod p").

INVESTIGATIONS

ON GROUP

945

CHARACTERS

Since everypL is congruentto one of the a (mod p-l1), and since 2,, is of the order1, = v(p - l)/'r, , we have A,,+ B,, = v2(p - 1)/r,,. Then (17) becomes UP = vpA,,- v3(p - 1)/r,,and (14) takes the form P Z1

A1 + pag A2 - (p Z2

(p - 1)g + p

(p -

rizi

r2 Z2

0)g =

Accordingto (13), the last two termsare divisibleby p3a-2 2a, can be neglected. We then have (18)

(mod p2a)

and since 3a

-2

>

gAi/zi+ gA2/z2 0 (mod pa).

If p is odd, we shall need the value of A,,, only if r,,is primeto p. The field K(,,) thenis containedin the fieldK(ep) wheree, is a primitivepth rootof unity. If p $( 0 (mod v), (p in e), thenforeveryL in 2,,the numbercPL is conjugate to e.'withregardto K(g;,). If JTappears in the formcpL, so do r E-

r

r p-1

e p,

e**

P

and each of them appears the same numberof times. Exactly one of these quantitiesis of the formeJ,(a in d). Hence, fora fixedp, one in everyp of the elementsL ofV,satisfiesthe conditionthat pL belongsto d. For p 3 0 (mod v), the numberof elementsL of V, , forwhichepL has a fixedvalue, is divisibleby pa-l. Hence, in the case (r,,, p) = 1, we have (19)

A, _ I,,(v- 1)/p

V(p

-

1)(v - 1)/r,,p_ p2/r

(mod p'l)

The congruences(18) and (19) enableus to provethefollowinglemma: LEMMA 6: Let p be odd. If ?, and 2 satisfytheassumptions oflemma3, and if 4 0 0 then (mod pa-l) r2 (mod p). zi PROOF: From theorem 2, it follows that r1 $ 0 (mod p). Assume that r2 0 (mod p). If we choosefora the smallestvalue forwhich P and ?2 belong to the fieldK(e) obtained fromK by adjoining a path root of unity e, then r2 mustbe divisibleby pa-'. Theorem2 showsthat we have Z2$ 0 (mod pa-a+l) the second termin (18) is divisibleby pa. Since (ri, p) = 1, and, therefore, we may use (19) forA = 1 and findthat the firsttermin (18) is divisibleby pa-l only,whichgives a contradiction. Hence r2 4 0 (mod p) as was stated. For p = 2, we choose a _ 3. Here, r,,is a powerof 2. If r,,= 2', 1 < j < a - 2, thefieldK(e) has exactlythreesubfieldsofdegree2' over K. One ofthem, say r'?', is obtained by adjoining a 2j+lth root of unity to K. We denote the othertwo by rp5)and r 2), and set K(e) = r?) I, K = rp1). Then, by an argumentsimilarto the one used forodd p, we may prove (20)

A,,

(21)

A_=2

0 (mod 2a-1-J), if K(t,,) = r'?'. (mod 2a-1-i)'

i

= p'l)

or p(2)

LEMMA 7: Let p = 2, and supposethatP, and t2 satisfy theassumptionsoflemma 3. If riz1 0 (mod 2a1) and if K(P,) is notoftheformK(i) wherei is a primitive

946 2ftb

RICHARD BRAUER

rootofunityuithj 2 2, thenr2z2 0 (mod 2a-'), and K(t2) is notoftheform

K(f).

PROOF: From the assumptions,theorem2, and (21), it followsthat Aig/zi thenis also divisible is divisibleby 2a-1, but not by 2a. Accordingto (18), A2g/z2 by 2a-1 but not by 2a. If we set r2 = 2' and use (20), we see that K(?2) cannot If K(v2) = r and K(?2) = r2) we be one of the fieldsr'0), (j = 1,2, .). use (21) and concludethat z2 is divisibleby 2a-l-2,whichimpliesr2z2 0 (mod 2a-1).

6. Representationsof a degree whichis divisibleby pa-l 0 (mod pal). If z is divisible Let v be an irreduciblecharacterof degreez and we willexcludethiscase in thefollowing. by pa, thenv is ofthehighestkind,20 We shall firstassume that p is odd.2' The block B to whichr = t belongsis and not of the highestkind. Then B does not consistof the familyof ? only,22 finda charactert2 whichis not p-conjugateto ? = P, but has a we may therefore modular constituentin common with ?, (cf. the analogous argumentin ?4). If zAis the degree of A,, and r,,the numberof p-conjugate characters,then theorem2 and lemma6 show that ri 4 0, r2 4 0 (mod p). In (13*), lemma3, the firsttermon the leftside is divisibleby p but not by p2. The same must (mod pal), so that t2 hold forthe second term,and hence we must have Z2 satisfiesthe same assumptionas I. Continuingin this mannerwe see that the This gives degreeofeverycharacterofB is divisibleby pa-l. THEOREM 3: If thedegreeof one character of a blockB is divisibleby pal, the of B. same is trueforall characters In the notationofBN, the blockB is of the typea - 1. If t, and t2 are again two charactersof such a block B whichare not p-conjugatebut have a modular constituentin common,then on multiplying(13*) by rlr2zlz2/g= 0 (mod pa-2) the congruenceriz1+ r2z2 0 (mod pa) is obtained. Now any two characters OX, ;,,can be joined by a chain PX, ,p, . .*. , ;,,of characters,such that any the chain a terms of have two consecutive modularcharacterin commonwithout It follows that for any I, ofB, we have r,,z, -?riz1 (mod pa). beingp-conjugate. 0 and is for each Au onlyone of the signscan be used. Since riz1y (mod pa) p odd, Hence in two THEOREM 4: The characters ofa blockB of typea - 1 can be distributed subsetsB' and B" such thateverycharacterbelongsto exactlyone of thesesubsets. wemayfinda If z,.is thedegreeofa character ofB, and r,,thenumberofp-conjugates, rationalintegerN such thatr,,z, N (mod pa) for all X, of B' and r,,z,, -N (mod pa) forall X,. in B". If I, and t, are notp-conjugateand belongbothto the in common. same subsetB(', thentheyhaveno modularconstituent Finally,lemma 5 can be applied and yields =

=

'I

Cf. BN, Part II.

21The case p = 2 will be treated in ?8. 22 Cf. footnote 16.

947

INVESTIGATIONS ON GROUP CHARACTERS THEOREM 5: If r belongsto a blockB of typea 1. onlywiththemultiplicity modularconstituents

-

1, thenit containseach ofits

7. The matrixD, correspondingto a block B of typea

-

1

NOTATION: B = Bxis a block of type a - 1 consistingof the ordinarychar2 *- , p, . The degree , , and the modularcharacters(pi,~02, actersPi 2, ofX is zM,the numberof p-conjugatesr, . There are w familiesof p-conjugate characters?, in B, and the G are arrangedso that 1, 2 , ***, DWlie in different families,while the charactersof the subset BA (theorem4) come beforethe charactersoftheothersubsetBk'. For p-regularelementsG of(M,we have

(22)

Bs(G) =

E

dppv(G),

= 1, 2,

*.. ,

x).

,y) and denote the matrix We set Dx = (dp), (4 = 1, 2, ***, x; v = 1, 2, occupyingthe firstw rowsof Dx by DA, i.e.Dx = (d,,), (4 = 1, 2, ... , w; Y = 1, 2, ***, y). The matrixDA has the same rows as A , the uth row of A appearingr, times in DA, (1i = 1, 2, ... , w). The theorems4 and 5 give at once 1, one in a row THEOREM 6: Everycolumnof Dx containsexactlytwocoefficients to a to a characterPx of BA, and theotherin a row corresponding corresponding in thecolumnare zero. character A. ofBk'. All theothercoefficients We also can prove easily suchthat(DA = 0. THEOREM 7: Let t = (%i , * , ( bea rowwithw elements Thent is a scalarmultipleoftherow(61, 62, * , &W), whereby = 1 if X belongsto B and , = -1, if X belongstoBk'( = 1, 2, ** , w) ofB can be joined by a chain PROOF: Two charactersP, of charactersUp, (p = 1, 2, ... , w), such that two consecutivecharactersDi, ji of the chain have a modular characterin common. Then Pi, , belong to different subsets BA, B' (theorem4). There must be a column of A which 1 in the ith and the jth row. Then,{A = 0 implies containsthe coefficients 31, (%, * , t.) = 3(1,(i , and we obtain successively = 3 t = ,SW) whichprovesthe theorem. The rank of A is y.2' Hence we have COROLLARY 1: The numberw of familiesin B and thenumbery of modular characters bytheformula ofB are connected (23)

w = y+1.

We furtherstate whichare p-conjugateto COROLLARY 2: If I, is the sum of the r, characters ... = = then 1, 2, ofG ofO. , w), (,i 61l1(G) 6,,8A(G)forany p-singularelement Indeed, the numbersp1(G), p2(G),.- , w(G) forma solution of {A = 0, (cf. equation (11) of the paper mentionedin footnote16). Similarly,equation (6) of the same paper gives 23Cf. BN, ?9.

948

RICHARD BRAUER

3: Let d,,>be the higherdecomposition numbers,i > 0, and set d', wherethesum extendsoverall ther,,valuesp forwhich is' p-conjugate to , Then6adI = = 0 For p-regularelementsG, the formula(22), togetherwith (61, S W), gives G of (, we have COROLLARY 4: For p-regularelements COROLLARY

d,=

X 6?,(G)

(24)

J-1

= 0.

In particular, forG = 1, this gives to

(25)

E J 1

IZ = 0.

The matrixDA has w minorsof degree y = w - 1, and these, if properly arrangedand taken withsuitablesigns,forma solutionof M = 0. Let A be a fixedminor. Then, accordingto theorem7, every minorhas the value dA. The minorobtainedby removingthe lth columnof A will appear r1r2.*.- r/r On times as a minorof Dx. We now formthe determinantof CA = D.D. the one hand, thisdeterminanthas the value p.24 On the otherhand, its value is the sum of the squares ofall the minorsofD&. This gives E

j-1

A2(rir2...

rl/r,;)= p.

Consequently,we musthave A = d 1, and we find COROLLARY 5: The numbers r,,satisfytheequation (26)

r r2 ... rw 1 + \r,

r2

+ **

r

p = P.

The numbersr,,are divisorsof p - 1; (26) showsthat any two of themare relativelyprime. We associate w distinctpoints P,, J2, ***, Pw,with the characters 2, D2, . , ,w; we join Pi and Pi whenfi and Us have a modularcharacterin common. The linear graph thus obtained characterizesthe matrix A completely(apart of the columns). We prove fromthe arrangements COROLLARY 6: The linear graph associated witha blockB of typea - 1 is a tree.

PROOF: This followsfromthe facts that the graph is connectedand has one morevertexthan it has edges. Each pointP, whichlies only on one of the edges correspondsto a row of J5 which contains only one coefficient 1, and this means that I, is a modularirreduciblecharacter. 24 The determinant of C is a power of p, cf. R. Brauer, On theCartaninvariantsofgroups of finite order, Ann. of Math. 42, p. 53, 1941. It follows from the result of footnote 18, that the exponent must be ).

INVESTIGATIONS

ON GROUP

949

CHARACTERS

7: EveryblockB of typea - 1 containsat least twocharacters;, COROLLARY and whichare modular-irreducible whichare notp-conjugate characters. 8. The case p = 2 For p = 2, several of the precedingproofsdo not hold. We shall now treat kindofargument. thiscase, usinga different 0 (mod 2a1) but z # 0 Let r be an irreduciblecharacterof degreez wherez (mod 2a); herea is again thehighestexponentforwhich2a = pa dividesg. From theorem2, it followsthat s is p-conjugateonlyto itself,i.e. r = 1, and v belongs to K. Since K is not oftheformK(e) wheree is a primitive2ith rootofunitywith j ? 2, we may apply lemma 7, ?5. It followsthat if the characterA, has a modular constituentin commonwith tj then rAzA 0 (mod 2a1), (where z, denotes again the degreeof A,,,and r, the numberof 2-conjugates). Further, K(,,) also is not of the formK(e). Therefore,lemma 7 again can be applied to in commonwithA,without A, and any characterwhichhas a modularconstituent 0 (mod 2a1) beingp-conjugateto ,. We finallysee that the congruencerz, is trueforany characterA,of the block B.to whichv belongs. Let tj, 2, * .*, be the ordinarycharactersof B, and p , 'p2, * * , ' the modular charactersof B, and set ?2

(27)

2 z,

z=

(z', 2)

=

1.

It now followseasily that (28)

2a-1-T

Suppose that T1 = m is the smallestof the numbersTi, r2, We considerthe sums

, rx.

ko

S=

(29)

...

E

whereG, rangesover a systemof representatives forthe 2-regularclasses of 65, and whereg9denotes the numberof elementsin the class of Go. The value of = ,, is an algebraicinteger,it follows S,,,is a rationalinteger. Since g,,;,(GK)/z, easily that S,, is divisibleby z, and hence by 2'. On the otherhand, we have25 W$ CAW1. moduloa primeideal divisorof 2. Hence Ssv =

Z wZ EW~gY(G-1)

Z=

2 cou -1) M(G,

= Z S1,(mod2V+i).

On applyingthe same argumentto S1, = Sl, we arriveat (30)

S,,=

(31)

S-S_ =

25

BN, ?9.

0 (mod 2m), _

ZZP ZI

I,

S1

(mod 2T+1).

950

RICHARD

BRAUER

If the blockB is the Xth of the blocksB1, B2, ... of (M,and ifD% = (d,), Con= (cV) are the parts26of the matricesD and C correspondingto the block BA, then (22) gives forthe matrix(S,,Y),, = (I (S~v~v

it E

()

g.d6pfp(GK

POl

,vs

=

Dx(gC-)D

'

27

The relation(31) can be writtenin theform (32)

(So')'

= Dx(gCAj)D1= Si, -

Z1

+ H #

whereH is a matrixin whicheverycoefficient in both the ,th row and Mth column is divisibleby 2"M+'. The firstmatrixon the rightside in (32) has the rank 1; in both the/th rowand Mth columnis divisibleby 2T. everycoefficient We now proceedto discuss the elementarydivisorsof (See), choosingthe ring of all 2-integersas the underlyingdomain. If C,,has the elementarydivisors el, e2, ... ., eywithe, = 2v, then

|CA |

=

ele2 ...

ey

=

2e1+e2+.

.+ey

28

The elementarydivisorsof gC-jare 2a"y, ... , 2a-, 2aE1 and, sincethe columns of DAare linearlyindependent(mod 2),29the elementarydivisorsof Dx\(g9C-)D are given by (33)

2

i, 2

2d-0. .

..

a)

0.

On the otherhand, the rightside of (32) can also be used fora discussionof these elementarydivisors. If M is a minorof degreej of (SyV)involvingthe A, i, then a simple computationshows that M is divisible by rowsA, I, * 2 to the exponent E Tr + j - 1, (A = Al, M2, ...*, ) because of the properties of the matriceson the rightside of (32). If the j characters M(G), (A =Al, , Ai), are linearlydependentfor2-regularelementsG of 6, thenM = 0, 2 *... as followsfrom(29). Let us now choose a maximal systemof charactersof B which are linearly independentfor2-regularelements6. Such a systemconsistsof y characters. We firsttake as many charactersas possible with a, = m, then as many as possiblewith Tp = m + 1, etc. We may assume that the characterschosenare , t, . Let 3p be the number of characters l, *.. , Py for which rT, = ., m + p- 1 (p = 1, 2, ... Is), such that B1 > 0, 02 _ ... y0,$-1 -? , 0 > ?. Then rl = m,

TY = m + S-1

m < T < m + sl-

for , = I, 2, ... , y.

26BN, ?9, (28). 27 Cf. BN, (21). 28 The determinant is actually a power of 2, cf. footnote 24. 29 BN, ?19.

951

CHARACTERS

ON GROUP

INVESTIGATIONS

Of course, we have 1+ 02 +

+ f-

=

Y.

is divisibleby 21with It now followsthateveryminorofdegreey of (SMV) l-

1Mm+

2(m+1)+

S

+

I = my-1

*--

2+

++t3M(m+s-1)+31+

+

S,

o3..

Ej ar=l

Similarly,the minorsof degreey -- 1 are divisibleby 2" with /=tlm

+ 02(m +)

+fl3Oi(m + s-2)

+

+ (O.,- 1)(m + s-

' = m(y -1)-1-s

) +

1+

OS-2,

+

+ E odd. a=1

On comparingthis with (33), we find ya- (1

(y - )a - (C2

+

+

E2

C-3 +

From (28) we see that rjr2... (a-1-m)lI

+

(a-2-m)f32+

my- 1 + Ea~

+

+

V)

**+

(V)

_-! (Y-

1)m

-1-S

, 3 + E age.

ryis the powerof 2 withthe exponent + (a

-s-m),3 = (a-m)y-E

a,3.

Therefore,the two inequalitiescan be writtenin the form (35)

|CA = eie2*.. ey < 2r1r2*..

(36)

e2es... ey?

ryX

2m-a+S+lrlr2...

ry

DVcan be 2-conjugate.Lettingw No two of the characters Pj, D2, ..-, be the numberof familiesof 2-conjugatecharacterswhichbelong to the block in such a mannerthat B = BA, we have w > y.3' We arrange ,+ , ... , ... the Denote in families. lie different all IAth rowof DAby bM, P D1, 2, *, and let t2

30This holdsin the case y = 1 also; we have heres = 1, 0, = 1, and e2 + be taken equal to 0, and e2 ... eyequal to 1. 31Cf. footnote 16.

+

ey

is to

952

RICHARD

BRAUER

so that DX contains the firstw rows of Dx, and T the firsty rows. Then DX has the same rowsas DA, the 14th row of Dx appearingri timesin Dx. Let Al, be the minorsof degreey of DA ; the determinant T I appears rlr2 ... r, A2 , ... times among these minors. Consequently,we have (37)

ICxI

|DADAI = ZA2

=

> rlr2

. .

ryj 7 12.

Here, I T l $ 0, since P,(G), ***, Dy(G)are linearlyindependenteven if G ranges over the 2-regularelementsonly. If all the minorsof degreey of DAexcept I T I vanish, then by+,= 0, which is impossible. Hence the inequality sign must hold in (37). On comparing(35) and (37), we find I T I = ?1,

(38)

and, since everyr, and e, is a powerof 2, we have (39)

ICA |

2rlr2.. .ry

Then (36) yields el > 2

(40) Because of (38), we may set

bY= hklbi+ **+

h,,vbv,

wherethe h, are rationalintegers. Then Dx = HT withH = (hp). It follows that C- = T'H'HT, and this shows that H'H has the same elementarydivisors in the yth row,yth columnof H'H must be as Cx. In particular,the coefficient divisibleby el, and hence, by (40), we have 2

(41)

E

v 1

h1V 0 (mod 2a-8).

When t, is 2-conjugateto Dy, then b, = byand ha = 1. There must exist characterst, whichare not 2-conjugateto Dyand for which h $ 0. 32 Using (34), we obtain ,1

hV >

ry = 2a1v

-

and, therefore,(41) yields 2

(42)

EhZ > 2 v1

= 2ry.

Considernow the minorAXconsistingof the rows1, 2, * , y - 1, P. Its value is hVy I T I = -hv. If D, is 2-conjugateto one of the characters l , ***, , 32 In ?5 of the paper mentioned in footnote 16, it is shown that it is impossible to find a linear combination of the columns of D>x , such that exactly one row contains a term # 0. This implies the statement.

INVESTIGATIONS

ON GROUP

953

CHARACTERS

then the minorvanishes. It is easily seen that there are r1r2... ry-1minors of DA with the value ?h~, ; and (37), (39) and (42) implythat (43)

2rr2

r=I

CAI = X

rr2 r... 2rr

h2

?

*

Since actually the equality sign holds,it followsthat those minors,whichhave not been taken into account, must all vanish. If y > 1, it followsthat the minorsformedby means of the rows 2, 3, ***, y, v' must vanish unless D, is 2-conjugateto t, . The value of such a minoris ?hAi, and we obtain a contradiction,since theremust exist charactersi, whichare not 2-conjugateto ?, forwhichh., # 0.32 in It followsthat y = 1, i.e. that DA has only one column. The coefficient the firstrow must be 1, because of (38). We set hay= h^, 1\ D=

h2

The relation(43), (28), and (34) then imply (44)

1 + h2

+ .-

+ h2 = 2ry= 2r, = 2a-m.

Since ?, has degreez1 = 2mz', the degreez2 = 2124 of t2 must be equal to h2z, whichshowsthat h2 is divisibleby 212 m. The leftside of (44) is at least equal to ri + r2h, since ri terms1 and r2termsh2appear. Because of (28) and (34) we have 2a-m > 2~" _ ri + r2h2> 2a-1-m + 2a-1->222T2-2m We readilysee that w = 2, T2 = m, ri = r2 = 2a-1-m, h2 = 1. The block BA consistsof two families. All of its charactershave the same degreez1 = 2mz'.33 Since it was assumed that B containsa characterof degree2a-1z,, ((z', 2) = 1), = 1, it followsthat m = a - 1 and hence ri = Do = (1).

This provesthe theorems3, 4, 5, 6, and 7, and the corollariesof ?7, forp = 2. 9. Applications THEOREM 8: If 05 is a groupof orderg = paq brc, (p, q, r distinctprimes)uith a < 2, then65possessesirreduciblerepresentations (3 besidesthe 1-representation [1] whosedegreez is a powerof q, and also irreduciblerepresentations 31 # [1] whosedegreez1 is a powerof r. PROOF: Considerthe p-blockB of representationswhich contains the repre33 All the facts derived so far hold for any block B which contains a character t. with the following properties: (1) t, is not of the highest kind. (2) rz,= 0 (mod 2a-1). (3) K(?') cannot be obtained fromK by adjoining a 2ith root of unity with j _ 2.

954

RICHARD

BRAUER

sentation[1. This B is not of the highestkind,and if a = 2, it is not of the = 1. It followsfromtheorem3 that the degreesof all representatypea-1 B tionsof are primeto p. We apply the relation34E MR)M(S) = 0 whereju rangesovertheindicesbelongingto B, and whereR is p-regularand S is p-singuto the character lar. For R = 1, we have ,(R) = z, . The termcorresponding [1] is 1. At least one otherterm ;,(1)%(S) = z,~,(S) must be primeto r, and then z, must be a power of q. In the same mannerwe see that a character $ [1] has a degreez, whichis a powerof r. 96 If @ is equal to its commutatorgroup@', thenz, and z, cannotbe 1 since [1] of @. In particular,this will be so, if G is is the only linear representation simple. of one of Assumethat rc = 4 or rc = 3. Then @ musthave a representation we the degrees2, 3, or 4. Since all lineargroupsof thesedegreesare known,35 obtain easily THEOREM 9: The onlysimplegroupofan order4paqb, (p, q primes)witha ? 2 groupW6oforder60. The onlysimplegroupsof an order3paqb, is thealternating (p, q primes)witha < 2 are thegroupsAs = LF(2, 5) of order60 and thegroup LF(2, 7) oforder168. representations 10. On $3-similiar in an coefficients of @ with 53-integral Let 3 be an irreduciblerepresentation algebraic numberfield Q, where $3 is a prime ideal divisor of p. If .81 is a in the same field Q, it may with 15-integral coefficients second representation modularreprehappen that 3 and 31 are similar,but that the corresponding sentations8 and 81 are not similar. Indeed, if (45)

P-13P = Si ,

which are not all of P are 83-integers we may assume that the coefficients divisibleby $3. Going over to residueclasses mod $3 (whichagain will be indicated by a bar) we obtain SP = PS, and P $ 0, so that 8 and 81 are intertwined. However,this proves 83 & onlyif the determinantof P is not divisibleby $3. We shall say that 3 and 31 are 1-similar,if P in (45) can be chosen in accordancewith these conditions, and a determinant0 0 (mod $). The class of coefficients i.e. with 83-integral all representations81 which are similarto 3 and have 1-integralcoefficients representations. thus breaks up into subclasses of 53-similar of @3 in the fieldQ of residueclasses Conversely,if ? is any representation 81, whichis (mod ), whichis similarto 3, thenwe may finda representation = $--similarto 3, such that 381 ?. 34R. Brauer and C. Nesbitt, University of Toronto Studies, Math. Ser. No. 4, 1937, theorem VIII. 35 H F. Blichfeldt, Finite collineation group8,Chicago 1917.

INVESTIGATIONS

ON GROUP

955

CHARACTERS

In the general case of (45), we may assume that P appears in the normal formof the theoryof elementarydivisors (with regard to the domain of all representations.We thus may ?-integers)ifwe replace 3 and 31 by ?3-similar set P lr=

(46)

?r7'I9a)

=

(W8K6+X,8lImx)gX,

whereml > 0, m2 > 0, . **, m8_1> 0, m. > 0 are rationalintegers,Imis the whichsatisfiesr _= 0 (mod ?3), unit matrixof degreem, and 7r is a ?3-integer If we set or 0 (mod $2).

(11 t12

(47)

3

2122

2121

8

I

s18\

(**

21,

- -*

188

*.

212

/311 =2321

012

...

.18

022

..

028)

2 \81

82

...*

8.

columns,and Q3,xhas m, rows and mx where 2Kghas m1-K+1 rows and m8_x+l columns,then (45) gives (48)

7r

8+1-ic8

-l

=

This shows that in (47), all the termsabove the main diagonal in 3 and 31 are congruentto 0 (mod ?3). In the modular sense, 3 and 31 split into the (reducibleor irreducible)constituents fl

=

288.

2!922 =

Z8-1,8-1

X *

2 =88

=

If s = 1, then 3 and 31 are $-similar. We certainlyhave this case when 3 in Q. is modular-irreducible 10: If therepresentation THEOREM 3 remainsirreducibleas a modularreprewhichis similar coefficients, theneveryrepresentation sentation, 3'1with2$-integral to 38,is ?3-similarto 3 . Remark: This theoremcan always be applied if the degreez of 3 is divisible by pa. of a representation $-similar Further,if it is knownthat no fixedcoefficient to 3 is divisibleby ?' for everyG in 65,then (48) showsthat s < 1. 11. On the arrangementof the modularconstituentsof an irreducible 3 of typea - 1 representation We shall say that the algebraicsplittingfieldQ of the representation3 is a if Q is normalover the fieldof rational normalsplitting fieldof leastramification, of the fixedrationalprimep is the ramification numbersP, and if the orderof obtained by adjoiningthe characterv of 3 same for U as forthe subfieldP(?) fromlemma 1. Denote by K the follows fields Qi to P. The existenceof such of ideal divisor $3 p. Then fieldof inertiaof the prime ?r = P, (p) = pq, with (p, q) = (1),

956

RICHARD

BRAUER

wherep is a primeideal divisorofp in K, and wherer is the numberofcharacters whichare p-conjugateto A. As is easily seen, the degreeof K(v) over K is r, and hence K(v) = Q. We now prove ofdegreez withz 0 (mod pa1) representation LEMMA 8: Let 3 bean irreducible G lie in the such thatall thecoefficients a,(X(G)of thematrix 3(G) representing field Q of least ramification.For each pair (K, X) thereexistsa normalsplitting groupelementGo such thatakx(GO)and ax,(Go) are not bothdivisibleby I. For Go suchthateitheraX(GO) is nota 3-integer each (K, X),thereexistsa groupelement or ax,(G*) $ 0 (mod 32). 0 (mod A) for every G in 5. PROOF: (a) Assume that aKx(G) -- ax,(G) WVenow apply the methodof ?2 settingj3(G) = a,,(G), -y(G) = ax,(G),w = 1. We have here (r, p) = 1 because of theorem2, and hence r p- 1, m = V(p). 0 (mod p), and (4) gives a contradiction. Then 41(G) =O0, 2(G) (b) If ax(G) is a $-integerand axk(G) 0 (mod 32), we multiplythe Kth row by an elementir =0 (mod A), forwhichir 0 0 (mod 2), and divide the Kth columnby or. The similarrepresentationthus obtained satisfiesthe assumption of the part (a) of this proof. Therefore,we again obtain a contradiction. of 3 are $-integers. It followsat once Suppose now that the coefficients fromlemma 8 that for the correspondingmodular representation3 all the are irreducible. This impliesthat the arrangementof the Loewy constituents36 irreducible(modular) constituentsof 3 is uniquely determined. coefficients and are similar,then Further,if 3 and 31 both have 2$-integral it followsfromlemma8 that s < 2 in the notationof ?10, (46), (47). If s = 1, 3 and 31 are 3-similar.If s = 2, we have by (48), (49)

1( 2{1

)r-i 2 )

P

7r(

m2)

3

(2i22

721

constituentsof 31 are, of course, the same as those The modular-irreducible of 3, but we see that the arrangementin which they appear is a cyclic permutationH of the arrangementin 3; H $ 1. Since all the modularconstituents of 3 are distinct(theorem5), it followsthat 3 and 31 are not $3-similar. coefficients breaksup intotwomodular If therepresentation3 with2$-integral constituents,then we may set 3 in the formgiven by the firstequation (49). If we defineP, 31 by the otherequations (49), then 31 is similarto 3. This shows that any cyclic permutationof the modular constituentsof 3 can be effectedby a transitionto a similarrepresentation318 0 (mod pa THEOREM 11: Let 3 be an irreduciblerepresentationofa degreez 'whosecoefficients are S3integersof a normal splittingfield Q of least ramification. constituents Let j be the numberof modular-irreducible of 3. The class of all 1)

are $-integers of Q and which are similar representationsof 5, whose coefficients to 3, splits into j subclasses of 9$-similar representations. In all representations 36

Cf. e.g. R. Brauer, Trans. Amer. Math. Soc. 49, p. 502, 1941.

INVESTIGATIONS

ON GROUP

957

CHARACTERS

appear in thesame arrangement.The ofa fixedsubclass,themodularconstituents

j possiblearrangements in 83by thej cyclic are obtainedfromthearrangement

permutations. It is clear that lemma 8 will,in general,not remainvalid if Q is replacedby an extensionfield. Then theorem11 also will not hold. mentionedin lemma1; Let Q be the normalsplittingfieldofleast ramification T is a root of unityof an orderprimeto p. The splittinggroup of the prime ideal $ containsthe substitutionT whichtransformsT into Tr and leaves the th rootsof unityfixed. Then T transforms any (ordinaryor modular) charpa p~~~~~~~~~~~~ acter x into a conjugate characterXT. It is easily seen that if an irreducible splits into the modular characters sp X, spa ment, then AT will split into the modular characters r character

v

... * * Xsp

in this arrange-

T. Assume again that the degreez of v is divisibleby pa-l wherep is odd. If one of the constituentsso of r is leftinvariantby T',then ? and AT have a modularconstituent (p in common. Since they have the same degree,they must be p-conjugate (cf. theorem437). It thenfollowseasily that all the modular constituents of v willadmit the substitutionTV. The same will hold forall the charactersX whichhave a modularconstituentin commonwith A,and finallyforall the A i,

*.

*,

of the block B of P. Hence THEOREM 12: Let B be a block of type a - 1, and denote by T the substitution which replaces the gith roots of unity by theirpth powers but leaves the path roots of unity invariant. If a power TK of T transformsone of the modular charactersof B into itself, it transformsevery modular character of B into itself; every ordinary characterv of B is transformedinto a p-conjugate character by T' (p odd).

12. Real charactersof typea

-

1

representationS3* To everyrepresentation23,therebelongsa contragredient of the same degreez; the charactersv and A*of 03and 3*are conjugatecomplex, A= A. If z is divisibleby pa-l, p odd,38then eitherv and A* are p-conjugate, or they have no modularconstituentin common,as followsfromtheorem4.37 Consequently,if ? containsa modularcharacter(p with(* = (p,then r and * mustbe p-conjugate. It is easily seen that,ifthe modularconstituentsof 03are (49)

al

2

X*

i

in this arrangement, those of S3*are (50)

a1 2

Ointhe otherhand, using a suitable splittingfieldQ. we easily findthat in any representation p-conjugateto 23,the modularconstituentsappear in the same arrangementas in S3. It now follows fromtheorem11 that if ? (G) is real forp-regularG the constituents(49) and (50) must be the same apart froma 37 The 38 The

number of p-conjugate characters is the same for both characters following theorems 13 and 14 are trivial for p = 2; cf. ?8.

958

RICHARD

BRAUER

then a, 0*+i- where we set0i+M = cyclicpermutation. If a *, For odd j, therewill be exactlyone value of v forwhichv =_ p + 1 - v (modj), . For evenj, we have eithertwo values v or no value P. Hence i.e. a, ~-'* 13: Let ,3 be a representation of degreez _0 (mod pa-l) withthe THEOREM characterP. If ? is notreal for p-regularelements,noneof themodularconstituentsofv is real. If v is realforp-regularelementsand containsj modularconstituis real; if j is even, ents,then,for odd j, exactlyone of thesemodularconstituents eithertwoor noneof themare real. If, in particular,all the modularcharactersof the block are real, theneach r contains one or two modular constituents. In the tree correspondingto the block B of A,each vertexlies on at most two sides. Hence of a blockB of typea - 1 are real, 14: If all themodularcharacters THEOREM treeis an open polygon. thenthecorresponding .

UNIVERSITY OF TORONTO TORONTO, CANADA

ON GROUPS WHOSE ORDER CONTAINS A PRIME NUMBER TO THE FIRST POWER I.* By

RICHARD BRAUER.

To the memory of I. Schur.

Introduction. Since the foundation of the theory of group characters by Frobenius in 1896, the characters of many special groups (Mof finite order have been determined. In, these examples regularities appear which, it seems, are not explained by the general theory. Their occurrence depends on certain assumptions concerning (; it would be extremely difficult to determine the exact nature of these assumptions and their consequences from the known examples. In order to derive new properties of group characters we can use the theory of the modular representations of a group as a possible approach, particularly when we are interested in questions of a somewhat more arithmetical nature. The relationship between the ordinary and the modular theory can be described in the following manner: Let K be an algebraic number field. The group (Mdefines an associative algebra, the group ring r, and the investigation of the structure of P is the main subject of the ordinary theory of group characters. If we consider only elements of r which are linear combinations of group elements with integral coefficients we obtain an "integral domain " J in r which is, in general, not a maximal domain. We may study the group ring from an arithmetical point of view; in particular, we are interested in the prime ideal divisors of a fixed rational prime p. The study of this question leads to the modular theory of group characters. There is a close connection between the algebraic and arithmetical structure of r which implies that the theories of ordinary and modular group characters are interrelated. Thus the "modular" theory provides a new approach to the "ordinary " theory of group characters.1 In the case in which we are interested the prime is a discriminant divisor, i. e., p divides the order g of the group (M. We shall assume in this paper that g contains p to the first power onlv, i. e., ()

g ==pg'

(g',P)=1

* Received April 4, 1941; Presented to the American Mathematical Society April 7, 1939. 1 For this point of view, cf. R. Brauer, Proceedings of the National Academy of Sciences, vol. 25 (1939), p. 252.

401

402

RICHARD BRAUER.

To be sure, this assumption is of a very restrictive nature. It may, however, be mentioned that Dickson's list 2 of 78 simple groups of an order smaller 'than 10' contains only one group for which there is no prime p such that (*) holds. The simplification imposed by the assumption (*) is threefold. In the first place, the arithmetical structure is far simpler than in the general case, and the modular theory as developed in a number of previous papers 3 gives complete results. Furthermore, the group-theoretical situation is greatly simplified compared with the general case because of the simple structure of the p-Sylow-subgroup of (M. Finally, we have simplifications of a somewhat more analytical nature; in certain inequalities, as they occur in the theory of group characters, the terms are small so that it is easy to handle them. Let K be the number of classes of conjugate elements in (M. The characters of (Mcan be arranged in the form of a matrix Z of degree K. Each row correspondsto a fixed character, each column to a fixed class of conjugate elements. We arrange the columns so that the classes with p-regular elements 4 come before the other classes, and we arrange the rows so that the characters of a degree prime to p come before the characters of a degree divisible by p. Thus if (**)

(z

Z1 kZ3 p-reguflar classes

degrees prime to p,

Z2 }

Z4)

degreesdivisibleby p,

p-singular classes

0. Our main result states that the matrix then we have first of all that Z4 of the normalizer 9 of $, the p-Sylowstructure the is determined by Z2 only some ? signs remaining undetermined. In particular, subgroup of @M, if we replace (Mby 9N, the matrix Z2 remains essentially the same, only the signs of some of its rows have to be changed. In the case of the group 9, the second row in (**) is missing. This implies that the number of characters of (Mwhose degree is prime to p is equal to the number of classes of conjugate elements of W. We obtain these results in a somewhat more explicit form (cf. Theorems 1, 2, 3, 4, and 5). We show that the values of the characters of (M for p-singular elements can be written down (apart from ? signs which remain undetermined), provided that the characters of a certain subgroup I are known. This N is a subgroup of the normalizer 9U of the p-Sylow-subgroup; =

2 L. E. Dickson, Linear groups, Leipzig

(1901), pp. 309-310.

3 Cf. the papers mentioned in the bibliography. 4

Cf. the list of notations at the end of the Introduction.

GROUPS WHOSE

ORDER CONTAINS

A PRIME

TO THE FIRST POWER i.

403

the order of B is prime to p, and hence the construction of its characters can certainly be considered as a more elementary problem than the corresponding problem for (M. As a matter of fact, in most of the applications of the theory developed here the group B is of a very simple structure, and its characters can be written down at once. Very often, B is a cyclic group. From a knowledge of the matrices Z2 and Z4 in (**), some information concerning Z1 and Z3 can be obtained. First, the orthogonality relatiofis for group characters can be applied. Secondly, we may obtain the values (mod p) of the characters for those classes whose elements commute with elements of order p. In particular, this gives a set of new conditions for the degrees of the characters which in many cases is sufficient for the computation of the degrees. Finally, from our knowledge of Z2 and Z4 we obtain conditions for the multiplication of the characters which also can be used to gain new information concerning Z1 and Z3. In many cases it is possible to derive the complete table of characters in this manner; the following examples may show the significance of the method. Heretofore it has not been known whether more than one simple group exists 6048. Assuming that (Mis a simple group 5616 and 9g for the orders 9g of one of these orders, we may find by elementary methods the structure of 9XY, where p -

13 in the first case and p-= 7 in the second case. It turns out

that the results obtained are sufficient to construct the complete table of group characters in either case. From this table we can derive the modular group characters of (Mfor the prime p = 3 (here the theorem as given above does not hold since 3 divides the order of the group to a higher power than the first). The values of the modular characters show that a simple group (Mof order 5616 must have an absolutely irreducible representation of degree 3 in the Galois field GF(3). Hence it is a subgroup of LF(3, 3), and since S and LF(3, 3) have the same order, we find S = LF(3, 3). Similarly, it can be shown that a simple group of order 6048 is isomorphic with HO (3, 9) since it must have a unitary representation of degree 3 in GF(9).5 This shows 6 that the list of known simple groups is actually complete up to the order 6232. Some applications of our results of a more general nature have been given without proof elsewhere.7 r It is not possible to give the details of the computation in this paper. 6 F. N. Cole, Bulletin of the American Mathemratical Society, vol. 30 (1924), p. 489. Actually, it follows from a theorem of Frobenius Sitzung.ber. Preuss. Akad. (1895), p. 1041) that no simple group of order 6232 can exist, since 6232 = 2 . 2 * 2 . 19 . 41 is a product of five primes. It would be very easy to replace 6232 by a larger number using the methods of this paper. 7 R. Brauer, Proceedings of the National Academy of Sciences, vol. 25 (1939), p. 290.

404

1t1CHARD BRAUER.

Notation. (Mis a group of order g = pg', where p is a prime and g' is not divisible by p. Let ti, 2, be the absolutely irreducible characters ,K of (Mand 1k,02, * * *, pokthe irreducible modular characters. Then K is the number of classes of (M,and ko is the number of classes consisting of p-regular elements. Here an element G of (Mis termed p-regular, if its order is prime to p, otherwise G is p-singular. The characters g of (Mare distributed into " blocks" 8 Bx (A = 1, 2,

*), and we may speak of the modular characters

belonging to the block Bx. The degree of *til will be denoted by zM. Each character t determines a family of p-conjugate characters,9 we denote the number of members of this family by r1i. 1. The p-singular elements of (. We first formulate some elementary facts concerning the type of groups we propose to consider in this paper. LEMMA

(1)

1. Let (Mbe a group of order 9Pg =pg"('P g),

where p is a prime, and let P be an element of order p. The normalizer 9X(P) is a direct product of the Sylow subgroup, = {P}, and a group S of order v which is prime to p, i. e., =

FX

(2)

This follows at once from a theorem of Burnside or a more general theorem of Frobenius.10 Using Sylow's theorem, we obtain easily LEMMA

2. If two elements of 9X are conjugate in (M, then they are

conjugate in the normalizer 9) X= 9($)

of the Sylow-subgroup

$.

There is no difficulty in discussing the structure of W. We have 8 Cf. [1], ? 9. 9 Cf. [2], ? 4. 10W. Burnside, Proceedings of the London Mathematical Society, vol. 33 (1901), pp. 163, 257; G. Frobenius, Sitzungsber. Preuss. Akad. (1901), pp. 1216, 1324. We may also prove the statement in our case in the following fashion. Assume first that 9 has an irreducible representation of degree f prime to p which does not represent P by the unit matrix. Then the elements of X foir which the determinant is a g'-th root of unity, form a normal subgroup S, and P does not lie in 2, whence we easily obtain (2). If Lemma 1 is not true, then every irreducible representation ! of 9, whose But adding the squares degree f is prime to p, actually is a representation of T/{P}. of the degrees of all irreducible representations, we obtain the order of the group, and are congruent (mod p2), which is it would follow that the orders of g and of /{P} not the case.

GROUPS WHOS~ ORDER CONTAINS A PRIME TO THE FIRST POWER L LEMMA

405

The group ?!11/ilc is cyclic, its order q is a divisor of p -1,

3.

say

(&)

p-l=qt.

If M in ID1 corresponds to a generat'in[! element of ID1f.)1, then M- 1 PM =PY'

(4)

where y is a primitive root (mod p). The order of M is prime to p. t The elements (5)

Po= 1,

P1

=

P,

P 2 = PY, · · ·,Pt= py•-1

form a complete system of representatives of the classes of conjugate elements in @ in which the order is a power of p. They can be taken as the elements P, in [2], § 1. Every p-singular class O of @ contains an element pav with ex~ O (modp), V:in fil. We readily obtain LEMMA 4. Le.t V1 , V 2 , • • · , V1 be a maximal system of elements of fil such that no two of them are conjugaü, in Wl. Every p-sif!-gular class of @ contains an element pa V;>,. with ex ~ 0 ( mod p), and )l. is uniquely determined by the class. All the classes for which )l. has the same value belong to -0ne f amily 11 F:>..; we have l families of p-singular classes.

Let T;>,. be the least positive integer· for w:hich .ll(-T;>,.V :>..M"). and V:>.. are conjugate in fil. Then the class of V:>.. in ID1 = {fil, P, M} splits into T;>,. sub-class.es of elements which are conjugate in_.. fil. The number T;>,. must be a divisor of q, since Mq lies in 91 and, therefore, M-qV:>..Mq and V;>,. are conjugate in fil. Suppose that J>"V;>,. and PßVA are conjugate in@, say a-1 paV;>,.G = PßVA; Then G-1 paG = Pß, G-1 VAG = V;>,., Hence G lies' in one Qf the cosets 91MP. Sjnce M-PV:>..MP and V:>.. are conjugate in m,. the exponent p must be· a multiple of r">,.. Furthermore, ( 4) implies that cxyP' ß ( mod p). Conversely, if this congruence holds for a multiple p of T;>,., then: pav;>,. and PßV;>,. are conjugate in @. lt follows that the elements pav:>.., ex= 1, 2, · · ·, p-1, belong to h = tT;>,. different classes in @.

=

·1"d"'1ts uo..c..tl-a

5. Suppose that the cl~ of V:>.. in @ f;!'B8~ i1'11t9o T;>,. subclasses of efoments which are conjugate ~ fil, and set h = tn, The family FA consists of t;>,. classes. of conjugate elenients ( with regard to @). LEMMA

lt follows from ( 3) and

T;>,. 1q

that h divides p - 1. Two elements pav:>.

ii. Cf. [2], § 7.

t

Fov-

t. :=; r-1, +\...',s \s

--h-, .....E!

~.... s"'"'~lo\q_ c:..\.....·,tSi ('R,"g ::t

.,,..,°"'et +....\;e. M : 1. \,. . 1-i: b'1 li?.\E!. ~~""4-s o~ i..,JE!

442

c~. ~ :J

FINITE GROUPS

of-

Mj ~-~·,

406

RICHARD BRAUER.

alndPJ3VIare conjugate in 63, if and only if Pa and PI are conjugate in the normalizer 91( Vx) of Vx. Hence we have tx classes of elements of 91( Vx) which contain elements of order p. Each of these classes contains n(Vx)/n(VxP) elements, where n(G) denotes the order of the normalizer of an element G in 63. Hence 91( V) contains txn( Vx)/n( V,P) elements of order p. Together with the unit element, they form a complete set of solutions of the equation XP = 1. Using a theorem of Frobenius,12we now obtain LE.MMA6. Let n(G) be the order of the normalizer of the element G in 63. The number tx is characterized completely by the two conditions p- 1

(6)

0(mod tx),

1 + tx >(VA))j

O (modp).

2. The blocks of characters. A block B is either of type 13 1 or of type 0, according as the degrees of the characters in B are divisible by p or are prime to p. In the 'firstcase, B is of the highest type. Therefore,14B consists of one character CI only. Further, tp is modular irreducible, it lies in the field of the g'-th roots of unity, and it vanishes for all p-singular elements of 63. The corresponding generalized decomposition numbers d$pvvanish for i > 0 and all v.15 In the notation of Lemma 4, there exist I classes of conjugate elements in 6 such that the elements of the class are p-regular and that the number of elements in the class is prime to p. This implies,"' that exactly I of the blocks of 63 are of the lowest type, i. e., of type 0. , Ba of lowest 1. The group 63 possesses I blocks B1, Bon. type where 1 is defined in Lemma 4. All the other blocks Bl+1, B1+2y ... are of highest type; they consist of one character g,z, which ts p-conjugate only to itself, which is modular irreducible, and which vanishes for all p-singular elements.

THEOREM

Hence the blocks If we set g - pog' with (g', p) - 1, then a --1. 1. Consequently, the results of [3] will hold for them. We shall apply these later. B1, B2, * * *, B are of degree a -

3. The matrix of the generalized decomposition numbers. In order to discuss the decomposition numbers d$lv -with i > 0, we have to form the 12

Frobenius, Sitzungsber. Preuss. Akad. (1895), p. 981.

13 [1], 14 15

? 17.

Cf. [1], Part II. [2], Theorem 10.

16 [11, Theorem 2.

GROUPS WHOSE ORDER CONTAINS A PRIME TO THE FIRST POWER I.

407

normalizer 9Z(Pi) of the element Pi (cf. (5)). This normalizer is identical with the normalizer 9Zof P, and (2) now shows that the irreducible modular characters 01p, IP2I, * of 9Z(Pi) are identical with the ordinary irreducible characters

01, 62, *

of B. The corresponding Cartan invariants of 9(P),

then, are given by (7)

(for i > O).

C pa po-p17

This implies that the indecomposable character Pp of 9 (Pi) associated with opt is connected with cpt by the equation pbpi.

(')Pi =

Then the formulae [2], (1) and (2) defining the decomposition numbers difv become

~

(8) (9)

tt(Pi V)= dilv=

(1/v)

(V in ~,i> 0).

d'i0v(TV) gv(PiV)0OL(VT1)

z

for

V in Z

i>

0.18

According to (7) and [2], Theorem 4, we have K

,d'svdjyp= O if iz j or if vp.

(10 a)

,U=1

K

_

, diyvdiuv- p.

(10 b)

A=1

If in (10 b) the index pt ranges only over the values corresponding to a fixed block Bx, then the sum on the left-hand side is a rational integer which is divisible by p (cf. [2], Theorem 7). It follows that for every possible combination (i, v) with i > 0 there exists exactly one block Bx of lowest kind 0 for all ptnot belonging to Bx. We denote this block Bx by such that di, B (i, v). If two d-columns ditv and djyp are algebraically conjugate, then B (j, p). The same block Bx correspondsto all the members obviously B (i, v) =

of a family of algebraically conjugate d-columns.

Every block Bx of lowest type appears, conversely, in the form B (i, v) for at least one combination (i, v) with i > 0. If this were not so, we would have diiv - 0 for the values ti belonging to Bx and for all i > 0 and all v. Then (8) shows that the characters g,uof Bx vanish for all p-singular elements, and this would imply 19 that 4ptbelongs to a block of highest type and not to a block Bx of type O. Cf. ['1], ? 29.-Of course, 6pa = 0 for p ; a- and 8pp = 1. make use of the fact that v is the p-th part of the order of P0O1is the indecomposable character .1 17

18 We 19

Cf. [1], Theorem 10.

X

(p)

and that

408

RICHARD

BRAUER.

According to [2], Theorem 11, the number of families of algebraically conjugate d-columns is equal to the number of families of classes of conjugate elements. Each d-column with i = 0 forms its own family, and so does each p-regular class. Further, we have as many d-columns with i = 0 as we have p-regular classes. It follows that the number of families of d-columns with i > 0 is equal to the number of families of p-singular classes. This latter number is I (cf. Lemma 4). Hence we have as many familiesi of algebraically conjugate d-columns with i > 0 as we have blocks B1,B2, * *, B, of lowest type. The mapping d -> B (i, v) necessarily defines a (1-1) -correspondence between families of d-columns with i > 0 and blocks of the lowest type. We shall say that the d-column d,(iv belongs to the block Bx, if B (i, v) = Bx. Arrange now the characters t,, t,, * * - , g, of 0 so that the characters of B1 come first, then those of B2,, * , then those of B , and then those of the ... of highest type. blocks B1+i,B1+2,~ On the other hand, we choose a suitable arrangement of the columns of the matrix D. formed by the decomposition numbers (cf. [2], (3) ). We start with the d-columns with i = 0, taking them in such an order that the matrix i = DObreaks up in the form [1], (28) (i. e., arranging the modular characters l.02,2 - - of 6 so that the characters of B1 come first, then those of B2, etc.). After the d-columns with i = 0 we take the columns belonging to B1, then those of B2, * *, finally those of B . From our results above it follows that D has the following form

(11)

Di O . . .O 0

Q, 0 . . .0

O

O

LD-.

A2*

O O .

0

0

0

I= 0 0 -

Q2..

0

00.Q

0

0 (mod p) and I1 z= D to the left of the vertical of m. The part of matrix degree unit denotes the =d numbers pv i. e., the matrix d,uv line contains the ordinary decomposition D = DO. The columns of each fixed Qx are algebraically conjugate. Since D is non-degenerate, it follows that these columns are linearly independent. Here, mnis the number of characters ;, of a degree

4. The number of characters in the blocks of lowest kind. Denote the number of columns of Qx by t'x; the corresponding columns of D form a *,1). Besides these I families of d-columns, family of d-columns (X 1, 2, we have 1cofurther families of d-columns dtv with i = 0 where ko is the niumberof classes of p-regular elements in (M. Each of these families consists

GROUPS WHOSE

ORDER CONTAINS

A PRIME

TO THE FIRST POWER I.

409

of one column. Hence the number of members in the different families of d-columns are

(12)

t'l1.2 e

*

2

ttl

I

I

ko

respectively. We now apply [2], Theorem 12.20 We have 1 + kcofamilies of p-conjugate characters, and if the number of members of these families are (13)

r1, r2, * * , rz+ko

respectively, then the sets of numbers (12) and (13) coincide. Further, the elasses of conjugate elements are also distributed into families. According to Lemma 4, we have I families containing p-singular classes, and the A-th family contains tx classes (Lemma 5). Each p-regular class forms a family by itself; there are ko such families. Hence the number of members of the different families is . . In*.l (14) ti, tonp ko

respectively. Again, [2], Theorem 12, shows that the system (14) coincides with each of the systems (12) and (13). In particular, after changing the order of the I elements Vl, V2,*. *, VI if necessary, we may assume that (15)

I, 2,

t>.=t'., (At

Let e be a primitive g-th root of unity, and choose a rational iliteger -y such that y is a primitive root (mod p) and y - 1 (mod g'). The substitution T:

(16)

e-* 'y

transforms Qx (cf. (11)) into a matrix QxT. There are two ways of obtaining QxT from QX: (1) the substitution T permutes the d-columns and this induces a permutation of the columns of Qx. The coefficients of Qx belong to the field of the p-th root of unity and the tx columns of Qx are algebraically conjugate. It follows easily that QxT can be obtained from Qx by a cyclic permutation of the columns.21 The substitution T permutes the characters l, C2, * *; actually, T induces a cyclic permutation of the characters of each family. This shows that QxT can be obtained from Qx by a certain permutation of the rows. The assumptions of [2], Lemma 2, are satisfied since the columns of Qx are linearly independent as observed above. It follows that the block Bx must For the case p = 2, cf. footnote 21 of [2]. According to [2], ? 1, the generalized decomposition numbers lie in the field of the, p-th roots of unity, since we have a = 1. 20

21

410

RICHARD BRAUER.

contain at. least one character tp, such that the number r1, of members of the family of tj4 is divisible by tx. But the systems (14) and (15) both consist of the same numbers. We readily see that the block B, contains a family consisting of exactly tx characters; all the other families belonging to Bx consist of one character (A 1, 2, * * *, 1). If Bx contains wx families of characters and if ra, rfi,. . . ,rK, respectively, are the numbers of members of these wx families, then one of the numbers, say rac,is equal to tx whereas the other 1 numbers are equal to 1, i. e., r0 w1. On the other hand, rK we have 22 p. raro * rK(1/ra + I/rfi + * * + IrK This now gives tx(I/t + (wx ) ) p, (17) wx -1 (p 1)/tx. =

=

If a character gy is p-conjugate only to itself, i. e., rM 1, then ; lies in the field of the g'-th roots of unity. Hence =

THEOREM2. Let tl, t2,.* , t1 have the same significance as in Lemmas 5 and 6. If these I numbers are taken in a suitable order, then the block Bx of lowest kind consists (a) of one family of tx p-conjugate characters, the ",exceptional family" of Bx and (b) (p - 1) /tx further characters gE,which belong to the field of the g'-th roots of unity, i. e., each such giJ is p-conjugate only to itself (A 1, 2, * * *1).

On account of [3], ? 7, Corollaries 1 and 6, and [3], Theorem 5, we also have (A =,

THEOREM3. The block B, contains (p- 1)/tx modular characters 2 *. . , 1). The structure of B, is characterized by a tree TX with

1 + (p- 1)/tx vertices and with (p- 1)/tx edges. Each of these edges Ev corresponds to a modular character 4v of Bx; each of the vertices V,J corresponds to a family of p-conjugate characters. A modular character 'pv of B, appears as a modular constituent of an ordinary character gs, of B., if the edge Ev contains the vertex corresponding to g,L;the multiplicity of 'v in g then is 1. We may describe the tree Tx in the customary manner 23 by a matrix with 1 + (p - 1)/tx rows and (p - 1)/tx columns. Each row corresponds to a vertex, and each column corresponds to an edge. The coefficient in the i-th row and the j-th column is 1, if the j-th edge contains the i-th vertex. 22 23

Cf. [3], Corollary 5 in ? 7. Cf. 0. Veblen, Analysis situs, second ed., p. 12.

GROUPS WHOSE ORDER CONTAINS A PRIME- TO THE FIRST POWER I.

411

In the other case, the coefficient is 0. Let D5 be this matrix. Suppose that the numbering of the vertices is such that the last vertex corresponds to the family with tx members. Then the matrix D>. in (11) is obtained from Dx by adding tx. 1 further rows all of which are equal to the last row of D]5. Since Cx = D'xDx, we can find the ordinary decomposition numbers and the Cartan invariants corresponding to the block Bx, if we know the tree Tx, the r,umber tx and the vertex corresponding to the family with tx members. 5. The coefficients, of the matrix Qx. Consider a fixed block Bx , 1). The corresponding matrix Qx in (11) has tx columns, (A 1, 2< (cf. (15)), which are algebraically conjugate. The substitution (16) induces a cyclic permutation of these columns; we may arrange the columns so that the first column is carried into the second, the second into the third etc., finally the last column into the 'first. As we have seen, the block Bx contains (p - 1)/tx characters gy which belong to the field of the g'-th roots of unity. It follows from (9) that the corresponding numbers dt,Mvlie in the same field. On the other hand, the generalized decomposition numbers lie in the field of the p-th roots of unity.24 Consequently, the numbers dt,Mvwith the fixed 'first suffix ju are rational integers, therefore dt,[vis algebraically conjugate only with itself. The formula (11) now shows that the row of Qx, which corresponds to gy contains tx equal coefficients. Set h = (p - 1) /tx and arrange the characters gl, of Bx so that the h characters lying in the field of the g'-th roots of unity are taken first and are followed by the tx characters of the exceptional family of BX. If the order of these last tx characters is chosen suitably, the matrix Qx will have the form =

Qx

(18)

al

a, * * * al

a2

a2

ah

ah .

.ah

a

a

.(tX-1)

aX(tX71)

aZ

. .

.

*

a2

..(tX-2

, ah are rational integers, while a is an integer of the field of Here a1, a2, a (t-1-) with the p-th roots of unity which has exactly tx conjugates a, a',* P numbers. of rational field the regard to

24 Cf.

[2], ? 1.

412

RICHARD BRAUER.

The relations (10a) and (lOb) can be used to determine the coefficients 0 the of Qx. Because of the form (11) of D, we obtain from (10a) for j matrix equation (tal))Dx =- O. (19) *,ah, a,a',*,t (a1, a2,* Secondly, (10a) applied to two suitable columns of D passing through Qx yields h

E

(20)

0

av2+ tr (aa($))

(t==1,2,

l/=1

where tr (t) denotes the trace of a number w of the field P (a) with regard to P. Finally, the equation (lOb) gives h

Eav2 +tr (ca) - p.

(21)

l/=1

The last tx rows of Dx are equal (cf. ? 4). Hence (19) can be written in the form *,ah, tr (a) )Dx ==, (22) (al, a2, where Dx, as in ? 4, is the matrix describing the tree TX. We set a, += tr (a). From (22), it follows that if the edge Eg of TX is bounded by the vertices VK and Vv, then aK+ av- 0. On comparing these equations for all h edges of Tx, we see that (23)

a-,

a=

tr (a)

ah+l

, h + 1. In (23), the + sign is to be used, when the verfor i, j 1,2, tices VI and Vj are connected by an even number of edges, in the other case =

the

-

sign is to be taken.

We now add the equations (20) for i=1, we obtain

tx - 1 and (21). Thus

2,

h

tx av=+trl

r(i

p=1~~~j=

c

p,

or (24)

aV2

t;
0 not appearing in (42) vanish as follows from the form (11) of D. Hence (8) reads TX-1

(43)

t,(Pi V)

di,V+KOV+K

E=

(v),

K=O TX-1

CI

(PiV)

diM,V+KOV

(M

KVMK),

(i>

O, V in F3).

K=O

The row (42) of Qx has been determined in ? 5. If Clilies in the fields of the g'-th roots of unity, then all the coefficients (42) have the same value 81, ? 1 so that (42) has the form

8. 811,811,.~sl

(44)

If Cilis a suitable one of the tx p-conjugate characters of Bx, then (42) is identical with syq, 8tqli, * , 8qtx-,. (45) (cf. (31*)). The index v is determined by the fact that the first column of Qx in (11) and Qx also contains the d-columns dlp v+j ... * dlli,v+Tx-j is the d-column d1Mv, belonging to characters Opof F8which are associated with Ov. In this manner, Bx determines a class Cx of associated characters of B8. To another block Ba' with I ? X'? 1, there must correspond a class Cx' Cx. Every class of associated characters of F8necessarily appears in the form Cx. We now change the notation and denote the character Ovof F8belonging to the first column of , 1. No two of the characters 61,02, * * *, 01 of 2 then Qx by Ox,A = 1, 2, are associated, but every irreducible character of F8is associated with one of these I characters. obtained Collecting the results: we substitute in (43) the values of dipLv by comparing (42) with (44), (45) respectively and express Bp by means of a primitive p-th root of unity using (26) and (27). We now obtain 4. To each of the blocks Bx of the lowest kind there correTHEOREM O ? sponds an irreducible character Oxof B8. No two characters 01, *2 .. * . F8 are associactedwith regard to V, but every irreducible character of is assotx/t characters ciated with one of the characters Vx. There are exactly TX are with associated OX;they . . *, rx -) (K = O 12, =

Ox(M-KVMK)

GROUPS WHOSE

ORDER CONTAINS

A PRIME

TO THE FIRST POWER I.

417

where V razges over F; we have oX(M TXVMTX)=ox((V)

(46)

V in B.

If C,,is a character belonging to Bx which is p-conjugate only to itself, then TX-i

CA(PPV)

(47a)

8=

,K=O

OX(M-KVMK),

(TX=

for p + O (mod p) and V in 93. Here S3 + 1 or S,acter of the exceptional family of Bx, we have

tx/t)

1.

If C,is a char-

q-1

(47b)

Cp(PPIV) =-

S1 p'YP-Kt6 (M-KVMK) Kc=0

(q

(p -)/t)

where p * 0 (mod p) and V is in B3. Here E is a suitable primitive p-th root of - 1. unity, y is ccprimitive root (mod p) and Si= + 1 or 8 If the characters 0 of F8 are known, the formulae (47a) and (4Tb) together with Theorem 1 show that the values of the characters C of (5 for all 1 remain undep-singular elements can be obtained, only the signs SI, termined. If the tree Tx corresponding to Bx is known, then we obtain from (23) and the remark at the end of ? 5 If ti and gv are two characters of a block Bx of the lowest if and only if the corresponding vertices of the tree Tx kind, then 8p =v can be joined by an even number of edges. THEOREM

5.

Concerning the values of the, characters for p-regular elements we state THEOREM

6. Let Bx be a block of lowest type. For any p-regular elements

G of (5, we have

(48)

E

lt,(G) =0,

wvhere4pranges over a complete system of characters representing the different families of B, (i. e., gy ranges over the (p - 1)/tx characters of B, which lie in the field of the g'-th roots of unity and one of the tx p-conjugate characters). In particular, (49)

EIZMI =0,

where zMis the degree of 4y and p ranges over the samnevalues as irn (48). Proof. From the definition of the S at the end of ? 5 and from (22) it follows that NSdMv ? (50)

418

RICHARD BRAUER.

for v = 1, 2, , kco where the dMv d0M.vare the ordinary decomposition numbers of (Mand ,uranges over the same values as in (48). On multiplying (50) by the v-th irreducible modular character c)v(G) of ( and adding over v, we obtain (48). The equation (49) is derived from (48) by setting G = 1. =

7. Corollaries. Apart from the signs 8,t, the right hand sides in (4Ta) and (4Tb) are completely determined, if (a) the characters of S8 are known and (b) it is known in what manner these characters are associated in the group {, M}. This group {S,iMl has an order v (p - 1)/t which is prime to p, and we may consider the computation of its characters as a simpler problem than the determination of the characters of (M. In the applications of the theory developed here, the characters of {3, M} can usually be found without difficulty and then the answers to (a) and (b) are actually known. In particular, the right hand sides in (47a) and (4Tb) are determined completely by the structure of 9, apart from the signs . Except for these signs, the right hand side of these equations are the same for any two groups (Mand ($,, for which VW=g ({P}) has the same structure. In particular, we may take for 05, the group 9J itself. Since the characters of ( of highest kind vanish for all p-singular elements, and since 9= ( {P}) has no characters of highest kind,26 we obtain THEOREM 7. Arrange the matrix Z of the group characters of ( in such a manner that the characters of a degree prime to p occupy the upper rows and the other characters the lower rows, further arrange the columns of Z so that the classes of the p-regular elements are to the left of the p-singular elements. If Z thug is broken up in the form

-

Z1

Z2\ Z4,

vZ3 p-regular classes

}

degrees prime to p,

} degrees divisible by p,

p-singular classes

then Z4 0. If the matrix of group charactersZ* of the group 9J = 9 ( {P}) is arranged in a correspondimgmanner, then the lower part is missing, i. e., =

Z

-

(Z*1, Z*2),

and Z2 can be carriedinto Z*2, if certainof its rows are multipliedby,- 1. This implies THEOREM

* 26

8.

Cf. [1], ? 29.

The number of irreducible characters of (M of a degree

GROUPS WHOSE

ORDER CONTAINS

A PRIME

TO THE FIRST POWER I.

419

z 0 ( mod p) is equal to the number of classes of conjugate elements in the group 9Nh=9Z({P}). The characteristic roots of the matrix 8 (P1V) representing PV in a given representation a of ( can be obtained by multiplying the roots of ~ (P) with those of j (V), taken in a suitable arrangement. Hence, for the traces of the matrices, we have (mi-od (1-ec))

_-=tr (W~(V))

tr (!C(PV))

if e is a primitive p4th root of unity. In particular, gy (PV) _ Cy (1V)

(mod (I1 -E)).

Combining this with (47a), we find CM( V)

VMK) t K=0OX(M_K

C

(mod ( 1-)),

if C, is p-conjugate only to itself. Since both sides here lie in the field of the g'-th root of unity, this congruence must hold (mod p). Similarly, we have in the case of (4Tb) q-1

-8 M

gM(1V) -

(mod p).

OX(M1-VM) K=O

(K On the right hand side, every term OX(M-KVYM1IK) Hence of (46). because times appedrs (p 1)/tx

THEOREM

9.

0, 1,

,

TX-

1)

For an element V of Z, we have (rx = tx/t) TX-1

gM(V) -

(5la)

(mod p),

x 8 E OX(M-KVMK) K=O

,1) which is p-conjugate only to itself. if C, is a character of Bx (A 1, 2., If Cl,belongs to the exceptional family of Bx then =

(5lb)

g81(V)

Finally, if

(!

=

(

0/t)

TrX-1 E OX(M-KVMK)

(mod p)

is a character of a degree z. divisible by p, we have (1V) _

(51c)

(mod p).

O

As a corollary, we obtain THEOREM

10. Let fx be the degree of the character ONof Z3. The degree

z,u of a character g, of the block Bx (X = 1, 2 (mod p)

*

,

1) satisfies the congruence

420

RICHARD BRAUER.

(52a)

z1 = 8Mtxfx/t or

(52b)

z-

(mod p),

8fx/t

according as we have the case of (51a) or (-51b). These congruences, together with (49) and the fact that z1,divides g are in many cases sufficient to determine the degrees. If the degrees z5 and the degrees fx are known, then the sign 8 can be obtained from (52a), (52b), at least for an odd p. 8. The block of the 1-character. We consider now the 1-characterof (M. We may assume that the notation is so chosen that it is the character C (G), i. e., 1( G) 1 for every G in (M,and that it belongs to the block B1. The formula (47a) then gives =

1

'&

01 (M-KVMK)

K=0

for every V in 8, and since this is a linear relation between the characters 1, and 01(V)

of , we must have r1 =

=

1. Hence

11. If the block B1 containis the 1-character C, of @, then 01 THEOREM is the 1-character of 23. Fuvrthert1 = t. The degrees z, of the characters of B1 satisfy the congruences

(53)

z[L

8M= +?1 (modp)

or zL

8y/t =

1/t

(mod p)

according as we have the case of (51a) or (5ib). If ZlX2y(q (p -1)/t), represent the different families of B,, then

*nC+ln ,

=

(54)

1

+

82Z2 + *

=.

+ 8q+lZ+l

0

UNIVERSITY OF'TORONTO.

BIBLIOGRAPHY.

[1] R. Brauer-C. Nesbitt, " On the modular characters Mathematics, vol. 42 (1941), pp. 556-590. [2] R. Brauer, " On the connection between the ordinary acters of groups of finite order," Annals of Mathematics, vol. " Investigations on group characters," Annals [St] (1941) pp. 936-958.

of groups," Annals of and the rxhodular char42 (1941), pp. 926-935. of Mathematics, vol. 42

ON GROUPS WHOSE ORDER CONTAINS A PRIME NUMBER TO THE FIRST POWER II.* By RICHARDBiuUER.

Introduction. This paper is a continuation of a previous paper with the same title.1 Its aim is the proof of the following theorem: Let 8 be a finite group of linear transformations in n variables. Assume that the order g of S contains a prime factor p to the first power only, and that 8 has no normal subgroup of order p. Then we have p ? 2n + 1. The equality sign can hold only when 8, considered as a collineation group, is isomorphic with LF(2, p). 1, has a normal subgroup v If the group 3 of order g = pg', with (g', p) of order p, then 8 contains a normal subgroup X, such that X is the direct product of $ and a normal subgroup 3 of 8, while the factor group 3/X is , i. e. cyclic.2 For a primitive group 8 of this type, we can take 3 X 3; if 3 is primitive and unimodular,3 then the degree n must be 23 divisible by p. It has been proved by H. F. Blichfeldt4 that the order g of a primitive unimodular linear group 8 in n variables is not divisible by a prime number p which is greater than (2n + 1) (n - 1). Our theorem improves this result for primes p which divide g to the first power only. Since LF(2, p) has an irreducible representation by collineations in (p - 1)/2 homogeneous variables,5 the inequality p ? 2n + 1 cannot be improved further. The proof of the theorem is obtained by combining a rough estimate for the degrees of the characters (? 3) with a formula for the product of certain characters (? 4); both these formulae are derived from the results of [4]. The case p = 2n + 1 requires a somewhat complicated discussion. For the notation employed cf. [4]. =

=

=

*Received June 26, 1941. Cf. the paper [4] of the bibliography. 2-Cf. [41, ? 1. 3 This means that all the matrices of 8 have determinant 1. 4H. F. Blichfeldt, Finite Coltineation Groups, University of Chicago Press, Chicago, 1917, p. 89, Theorem 5. - 1 (mod 4), this representation has been discovered by F. Klein, 6 For pMathematische Annalen, vol. 15 (1879), p. 275; for p =1 (mod 4) by I. Schur, Journ. f. d. reine u. angew. Math., vol. 132 (1907), p. 135. 1

13

421

422

RICHARD BRAUER.

1. Remarks on characters. Let ( be a group of finite order. As is well known, the linear combinations

et alg,+

a2C2 +

* -*+ alt,

(ax rational integers)

, t of ( form a ring Z. of the (ordinary) irreducible characters gl, t2, . If Zo denotes the subset of elements e for which all ax ? 0, then to is closed under addition and multiplication; the elements of Zo are the (reducible and Every element e of Zo can be obtained by irreducible) characters of (. , , and this additive repreaddition from the irreducible characters Cl, t2p . sentation of e is unique. Every element of Z is a difference of two elemenits of . It may be remarked that we have a certain resemblalnceto the familiar type of arithmetic; however, the roles of addition and multiplication are interchanged. We shall say that an element e of T contains aln element el of belong to Zo. We then write: e D el. On using the orthogonality Z, if e -e relations for characters, we easily obtain the following:

1. Let CK, gx, gp be irreducible characters of (. If gKgX D hAL, where h > 0 is a rational integer, then gKgL_Dhex, gK denoting the conjmugate complex character of CK. LEMMA

Indeed, both relations are equivalent to the fact that 1. times the 1-character t

gKgCXg[contains

h

=

2.

Summary of the results of [4].

Let ( be a group of order (p,g')=1,

g =pg,

where p is a prime number. Let P be an element of order p. Its normalizer XZ(P) is of the form S XX {P}; =

the normalizer 92Iof the p-Sylow-subgroup $ {P} contains both 9Z and S as normal subgroups. The factor group WI/ is cyclic. If M in 9 corresponds to a generating element, then M-1PM Plyt, (1) =

q is the order of W/N. where y is a primitive root (mod p), and (p - 1)/t 1, t3, * * be the (ordinary) irreducible characters of (, and Let gl denote by zM Dg (g) the degree of gy and by ru the number of p-conjugate characters. These characters appear distributed into blocks Bl, B2P * of characters. A block Bx is either /of the type 1 (highest type) or of type 0. In the first case, Bx consists of exactly one character g; we have zM 0 (mod p) and rM 1. In the second case, all the characters of Bx have degrees =

lp

.

-

=

GROUPS WHOSE

TO T:HE FIRST POWER II.

A PRIME

ORDER CONTAINS

423

which are relatively prime to p. Let B1, B,, - *, Bt be the blocks of type 0. To each of these I blocks, there corresponds a certain multiple t. > 0 of t. The block Bx then consists of qx =

1) /tx characters gy for which ry =

-

(p

1,

and one "exceptional" family of tx prconjugate characters.6 Set tl/t rX. To each block 1, of type 0 there corresponds a class of irreducible characters of 3, say =

, 0x(T-1)(V)-

OX(V), C\(1) (V) = OX(M11VM),

OX(M-(ITx-)VM7Xl),

which are associated in 9X; we have OX(M-TI

vMT\x)

=

OX( V)

Each irreducible character of Z appears exactly once in the form

- 1, 2,

Ox

, 1; K

0,

1, 2,

,rx-

For each gy belonging to a block B. of type 0 a sign 81= + 1 is defined such that the value of gy for p-singular elements G P$V (i 0 (mod p), V in 3) in the can be obtained following forms: =

CASE I:

=ll

St,1M 1, zMtw O (modp) ,X_1.

(,I)

CIL(P$V

CASE II.

r-=l1,

(2,II)

z1 X0

-1;

S,

, OX.(o (V);

K=O

(mod p) (V);

O\(K)

-

CL(P$iV)

K=O

CASE III.

ry=

tx > 1,

1; zM-O

8H

(modp)

TrX-i

(2, II)(P$fV)

'E(yta

0(K) (V) OX

-

K=O

aJ

where E is a suitable primitive p-th root of unity, and mod rx), 0 ? a' < q; values with a'-K( CASE IV. (2, IV)

riL

ranges over all the

1; zMEAO (mod p)

tx > 1, St, 4L(PiV)

a'

0( OX

=C=0

(V)

z

EVY

f0

where E, a' have the same significance as in Case III. 6If tx ==], then BX consists of 1 + (p -1) tx = p characters t, and for all of them we have r = 1. An arbitrary one of these characters can be selected as the only member of the exceptional family.

GROUPS WHOSE

II.

E Ox"" E

gy

Tx-i

E

(E)p +

p=1

K=O

CASE JII.7

425

v-1

Tx-i

CASE

TO THE FIRST POWER II.

A PRIME

ORDER CONTAINS

I"

(E)P

0x()

K=0

+1 S,

p

where p ranges over the values 0 ? p < p for which the qx-th power of p is not (p - 1)/tx); congruent (mod p) to the qx-th power of ytK, (qx =

CASE

IV.7

g

=

E ox,,,

."(E)P

+ S,

p

K=O

where p ranges over the values 0 ? p < p for which the qx-th power of p is congruent (mod p) to the qx-th power ytK CASE V.

=

S.

Here, (E) denotes a character of the cyclic group {P} which is not the 1character. The expression S is a linear combination of the characters of eJ with non-negative integral rational coefficients such that ox(K) (V) (e)) " 8\(K) (1V) 1 and (V) (e) OX(K) always have the same coefficient. ox(K) As a corollary, we have COROLLARY

1.

If Oxhas the degree fx, then the degree zM of gt has the

following forms: CASE I.

Z

fxtx/t + ps;

CASE II.

ZU

fxtx (p-1)/t

CASE III.

ZM=

fxtxt-(p

CASE IV.

z,t=

(p-1)fx/t

CASE V.

Z=

ps.

-(p

+ ps; -

1)/t

) + ps;

+ps;

where s > 0 is a rational integer, and s > 1 in Case V. If the degree z, is smaller than p, the expression S in Theorem 1 must vanish. In Case I, this implies that g (P) = z1u,and therefore P is represented by the unit matrix. In Case II, we must have zM (p - 1), fx = 1, t= t. Then every element V of $ is represented by a scalar multiple of the =

unit matrix:

V -> Ox(1V) I.

In Case III,

we must have fx

=

1, tx

t,

p- (p- 1) /t ? (p + 1) /2, since tx > 2. Again, every element of S z= is represented by a scalar multiple Ox(V) of the unit matrix. In Case IV, (p - 1) fV/t; Case V is not possible. Hence we must have z=

7If the character (e) is chosen in a fixed manner, then this formula holds for a character chosen suitably from the exceptionial family of Bx.

RICHARD BRAUER.

426

COROLLARY 2. If the degree z,uof the irreducible represettation is smaller than pI then we have one of the following cases: CASEI. zj

(

fxtk/t. The element P is represented by the unit matrix I;

=

CASE II. Z,u p - 1, fx by scalar multiples of I; =

=

1, tx

t. The elements of 1 ar-erepresented

t=

CASEIII. z,= P-(p -1)/t (p + 1)/2, fx =1, elements of 5 are repiresentedby scalar multiples of I; CASE IV.

3&Uof

(p -1)

Z,=

t > 2. The

fX/t.

The expression S in Theorem 1 vanishes in all these cases. 4. A formula for the multiplication of certain characters. Let w be an irreducible character of (Msuch that w possesses a p-conjugate character w' with ' 7 . Then w will belong to one of the I blocks BXof type 0. We must have tx > 2, and o must be a member of the exceptional family of Bx. Clearly, we have (for p-regular elements G of (M5). w(G) (G) =

Consider a character gy belonging to the same block B1 as the character =

1.

Since 01 =

t, the formulae (2) of ? 2, applied to gy, yield

1, t1

CASES I, II. CASES III,.

g

)=8

8( (G)

(for p-singular elements of C),

where the sum is extended over all the charactersgo,which are p-conjugate to g. Accordingly, we have -

CASES 1,II.

((G)

CASES III, IV.

(w(G) -w '(G))

for every G in (M. In Case I? we have 8

'(G))((G)

wu_ Dw.

Case II, we have 8

o' '

+

' ='gy

Consequently, 1, (O4y+

Hence w

(E(o (G) -8)

=

O,

1, (04u +

This implies

) =0,

8-

UO =

_ gtt,

y

0).

(cf. Lemma 1).

+ '

D g

In Case III, we have (oEr

-S

(o='

(o'U

-+

(0.

Similarly, in

427

GROUPS WHOSE ORDER CONTAINS A PRIME TO THE FIRST POWER II.

For at least one g belonging to the exceptional family of B1, Dg and hence

wgo)

contains

(,

Similarly, in Case IV, the product w'~ contains at least one member of the exceptional family of B1. This now gives the following result: 2. Let w be an irreducible character of (Mwhich possesses a . Then we have one of the following two cases: p-conjugate character ' THEOREM

CASE

a.

(O

CASE 93.

>A;

D X+>D c D'

3DX x + I3

1 and which do not Here a ranges over the characters gy of B1, for which 8, belong to the exceptional family; /3 ranges over those gy in B1, for which - 1, and which do not belong to the exceptional family. Finally, X is a 8, character 4[, chosen suitably from the exceptional family of B1; we have CASE a if 8,, = 1 for this gy, and CASE 3 if 8M - 1. 5. Representations of a degree n < (p

-

1)/2.

Preliminary remarks.

We now state THEOREM

3. Let 5 be a group of order g = pg', with (p, g')

1, which

has no normalsubgroupof the prime orderp. The degreen of any (1 representation 3, of 5 is not smacllerthan (p- 1)/2.

-

1)-

Proof. (a) We first show that it is sufficient to prove the theorem in the case where the representation is irreducible. Let us assume that the theorem is correct for irreducible 3. Suppose now that 3 is a (1 - 1) -representation

of (S of a degreen < (p-

1)/2, whichis reducible;let 1,

2
1, and hence tx > 1,

(6)

tx-?t?2

If w belongs to the block Bx, the value of w for elements of

X

is

TX-1

(7)

" (E)P

Gx(I) O

'w

/C=o

(G in 9Z),

p

where p ranges over-the qx values between 0 and p, whose qx-th power is con(p - 1)/tx, rx gruent (mod p) to the qx-th power of yKtv (qx tx/t). Let w be the p-conjugate character obtained by replacing E by EY. For elements of X, we have =

( 8)

t

z

IS>, OX(K)/(

KC=O

( G in 9Z).

)

p

(e) P7 appearing in (8) can also appear in (7), since for the exponents < p-i. we have pq -1 p ( yp)q yq s I (mod p), because q= (p -)/t

No term

Using the formulae (7) and (8), we find

(9)

-

,

XOj)8X(K)c) l,I

p

r/'(e)

P'Y-T

0f

9 We include the equality sign in order to avoid a repetition further on.

430

RICHARD BRAUER.

for elements G belonging to 9Z. Expressing k ()0x(K) by the irreducible characters of Z, the product ~w' can be written as a linear combination of the irreducible characters of X. Since py o- (mod p'), every term is of the form Oh( * (E) P5with v X 0 (mod p). On the other hand, ow' contains A,3in Case C and X + X,8in Case 3 (for arbitrary elements G of (M). Again, we restrict ourselves to elements G belonging to XZand express all the charactersby the irreducible characters of %. It follows that every #3can contain only terms Oh1(J)* (e)> with v 0 (mod p); in Case #3 this will also hold for X. Consequently, if Theorem 1 is applied to /3 (and also to X in Case p3), it follows that the expression S vanishes. Since and X belong to B1, and t, t, we then have =

p-l

(10)

A

(10,0!)

X0X

01 E(E)Pv p=1

(in Case a3),

i (E)P,

where p ranges in (10, a3) over the values between 0 and p, for which pq - 1 (mod p), (q (p -)/t). The character 61 is the 1-character of Z; hence and X degree (p- 1)/t = q in Case S. ,8 has degree p-i, in (9) does not contain 01 if t K; if t ,K, the The character Ox(@)0x(K) product 0A(t)60(L) contains 01 exactly once. It now follows from (9) that w'; contains exactly -xqX2terms 01. (E) P. Let u be the number of characters 13. On comparing (10) and (9) , we find =

(:1:1) CASEa. CASE

1) C qx2 == (p + (p- 1)/t ?

u.(p-

1)2/ttX;

u,(p - 1)

3.

ixqX2

(p-

1)2/ttX.

The total number of characters a and pl together is q (p - 1) /t, (cf. ? 2), and we have, therefore, q - u characters a. One of them, the character a 1, is linear. As assumed above, all the others have a degree larger than 1. -Corollary 1 then shows that Dg (a) ? p + 1, where Dg (?) denotes and 13 separately, the degree of a character ?. We now treat the cases using the same method in each case. =

=

a

We have 10

CASE a.

Dg(x) + E Dg(a) =EDg(P), a

la

and Dg(x) ? p- (p- 1)/t p- q, according to Corollary 1, since we have Case III for X. Consequently =

.

I

12

a)

p

-

q

1

(q

10Cf. [4], Theorem 6, (49).

-

u

-

1)

1p

+

1)

7f]

:W

u

(p

-

ORDER CONTAINS

GROUPS WHOSE

431

TO THE FIRST POWER II.

A PRIME

and hence qp ? 2itp, q _ 2u. However, (11) implies u < (p - 1) /ttx Combining these two inequalities and using (6), we obtain (13, (1)

tx = 2,

In particular, p,

1 (mod 4).

2,

t

u6

(p

q/tN.

1 ),/4.

Further, in (12) the equality sign must hold,

i. e. we have

(14,a)

Dg(x) =(p+1)/2,

for

Dg(a)=p+1

a5t1.

The number of characters a of degree p + 1 is q -u

-1=

1)/4-

(p-

1)/2-

(p-

1=

(p- 5)/4.

From (7), we obtain

This shows that Dgg(w) then fx Dg(Ox) ==1.

=

1)/2

==fx(p'-

Dg(w) =fx7xqN ==fx(p-1)/t

(15)

(p

1)/2 is impossible. If n

n < (p-

1)/2,

CASE A3: Here

ED'g(a)

Dgy(X) +YDg(f3)l, n

a

(12,8)

l+

(q

u-u

)(p +l) (p

:f;Dg (X) + EDg (,8) 1)/t + ub(p- 1) == + u(p-

1)*

Hence qp - p ? 2up, (q - 1)/2 _ u, whereas (11) gives u ? q/tx - 1/t. Combining the two inequalities, we have qtx - tx ? 2q - 2,r, i. e., q (tx - 2) ? tx -2rx< t2. Now tx 2 by (6) and q Dg(x) > 2, since -==1

is the only linear character of B1. Thus 2 (tx - 2) ? tx - 2. Again, we find 2. The two inequalities for u now have the form tx= 2, which implies t u q/2 - 1/2. Corresponding to (13, a) we have here (q 1)/2 =

t =-2,

==(p-

Since u is an integer, this gives p sign holds. Consequently,

3 (mod 4).

tx ~2,

(134 3)

(14p 3)

Dg(a)

p+ 1 for

3)/4. In (12, 8) the equality

a

The number of characters a of degree p + 1 is here q -u-1

(p -

)/2-(p

-3)/4-

a.

1

(p-

3)/4.

n Again, it follows that Dg(0) The equation (15) holds as in Case < (p- 1)/2 is impossible. If n = (p- 1)/2, then fx 1. This finishes the proof of Lemma 2 and of Theorem 3.

RICHARD BRAUER.

432

At the same time, we obtain 'LEMMA3. Let ( be a group of order q pg' with (p, g')= 1 and assume 11 thIat g, is the only linear character in the block B. Sfuppose that (p - 1) /2. (M has an irreducible (1 - 1)-representation 3 of degree n Case a: If p 1 (mod 4), then the first block B1 of characters consists of 1, (p - 5)74 characters a of degree p + 1, two conjugate characters X, X' of degree (p + 1)J2, and (p- 1)/4 characters ,B of degree p-1. 1, (p- 3)/4 characters Case Af3:If pt 3,(mod 4), then Bl, consists of C and a of degree p + 1, two conjugate characters X, X' of degree (p -1)/2, (p- 3)/4 characters /3 of degree p- 1. In either case, we have t = 2, and tx 2, if the character of 8 belongs to the bloc7kBx. =

=

=

7.

The case n = (p-

1)/2.

Preliminary remarks. We now state

4. Let 0S be a group of order g = pg' with (p, g') - 1, THEOREM which has no normal subgroup of order p. If (X has a (1 - 1) -representation 3 of degree n - (p - 1)/2, then the factor group of @Smodulo the center z of (Mis isomorphic with LF(2, p). In other words: $3, considered as a collineation group, represents LF(2, p) isomorphically. Proof. (a) If 8 is reducible, the argument in ? 5 (a) can be applied. It follows from Theorem 3 that this case is impossible. We may, therefore, assume that 8 is irreducible. (b) We next deal with the case where the block B1 of characters contains a linear character besides g, = 1. We first assume that this linear character is one of the characters a. Theorem 1 and Corollary 2 show that a (G) =1 for all p-singular elements of (M. If R is the normal subgroup consisting of all elements of 5 with a (G)- 1, then .S contains P and all other elements of order p; further, it contains PV for any V in 5, and hence it contains V, i. e., _{P}X (16) The representation 3 of (Minduces a representation 3* of degree (p -1) /2 of R. If 3* is reducible, the remark in (a) shows that O)contains a normal subgroup $ of order p. According to Sylow's theorem, this subgroup is characteristic, and hence it is a normal subgroup of 6, in contradiction to the assumption. Suppose now that :3* is irreducible. We may assume that Theorem 4 is true for all groups whose order is smaller than g. Then the theorem holds modulo the center E,, of e~ is isomorphic with for R. The factor group / 11 Cf. the remark at the beginning of ? 6.

GROUPS WHOSE ORDER,CONTAINS A PRIME TO THE FIRST POWER II.

433

LF(2, p) . Since in LF(2, p) every element of order p is congruent to its 72-th power (where y again is a primitive root mod p), we may find an element H in t and a center element COsuch that H-1PH

=

F2Co.

But then P'Y'2Cis an element of order p, and since it commutes with P, it must

be a power of P. Consequently, COalso is al power of P. However, the center (E, of 0 cannot contain a subgroup of order p, because any normal subgroup of O of order p is a normal subgroup of (X. This shows that COcannot have order p; we have, then, CO-- 1, and P and P"Y2are conjugate in '. The elements P and PY are not conjugate in (, since this would mean t -1 and Theorem 1 and the corollaries show that then (Mcannot have a (1 - 1) representation of degree

IA,

since both sides have the same degree on account of Lemma 3 (cf. (12) which turned out to be an equality in our case). If, therefore, an element G of (M is represented by the unit matrix in all representations of B1, then (18) shows I. This implies that 3(G) can have only one characthat :3(G) X 3(G) cI, and G belongs to the center 3 of (S. teristic root c, i. e., 3(G) On the other hand, every irreducible representation of (S/2 defines an irreducible representation of S. If Z, is the corresponding character, we have for V in B. We see easily 12 that t, tv (PiV) = g (P) - zv (mod 1 46)

belongs to B1, provided that its degree zv is prime to p.

8. The characters of the group {P, MA,813}/Z. According to Lemma 3, we have t = 2 and (1) thus has the form M-1PM =Py For n = (p - 1)/2, the n-th power of M belongs to T. The elements M, P {P, Mt,B3}/Z of order np. Let t be a primi(mod 93) generate a group N tive n-th root of unity. Then =

M ->,

P

1

defines a linear representation of S. If (t) is its character, every linear n - 1). Besides. S has character of S is of the form (4)v, (v = 0, 1, two conjugate characters13 X and X' of degree n. These characters vanish for the elements MK (K = 1, 2,

*, n

-

1), i. e.,

We may apply the condition [1], (27) to t1 and xv. This can either be seen directly or as a special case of the results of [4]. For the group ), we have I = 1, t = 2. Since we have n = (p - 1) /2 linear characters, the two p-conjugate characters X, X' must have degree n as n + 2n2 = np is the order of j; 12

13

GROUPS WHOSE

X(1)

=X

OR1DER CONTAINS

(1) =n,

A PRIME

X(MK) = X (MK)

TO THE FIRST POWER II.

-0

for

KE

435

0 (mod n).

After interchanging X and X', if necessary, we may assume that X

E(E)P

=

(for the elements of {P}),

where p ranges over the n quadratic residues (mod p). In the representation corresponding to X, the element P is represented by a matrix with the characteristic roots OP,while M is represented by a matrix whose roots are the n-th roots of unity. The determinant of the former matrix is 1 and that of the latter is (- 1) n+1.

From Theorem 2, applied to the group Q, it follows that X[ contains all the characters (") . Clearly, XI D 2 (t) is impossible, since it would imply that X(t)V D 2X (cf. Lemma 1), and the left side has a smaller degree than the right side. The representations of 0 of the first block B1 represent the elements of S3 by the unit matrix (cf. ? 7). Consequently, these representations of @ induce representations of W. Restricting ourselves to elements of {P, M, B3}, the characters a, fS,X,Xl can be expressed as linear combinations of X, X', (t)); the coefficients are rational integers ? 0. On comparing the characteristic roots, we see easily that for elements of {P, M, 58} (19)

A

X+X';

a

(45)P for a

(t)#+

X+X'+

, - 1. The character X conwhere t, v are two of the numbers 0, 1, 2, tains one of the characters X, X'. It is equal to this character in Case 03 where Dg (x) = Dg(X) = n, while in Case one of the linear characters Dg(X) + 1. (t)K must be added, since Dg(x) =- n,+ 1 The representations of 0 belonging to B1 are all unimodular.14 Indeed, the determinants of the matrices of such a representation form a linear character A of (, and A/(V) -1 for V in B8. Then A itself belongs to B1 (cf. 12), 1. On comparing the determinants of the matrices repreand hence A =g - v in (19) while K n/2 in Case (1, i. e., senting M, we see that ,

a

=

=

(20)

o

(2a,C)

X

(21, 3)

X

+X'+ X +

(t)n/2

(0)#+

W-A.

or X

X' +

or X=X'

(t)n/2

in Case

a;

in Case X,

no character of type II occurs. For p-regular elements G, the value X (G) is the sum O 0 (mod in), while the value of the n linear characters, which gives X (Afk) = 0 for k of X(P) is obtained' directly from [4], Theorems 4 and 11. of 6 are represented by matrices whose deter14 That is to say, all the elements minant is 1.

436

RICHARD BRAUER.

for elements of {P, M, 58}. On forming XXfor the elements of the same subgroup, we obtain the same linear characters (4)" as are contained in X 15 in Case X, while a further character ($)0 1 appears in Case C. Using a remark above, we find

(22,C a)

X x2(

(22n,3)

XX

)?+ E ($) +

*

(Case

a),

fl-i

(0)> +

v=O

(Case p),

where the characters, not written down on the right side, are not linear. On the other hand, Theorem 2 applied to the character x of ( shows. that XXcontains all characters o, and also one of the characters X,X' in Case a. 1 =(0). According to Lemma 3 (? 6), we In Case a, one of the a is have (p -5)/4= (n/2) - 1 further characters a of degree p + 1. The formula (22) now shows that no two of these a can contain the same (t) . Furthermore, XX,which (as a character of () contains X or X'/ contains only (n - 1) /2 characone such constituent. In Case 8, we have (p - 3)74ters a of 'degree p + 1. Again, (22, 03) shows that no two of them contain the same Consider a again as a character of (. The conjugate complex character a also belonigsto the first block.16 According to (20), the same linear characters (4)" appear in a and oc, if considered as characters of S.

As we have

a. The character seen, different a never contain the same (a)". Hence a XXis real. Since it contains exactly one of the characters X,X', this constituent is real. The p-conjugate characters X,X' are either both real or both not real. Hence we have =

4. Under the assumptions of Theorem 4, the degrees of the characters of B1 have the values given in Lemma 3. All the characters a are real. In Case a, the characters x, x' also are real. LEMMA

9. The degrees of the modular characters of B1. We separate the two cases C and B. The methods applied will be essentially the same in both cases.

a: The subset B*1 of characters g[ of B1 with St= 1 consists of *, 1,, xx' and (p )/4 characters au of degree p + 1 ( l,2 17 modular and characters B1 contains The block (p 1)72 (p 5)/4). 4, each of them appears as a modular constituent in exactly one of the characters gl, X, ca. On the other hand, each p appears as a modular constituent of CASE

=

15

1G 17

XX and X'X' contain the same linear characters (t) ". This can be seen easily from ? 7 (d) and also by means of Cf. [4], Theorem 3.

12.

GROUPS WHOSE

ORDER CONTAINS

A PRIME

TO THE FIRST POWER II.

437

exactly one of the (p - 1) /4 characters /8 of degree p - 1. This shows that Dg(4) _p - 1. Furthermore, it follows that we have at most (p - 1)/4 modular characters 4 in B1 with Dg (4) > (p - 1)/2. In the modular sense, each of the (p - 5)/4 characters ac must be reducible. The constituents of the ac account for at least (p - 5)/2 modular characters of B1, the constituents of X and the character C, for at least two further modular characters. (p - 1) /2 is the full number of modular charactersof Since (p -5) /2 + 2 B1, it follows that every ac has exactly two modular constituents, while X is modular-irreducible. One of the constituents of au has at least the degree (p + 1)/2, and X also has the degree (p + 1)/2. This gives (p - 1)/4 characters ( of a degree > (p - 1) /2. As a remark above shows, the other (p - 1) /4 modular characters of B1 necessarily have a degree ? (p - 1) /2. In particular, every a splits into two modular constituents of different degrees. Since a is real (Lemma 4), and the modular constituents cannot be conjugate complex, both constituents are real. The character X also is real (Lemma 4). Hence all the modular characters of B1 are real. Now Theorem 14 of [3] can be applied, which shows that the tree corresponding to the block B1 is an open polygon. The two end points correspond to the modular-irreducible characters 1 and X. If the notation is chosen (p - 1)/4): suitably, the polygon has the following form (h

3 la (23,a)

1

l

31

l

a

12

ah-l

fh

X

The sides of this polygon correspond to the modular characters of B1. The sum of the degrees of the characters corresponding to adjoining sides is either p - 1 or p + 1, according as the common vertex is associated with an acuor with a flu. The side 13,81corresponds to the modular character 1, and we find successively, that the degrees of the modular characters of B1 are given by

(24,a)

1,p -2,3,p-

4,

(p+5)/2,(p

3)/2,(p+)/2.

CASEa3: Here, the decomposition of g- 1 and the (p - 3)/4 char1, 2, * *, (p - 3)/4) of degree p + 1 into modular constiacters ca (a tuents gives all the modular constituents ( of B1. Each of these constituents also appears either in X or in one of the (p - 3)/4 characters /,3 of degree and shows This implies Dg()) ! p-I, l,2y,* * , (p -3)/4). p-1 (ar a have 4 of the > (p-l) degree may /2. at most -3)/4 (p also that Since B1 contains (p 1)/2 modular characters 4, it follows as in Case a that each au splits into two modular-irreducible constituents, one of which has a degree _ (p + 1)/2, while the other must have a degree ? (p - 1)/2. 14

438

RICHARD BRAUER.

Again, it follows that all the modular characters of B1 are real. Therefore, the tree corresponding to the block B1 is an open polygon. According to Theorem 2, the character xRof (Mcontains all the characters a.

Since Dgg(X) = (p -

1)/2,

no other character of (M can appear

(cf. (18, 3). Hence Xx-i

+

.

This shows that xRcontains the modular 1-character once only. Consequently, X is modular-irreducible.18This shows that gl 1 and x correspond again to the end points of the polygon. Here we have for p > 3 (h (p - 3)/4) =

I

(23,83)

I

I 1

I

I

a1

32

l *PAh

l

l

ah

X

The degrees of the modular characters of B1 are given by 1'~p - 2~,3,~,p.-4,

(24~, )

, (p +3) /2~,(p-

1) /2.

The case p 3 is without interest, as the assumptions of Theorem 4 cannot be satisfied. We have therefore the result, =

LEMMA 5. If ( satisfies the assurmptionsof Theorem 4, then the first block B1 contains exactly one modular character of degree 3.

10. The modular representation of degree 3. Let a be the irreducible modular representation of ( of degree 3, whose character ( belongs to the first block B1 (cf. Lemma 5). We can easily show that H can be written with coefficients in the Galois field GF(p) with p elements. Indeed, if this were not so, the traces of the matrices $ ( G), representing G, could not lie in GF (p) for every G in .0.19 Then we could find a representation $* which is algebraically conjugate to H without being similar. It would belong to B, and have the degree 3 which contradicts Lemma 5. Since ( was seen to be real, it follows further that $ is similar to its contragredient representation

.

If He M-1$M,

where M is a non-singular matrix with coefficients in GU(p), we conclude easily that MM'-1 commutes with every matrix of ~. Hence MM'1 = cI with c=/ 0 in GF(p), M = cM', M' = cM, and consequently C2 = 1. On the 18 The product of two contragredient modular characters contains the 1-character, as in the ordinary theory. Hence if xX contains D, exactly once in the modular sense, then X is modular-irreducible. However, if X is modular-irreducible, XX may contain , (modular) more than once. 19 Cf. footnote 12 of [3].

GROUPS WHOSE ORDER CONT.AINS .A PRIME TO 'l'HE FIRST POWER II.

439

other hand, on forming the determinants of M and M', we find c8 = 1. This gives c = 1, i. e., M' = M. Consequently, iJ has a non-degenerate quadratic invariant. We may, therefore, assume that the matrices of F are orthogonal matrices of GF(p). The group of all orthogonal matrices of degree 3 with coefficients in GF (p) has a normal subgroup 0 1 ( 3, p) in Dickson's notation, 00 consisting of the unimodular orthogonal matrices. The factor group is cyclic of order J. ~ The elements of @, which are represented in iJ by matrices belonging to 0 1 ( 3, p), form a normal subgroup ~- This group ~ contains m; the factor group @/~ is cyclic. If ~ C @, we may construct a linear character 21 l;p. of @ such that (;p."'F (; 1 but { 11 (V) = 1 for V in m. Then (;µ would belong to B1 (§ 7 ( d)), while (;1 was the only linear character in B1, Hence all the matrices of iJ belong to 0 1 (3, p). The group 0 1 (3,p) has a normal subgroup F0(3,p) of index 2; we conclude in the same manner that the matrices of iJ belong to F0(3', p) =:::.LF(2,p).

Let

m*

be the normal subgroup of

@

consisting of the elements G which

iJ represents by the unit matrix I. Obviously, mCm*, and @/m* is isomorphic with a subgroup of LF(2,p). The modular representation iJ appears as a constituent of at least one 7 (cf. (23), (24) 3). For p 5, they are #3. 7, the characters are a, and X; for p and X. It follows that the order of (S/W* must be divisible by (p- 1) (p + 1) /2 for p > 7, since the group has irreducible representations of degree p - 1 and p + 1. The conclusion is also correct for p 7, where we have irreducible representations of degrees 8 and 3, and for p 5, where we have irreducible representations of degrees 4 and 3. Since (S/W* is isomorphic with a subgroup of LF(2, p), and since its order is divisible by p and by (p-) (p + 1)/2, it follows now that O/F8* -LF(2, p). (25)

a,

=

=

=

=

The group LF(2, p) has two classes of conjugate elements which contain elements of order p. Hence 24 LF(2, p) has (p- 1)/2 + 2 irreducible representations of degrees which are not divisible by p. This is the full number of ordinary representations of (Mbelonging to B1. Consequently, every representation of B1 represents the elements of F3*by the unit matrix. According to ? 7 (d), we then have * C1 , and hence (26)

F1

*.

The equations (17), (25), and (26) contain the proof of Theorem 4. UNIVERSITY OF WISCONSTN.

BIBLIOGRAPHY.

[1] R. Brauer-C. Nesbitt, " On the modular characters of groups," Annals of Mathematics, vol. 42 (1941), pp. 556-590. [2] R. Brauer, " On the connection between the ordinary and the modular characters of groups of finite order," Annals of Mathematics, vol. 42 (1941), pp. 926-935. [3] R. Brauer, " Investigations on group characters," Annals of Mathematics, vol. 42 (1941), pp. 936-958. [4] R. Brauer, " On. groups whose order contains a prime number to the first power I," American Journal of Mathematics, vol. 64 (1942), pp. 401-420.

24 Actually, the characters of LF (2, p) are known, but the fact used here follows at once from the results of [41 and does not require the knowledge of the characters of LP (2, p).

ANNALS OF MATHEMATICS Vol. 44, No. 1, January, 1943

ON PERMUTATION GROUPS OF PRIME DEGREE AND RELATED CLASSES OF GROUPS BY RICHARD

BRAUER*

(Received June 17, 1942)

Introduction The transitivepermutationgroups of prime degree p appear as the Galois groupsof the irreduciblealgebraicequationsf(x) = 0 of degreep. This is the reason that these groupshave been the subject of a large numberof investigations.' However,only fewresultsof a generalnaturehave been obtained. In the presentpaper, the theoryof group representations2 will be applied in order to derivesomenewtheoremsconcerning thestructureofthesegroups. Actually, the methodcan be used forthe studyof a widerclass of groups,viz. the groups 5 of finiteorderg whichhave the followingproperty: P ofprimeorderp whichcommuteonlywith (*) The group(Mcontainselements theirownpowersPi. It is clear that transitivepermutationgroupsof degreep have the property (*). Secondly,the doubly transitivepermutationgroupsof degreep - 1 are of this type.3 A thirdexampleis furnishedby the irreduciblelineargroupsin a p-dimensionalvector space whose centerconsistsof the unit elementonly, in particularby the simplelinearirreduciblegroupsin p dimensions(cf. section7). It is easily seen (section1) that the orderg of a group(Mwiththe property(*) is oftheform (1)

g = (p

-

l)p(l

+ np)/t

wheret and n are integersand wheret divides p - 1. The group (Mcontains exactly 1 + np conjugate subgroupsof orderp, and each of them has a normalizer of order p(p - 1)/t. In section 2, the normal subgroups of 65 are studied, in particularthe firstcommutator-subgroup (i' and the second com6" of 6. Two cases mustbe distinguished: mutator-subgroup CASE I. The group0Mcontainsa normalsubgroup( oforder1 + np. We shall show that 5/2ithen is a metacyclicgroup of order p(p - 1)/t; the group (2 possessesan outerautomorphismof orderp whichleaves only the * Fellow of the John Simon Guggenheim-Memorial Foundation. We may mention here the work of Mathieu, C. Jordan, Sylow, Frobenius, Burnside, G. A. Miller. Cf. also E. Pascal, Repertorium der h6heren Mathematik, Vol. I, part 1, 2nd German edition, Leipzig 1910. 2 In this paper, the notation "representation of a group" always means a representation of the group by linear transformationsof a vector space over the field of complex numbers (or an algebraically closed field of characteristic 0). By a "vector-space" we always mean a vector-space over this field. 3For this class of groups, cf. G. Frobenius, Sitzungsberichte der Preussischen Akademie, Berlin 1902, p. 351. I

57

58

RICHARD BRAUER

unit element fixed. For t < p - l, we have 0 = @", and @' has the order p(l + np). Fort = p - l, we have 0 = @'. Unless n is of the form n = u

(2)

+ m + ump

(u, m positive integers),

0 is a minimal normal subgroup of@ (for n =I= 0). This case I is of relatively small interest. In particular, when @ is a transitive permutation group of degree p, @5 consists in this case I only of the unit element 1. If @ is a doubly transitive group of degree p + l or an irreducible linear group in a p-dimensional space with center 1, then 0 must be abelian. CASE II. The group @ does not contain a normal subgroup of order l + np. Here, we shall have @' = @". The group @' itself satisfies the condition (*); its order g' is of the form (3)

g'

= (p - l)p(l + np)/t'

where n is the same number as in (1). The number t' divides p - l and is divisible by t; we have t' ,;e. p - l. If n is not of the form (2), in particular, if n < p + 2, then @' is simple. In the later sections, we shall assume that @, besides condition (*), satisfies the following condition (**) The commutator-subgroup @' of@ is equal to @. By this condition (**), groups@ for which we have case I are excluded. If we have case II, the group @' satisfies both conditions (*) and (**), and our theory can be applied to @'. From @', the group @ can be obtained by a cyclic extension; the value of n remains unchanged. Our main result (section 5), is: If a group @ satisfies the conditions (*) and (**), and if n ~ (p + 3)/2, then n can be represented by the following rational function F(p, u, h)

(4)

n

= F(

p, u,

h) = puh

+uu ++l u + 2

h

where u and h are positive integers, and where u + l divides h(p - 1). If@ satisfies the conditions (*) and (**), and if n < (p + 3)/2, we must have one of the following two cases: (a)

n = l,

(b)

r,i

=

(p -

t = 2,

3)/2,.

@

= LF(2, p),

t = (p - 1)/2,

(p @

= LF(2, 211 )

> 3).

where p = 211

+1

is a Fermat prime, p > 3. 4 In a later paper, the values n with (p + 3)/2 ~ n ~ p + 2 will be discussed. lt had been shown by Frobenius that LF(2, p) is the only simple group of order p(p - l)(p + 1)/2. In section 6 we drop the assumption (*) and prove that the groups@ = LF(2, p) and LF(2, 211 )> (2 11 + 1 = p) are the only simple 4 That permutation groups of degree p with the value n p, was mentioned by Frobenius, loc. cit.

479

= (p - 3) /2 exist for these primes

REPRINT OF [42]

59

ON PERMUTATION GROUPS OF PRIME DEGREE

+

+

groups of an order p(p - 1)(1 mp)/T with m < (p 3)/2, (p a prime, T, m not-negative integers, T 1 (p - 1)); if for a simple group of this order we have m ~ (p + 3)/2, then m must be of the form m = F(p, u, h) where u and h are positive integers; 1. Preliminary remarks Let@ be a group of finite order g which satisfies the condition (*), i.e. which contains elements P of prime order p whose centralizer consists of the powers of P only. If $ is a p-Sylow subgroup of@ which contains P, th~n the order of $ cannot be larger than- p, since otherwise the order of the centralizer of P in $ would be larger than p. Hence g '1- 0 (mod p 2), $ = {P). The number of subgroups conjugate to $ is of the form 1 + np where n is a non-negative integer. The order of the normalizer 91 = 91($) of $ then is g/(1 + np). But since $ is a cyclic group of order p, and since 91 also satisfies the condition (*), we readily see that 91 can be generated by $ and another element Q such that pP = 1,

(5)

Qq = 1;

Q-IpQ = p'Y'

where 'Y is a primitive root (mod p), and where t and q are positive integers such that (6)

tq=p-1.

The group @ then contains exactly t classes of conjugate elements of order p. For the order of @, we obtain (7)

g

=

(p - l)p(l

+ np)/t =

qp(l

+

np).

Hence we have THEOREM 1. lf @ is a group of finite order g which contains an element P of , prime .order p which commutes only with its own powers (condition (*)), then g = (p - l)p(l np)/t, where n and t are integers, and;t divides p - 1. The group @ contains exactly 1 + np subgroups of order p, and t is the number of classes of conjugate elements of order p in@. Since g contains the prime p only to the first power, the results of an earlier paper5 can be applied. For the sake of convenience we mention those facts which will be needed. The ordinary irreducible representations of @ are of four different types: (I) Representations ~P of a degree ap = upp + 1 1 (mod p). Denote by AP(G) the value of the character AP of ~P for an element G of @. Then

+

=

(8, I)

'1-

0 (modp)).

-1 (mod p ).

If B.,(G) is

(fori

II. Representations SB„ of a degree b., the character of SB.,, we have

= v„p -

1 -

5 R. Brauer, On groups whose order contains a prime number to the first power, American Journal of Mathematics vol. f4 (1942) part I p. 401, part II, p. 421. I refer to these papers as [1) and [2).

480

FINITE GROUPS

60

RICHARD

BRAUER

- (fori 4 0 (mod p)).

B,(Pt) = -1

(8, II)

(III) Representations(S ofa degreec whichis not congruentto 0, 1, -1 (mod p) (i-J), and 1'6 There exist exactly t such representationsA, S', .., L fort theyare algebraicallyconjugate. The degreec is of the form c = (wp + 6)It,

6 = 4-1

whereto is a positive integer. If e is a primitivepth root of unity,suitably chosen,we have forthe characterC(G) of Cs:

(-s)

C(p) =

(8, III)

q-1

Z fie"

fori

,U=o

4 0

(mod p).

We denote the expressionon the rightside by (-6)tqi so that Pi is a Gaussian periodof lengthq = (p - 1)/t. (IV) RepresentationZT of a degree dT = PXT 0 (mod p). If DT(G) is the characterof Z)T, then (8, IV)

If we have a representation2fp,p 2 ... ,we have 1, 2,

=

fori 4 0 (mod p).

0

=

DA(Pt)

, a, and /3representations

1, 2, *

F8,0

a +0=q

(9)

(p-1)/t.

forelementsG of an orderprimeto p, we have Furthermore, 3

a

Z A p(G) +

(10) In particular,forG (11)

=

C'Y'(G) = Ej B(G).

1, this gives

Z ap +

6c-

bo

Pa It is well knownthat the degreesap, b, , c, d1 divide the orderg of 5 and that g is equal to the sum of the squares of all the degrees,i.e. (12)

Ea p P

+

E

a

2

+ tC2+ E T

T

d2

= g

It is oftenconvenientto set (as above) (13)

ap = upp+ 1, b, = vqp - 1, c = (wp + S)/t,

dT

= XTP,

(

=

?1).

On substitutingthese values in (11) and taking (9) into account, we easily obtain (14)

p

U+

t -

v

6 In the case t = 1, G can be chosen arbitrarily among the p irreducible representations of degrees not divisible by p. We then choose Y so that its degree c is of the formc 3-1 (mod p). This is always possible. The results given in (III) remain valid for this C. We then have 6 = -1, and C(Pi) is rational.

61

ON PERMUTATION GROUPS OF PRIME DEGREE

Substitute the values (13) in (12) and use (9) and (14). tion gives (15)

'°' + '°' v„ + wt + '°' 2

~ Up

~

2

2

2 _

~ XT -

pn - tn

A simple computa-

+ 1.

2. Normal subgroups of @

The number of · representations of degree 1 of any group @ is equal to the index (@:@') of the commutator-subgroup @' of @. In our case, for t ~ 1, t ~ p _, 1, only the representations ~P can have degree 1. By (9), their number is at most (p - 1)/t. Fort = 1, we may choose (g: (p p(l + np). If s $ 0, theorem4 shows that s is of the form1 + up whereu is a positiveinteger. From (7) and (16) we obtain

1 + np = (1 + mp)(1 + up). Hence n = u + m + ump. Under our presentassumption,we must have m = 0, i.e. s = 1 + np and thisprovesthe corollary. We now distinguishtwo cases: oforder1 + np. CASE I. TheGroup05containsa normalsubgroup The group05 does not containa normalsubgroupof order1 + np. CASE II. In otherwords,in case I the orders* of S* is equal to 1 + np whilein case II s* is smallerthan 1 + np. THEOREM 5. We have case I, if and only if one of thefollowingtu-osets of conditionsholds (a) (b)

t = p-i. subgroupsOi' and 5" of (D t < p - 1, and thefirstand secondcommutator are different.

In case (a), @3'has theorder1 + np, and 3/3' is cyclic of orderp. In case (b), thegroup@3"has theorder1 + np and @3'has theorderp(l + np); 3/3" is and can be definedby theequations(5). metacyclic PROOF: The case t = p - 1 is trivial,cf. theorem2 and (7); we may assume t < p- 1. If S* is an invariantsubgroupof order1 + np in (D, then (5/S* is a groupoforderp(p - 1)/t,whichsatisfiescondition(*) and in whicht classes of conjugate elementscontain elementsof order p. Hence (5/S* contains a subgroupof type (5), and since this subgrouphas orderp(p - 1)/t,the group W5/S*itselfis a metacyclicgroup of type (5). In particular,(M/e* containsa normalsubgroup(Sh/(* of index (p - 1)/t. Then (S1 is a normalsubgroupof index (p - 1)/tof (D, and theorems2 and 3 now show that (S1 = (M'. We may apply theorem2 to (M'whichagain satisfiescondition(*). Since (i' containsa normalsubgroupS* of index p, this group A* must be the commutatorsubgroup(M"of W5'. Conversely,assume that (M' $ (M". Accordingto theorem2 we have (61:65') _ (p - 1)/t. The orderof (M'then is divisibleby p, and (i' also satisfiesthe condition(*). If the index (61': 6") was primeto p, the group 61" would have an orderdivisibleby p, and theorem3 would give 61" D 61', i.e. 61" = 61'. Hence (61': 3") is divisibleby p. Now theorem2, applied to 61', gives (61':(6") = p and, therefore,(: 61") ? p(p - 1)/t. However, theorem4 shows that the orderof the normalsubgroup61" of 61 must divide 1 + np. As 61has the orderp(p - 1)(1 + np)/t,we now see that 61" has the order1 + np. This completesthe proofof theorem5. COROLLARY 5. In case II, theorderg' ofthegroup61'is givenby (17)

9' = (p

-

1)p(l

+ np)/t'

64

RICHARD BRAUER

wheren is thesame numberas in (7) and t' denotesthenumberof classesofconjuof orderp. W~ehavet I t', t' I (p - 1), gateelements in (i' whichcontainelements = ? p (i' (M". The group(5/(5' is cyclic. 1. Furthermore, and t t' < PROOF: It followsfromtheorem5 that t < p - 1 and that 65' = 65". The group(M'also satisfiescondition(*), and since it containsall subgroupsof order p of (D, we obtain (17). The numbert divides t', because g' divides g. If we had t' = p - 1, theorem2 would give (M' 5 5". The elementQ in (5) has the propertythat its (t'/t)thpower is the firstpower whichbelongs to (5'. This showsthat(5/(5' is cyclic. COROLLARY6. If n is notof theformn = u + m + ump, (u, m positiveintegers),in particularif n < p + 2, then6i' is simplein case II. PROOF: Corollary4 shows that I* = {1}, and corollary3 now proves the statement.

Finally, we treat the three kinds of groups mentionedin the introduction. We prove permutation groupofdegreep (p a primenumTHEOREM 6. If (D is a transitive ber),then 5 does notcontainany normalsubgroupof an orderprimeto p and different from{ 1 }; thegroup(i' is simple(or oforder1). If (D is a doublytransitive groupof degreep + 1 or if (D is an irreduciblelineargroupwithcenter{ 1 } in a p-dimensionalspace, thenany normalsubgroupof an orderprimeto p is abelian. seriesof(D has at mostonenon-cyclic factorgroup. In all thesecases,thecomposition PROOF: If (D is a transitivepermutationgroupof degreep, then 6 possesses a reducible(1-1)-representation23of degreep, whose characterhas the value 0 forelementsof orderp. As shownby the formulas(8, I), this representation it cannot contain cannot consistof representations2f,exclusively;furthermore Theorem 4 and corollary2 now show that 23 has the any constitutentTr. kernelA*. However, 23was a (1-1)-representation. We thus find * = {1 }, and corollary3 showsthat T-is simple (or (i' = I1 }, ifg = p). Any doubly transitivepermutationgroup (Mof degree p + 1 possesses an irreducible(1-1)-representationof degree p. As condition (*) holds for any to sufficient such (D, we easily see that 65has the center {1 }. It is, therefore, treat irreduciblelinear groups with the center {1} in a p-dimensionalspace. It followshere that A*, consideredas a linear group,must be reducible since the dimensiondoes not divide the orders*. Since t* is a normal subgroup, it splitsinto constituentsof the same degreez. Then z = 1, and t* is abelian. The last statementof theorem6 followsfromcorollary3. lineargroupin a p-dimensional irreducible COROLLARY7. If (D is a primitive space and if (D has thecenter{ 1 }, then(i' is simple. PROOF: When (D is primitive,the normal abelian subgroup A* must lie in the center. We thenhave (5* = {1 underour assumptions. Now corollary 3 givesthestatement. There is a well known theoremof Burnside which states that a transitive permutationgroup (D of degreep is eitherdoubly transitiveor it is metacyclic of the type (5). With the methodsused here,this could be proved in the fol-

ON PERMUTATION

GROUPS

OF PRIME

65

DEGREE

lowingmanner. If (D is not doubly transitive,the permutationrepresentation 23splitsinto the 1-representation 2[l and at least two moreconstitutentsnot all of whichcan be of type 21,. Then one constituentat least is a G(S). Since the characteris rational,all the conjugate G(O)appear, and we find t-1

1+ E (',

3

t > 2.

C = (p -l)/t

If t > 2, the group (D must have a normalsubgroupof orderp,10and therefore (D is oftype (5). If t = 2, we have also to considerthe case that (M_ LF(2, p). to excludethis last possibility. However,the proofmay It is not very difficult be omitted. 3. Conditionsforthe degrees a, , b , c We now make use of the fact that the degreesa, , be, c (cf. (13)), divide the order g = (p - l)p(l + np)/t of 0. For the a,,, we certainlymust have 1)(1 + np). If the sign a has the value +1, the condition (upp + 1) I(pforc gives (up + 1) I (p - 1)(1 + np). The question we have to treat then is this: When is (up +

)

+ np)

(p-)(

(where u = u, or u = w)? Set up + 1 = mlm2 with ml I (p -1) 0 (mod ml) and hence and m2 (1 + np). Then p = 1, up + 1 0 (mod m2),and this 1 1 0 1+ np + u+ (mod ml). Further,up 0, 0 and hence n - u 0 (mod M2). It now followsthat gives (n - u)p up + 1 = mlM2 divides (u + 1)(n - u). We may set (u + 1)(n-u)

(18)

with an integral h. Then h(up + -1 (mod u + 1), this gives

= h(up + 1) 1)

=

0 (mod u +

1).

Since u

-

(u + 1) I h(p-1).

(19) Assume firstthat h gives

=

-h' < 0.

Then (u + 1)(u

-

n) = h'(up + 1) which

+ h' +n > u(h'p+ n 1). u(h'p + n Hence u > h'p + n - 1 while (19) yields h'(p - 1) ? u + 1, i.e. u ? h'p - h' - 1 ? h'p + n - 1. This is a contradiction,the case h < 0 is impossible. From (18) it followsthat n = (hup + U2 + u + h)/(u + 1). We have therefore LEMMA 1. DenotebyF(p, u, h) therationalfunction U2 =

F(p, u,h) = hup + u + u + h

(20) 10 cf.

[2], theorem 3.

66

RICHARD

BRAUER

ofp, u and h. If gt = (p - 1)p(1 + np) is divisiblebya number1 + up where thenthereexistsan integerh _ 0 suchthat u is a non-negative integer, (21)

n = F(p, u, h)

and that(u + 1) I h(p - 1). In a similarfashion,we have to findthe conditionthat vp - 1 divides gt = (p - l)p(l + np) where v > 0, (v = v.,, or v = w for a = -1). We set vp - 1 = mIm2withml I (p - 1) and m2I (1 + np) and derivev-1 0 (mod m2). Hence 0 (mod ml), v + n (v-

(22)

1)(v + n) = h(vp-

1)

forsome integerh > 0. For fixedn, p, h, this is a quadratic equation forv. The second root v' = (h - n)/v must be integraltoo. For h = 0, we have v' = -n/v ? 0. If h $ 0, replace v by 1 in (22). The leftside of (22) then vanishes while the rightside is positive. This again gives v' < 0. Set u = -v' = (n - h)/v; thenu > 0. Since v' satisfiesthe equation (22), this equation remainscorrectwhen v is replaced by - u. We thus come back to (18). Since (18) implied (19) and (21), we have proved LEMMA 2. If gt = (p - l)p(l + np) is divisiblebyvp - 1, wherev is a positiveinteger,thenthereexistsa non-negative integerh such thatu = (n - h)/vis integraland notnegative,and thattherelationshold n = F(p,u, h);

(23)

(u + 1) I h(p-1).

If u = 0, thenn = h, and (22) gives v = pn - n + 1. However,it follows from(15) that v,,= pn - n + 1 or w = pn - n + 1 are possible only when v, = 1 orw = 1,i.e. whenn = O. If h =O, thenu = nand v = 1. 7. If (D is a group satisfyingthe condition (*), thenfind THEOREM all representations of n in the form n = F(p, u(P) h(p)) = (hV'VuY')p + + u(P) + h('))/(u(P) + 1) withpositiveintegersu(') h('). The degreesof the u(j) irreducible representations of(D,as far as theyare primetop can onlyhave someof thevalues (24)

ap = 1,

ap = u(v)p + 1,

(25)

ha,= p - 1

ba = V(v)p - 1.

(26)

c = (np + 1)/t,

a = np + 1.

c = (u(v)p+ 1)/t,

c = (p -107

c = (v(F)p- 1)/t wherev(")is setequal to (rt-h )/u For h > 0 and variable u, we have (a/Ou)F(p, u, h) > 0. Since wc are only interestedin solutionsu, h of n = F(p, u, h) with 1 < u < h(p - 1) - 1, we must have (27)

F(p, 1, h)

-

hp + h + 2 ? n ? F(p, h(p 2

-

1)-1,

h) = 2ph - h -2.

ON PERMUTATION GROUPS OF PRIME DEGREE

67

This gives THEOREM

8.

In theorem 7, only values h = h 3). - 3 t= P , p = 2 + 1 > 3 a Fermatprime. Here, (5_ (b) n 2

LF(2,

p-

'-

2

1).

COROLLARY. If n < (p + 3)/2, then6 mustbe eitherof thetype(a) or of the type (b). PROOF: Suppose that n is not representablein the form (37). Then the of (5 are eitherdivisibleby p or have degreesof the irreduciblerepresentations one of the values (30). Because of condition (**) the degree 1 appears only once, say a, = 1. If t was odd, theorem9 shows that the degreep - 1 does not appear. It followsfrom(11) that this is impossible. Hence

(38)

t

0 (mod 2).

The degree (p - 1)/t is impossiblefort > 2;'1 fort = 2 it occursonly in the case 6 LF(2, p), i.e. in the case (a). Hence we may exclude this possibility. 14 This implies that p # 2. 15 If n is not of the form (37), it is not of the formn = ump + u + m with positive inte-

gers u and m, since otherwise we could set h = (u + 1)m and would obtain a representation (37). Then corollary 6 (section 2) shows that (' is simple. Because of conditions ("*), (5 is simple. Now, [2], theorems 3 and 4, can be applied.

ON PERMUTATION

GROUPS

OF PRIME

71

DEGREE

We now see that we must have a, = 1,

(39)

a2

b, = b2 =

=a3=

.: =b=

1

a.

*

p-

+ np,

c = (pn + l)/t

1,

and the sign 5 has the value +1. The values of a and A can be obtained from (9), (13), (14), and (39) which give (40)

a +

(41)

=(p-1)/t=q,

+ 1

)n+ 1n

(an-

In particular,n + 1 is divisibleby t; we set n + 1=

(42)

st,

and thenobtain (43)

1+(qt+1)(st-

1+np=

1)=tr

where r = qst + s-q

(44)

= ps-q.

By (43), the orderg of 65can be writtenin the form (45)

9

=

(p -

I)pr

The next step is to show that r and p - 1 (44), (42), (41), (40), we obtainsuccessively

s

-1,

n

0.

(s-

whenceit followsthat 1 s

q,

n

1)(1

qtpr.

=

qt are relativelyprime. Using

=

+ s _ 3,

. i.e. that (r, q) 0,

1. In a similarmanner,we have

=

(a - 1) (- 1) + s_,

_-1,

whichgives s + 1 _ s (mod (r, t)), i.e. (r, t) (r, p -

(46)

(mod (r, q))

ca + 0B-

=

8 (mod (r, t))

a + A-

1. Hence we have

1) = 1.

From (45) and (46), it followsthat forany prime1dividingp tersBo are of highestkind.'6 This implies

-

1 the charac-

Fg(L) = 0

(47)

forelementsL of 65whose orderis divisibleby 1. For the primesm dividingr the charactersof degrees 1 + pn = rt and (1 + pn)/t = r are of the highest kind. Hence (48) 16

A, (M)

=

0 for p # 1,

C(")(M)

0

Cf. R. Brauer and C. Nesbitt, Annals of Mathematics, vol. 42, p. 556 (1941), Chapter II.

72

RICHARD

BRAUER

forelementsM of k whose orderis divisibleby m. Because of the assumption (*), the order of the elementsL and M is not divisible by p, and hence the equation (10) holds for these elements. If an element G of M would be an elementL and an elementM at the same time,then everytermin (10) except A1(G) = 1 would vanish,and this is impossible. Hence the elementsof k are distributedinto four disjoint sets: (I) The 1-element,(II) the elements of order p, (III) the elementsL whose order is divisible by at least one prime factorof p - 1, (IV) the elementsM whose orderis divisibleby at least one primefactorof r. Considernow the followingelementof the groupring r belongingto k (49)

T=

q-1

Qj.

We wish to show that (p now will always denote one of the values 2, 3,

(50) A1(T) = q-

1,

Bo(T)

Ap(T) = -(n + 1), C(V)(T) = -(n

+ 1)/It

0, D,(T)

The proofof (50) can be obtained fromthe resultsof section4. (13) we have U1 =

U2

=

=a=n,

Vi=

a), )

= (q

-

By (39) and

w = n,

=v=1,

1)x .

a=1.

Lemma 3 shows that A1(N) containsexactly one wc, A p(N) withp > 1 contains exactlyn + 1 of the WM, B(N) does not contain any Wc,,C(N) contains exactly (n + 1)/Itof the wc,,D,(N) contains xT of the w, . Here, N is an elementof T = 9($), cf. (5). It now followsfrom(35) that each of the wcappears in one of the charactersA1(N), A2(N), * *, A (N), C(N), while (36) shows that the w,,appearing in Ap(N), p > 1, do not occur in any othercharacter,and that the w,,appearingin C(N) occur onlyin the C(t)(N). Since, obviously,A1(N) = 1 = wo(N),this implies that the Ap(N) with p > 1 and C(N) contain only W,, forwhich1A? 1 whereasthe D, contain only wo(N). The elementsQj belongto W. On account of (31) and (32), we have wo(T) = q -

1,

Wc,(T) = -1

for 1A$ 0,

The first formula (50) is obvious, as A1(N) =

YP) (T) = 0.

1 = wo(N).

Since Ap(N) for

p > 1 is a sum ofn + 1 termsco,with1A> 0 and oftermsY(v),we findAp(T) = - (n + 1). The remainingformulas(50) followin the same mannerusing the facts that, apart fromterms Y(v), the character C(^)(N) is a sum of exactly (n + 1)/ttermsco,,with1A> 0, the character D,(N) containsonlyXTo , and the charactersBol(N) do not containany termexcept termsY(v). Let v range over all the charactersA1 , Ap, B , C(V),D, of 65. If L again is an elementwhose ordercontainsat least one primefactorof p - 1, then (47) and (50) give

73

ON PERMUTATION GROUPS OF PRIME DEGREE

L

t(Tk(L)

=

(q -

1) - (n

+

a

L

1)

AP(L)

p=2

(51)

- (n

+

+

l)C(L)

(q - 1)

L x,D.(L) T

since the t characters c p > (p - 1)/r. We have a contradiction,and theorem11 is proved. 7. Examples of groupswhichsatisfythe condition(*) First, let (Mbe a transitivepermutationgroup on p letters,where p is a prime. The orderg of (D then is divisibleby p, and (Mcontains elementsP of order p. Each such element P is representedby a simple cycle of lengthp. It now followseasily that P commutesonlywithits own powers;i.e. 5 satisfies the condition(*). In a similarmanner,we can show that a doubly transitive group of degreep - 1 satisfiesthe condition(*). We next consider irreduciblegroups 5 of linear transformationsof a pdimensionalvectorspace21 wherep again is a prime. Assumethat 65is of finite 19 cf. [1], theorems 4 and 11. 20 21

cf. [11,theorems 4 and 11. cf. footnote 2.

78

RICHARD BRAUER

orderg and that its centerconsistsof the unit elementonly. The orderg is divisibleby the degreep of the irreduciblegroup. Let $ be a Sylow-subgroup from1. Then of (D, and let P0 be an invariantelementof $ whichis different Po cannotbe a scalar multipleof the unit matrix,since it does not belongto the center{ 1 of(M. But P0 commuteswitheveryelementof $, and Schur's lemma impliesthat the linear group $ is reducible. The degree of every irreducible constituentof $ must be 1, since it divides the orderof $3. Hence $ can be taken as a set of diagonal matrices,i.e. $ is an abelian group. We now prove LEMMA 5.2 Let (D be a groupof orderg = pag* with(p, g*) = 1, and assume that(Mdoes notcontaininvariantelementsof orderp, and thattheSylow-subgroup of 5 $ of orderpa in (Mis abelian. If 3 is an irreducible(1-1) representation thenu = a. ofdegreepTM, Let v be the characterof 8. If G is an elementof (M which has PROOF. j (G)/pM is exactlyj conjugateelementsthenit is well knownthatj (G)/ (1) an algebraicinteger. Since $ is abelian,thenumberj is primeto p when G lies 0 (mod pi) forG in $. On the otherhand, P(G) is a sum in $. Hence P(G) If G $ 1, not all theserootsofunitycan be equal. Hence ofpTrootsofunity. all its algebraicconjugatesare smallerthan 1 in absolute and the integerP(G)/pM value whichimpliesP(G) = 0 forevery G 5 1 in $.23 Then the characterv is of the highest kind,24i.e. pT =

(1)

stated.

0 (mod pa) which yield M = a, as was

LEMMA 6. Let (Mbe a groupoforderg = p ag* with(p, g*) = 1. If thecenter of 3 consistsof theunit elementonly,and if (Mhas a (1-1)-representation 3 of degreepa, thenthecentralizer of a Sylowsubgroup$ oforderpa is containedin $. The characters of 3 has the values25 PROOF. pa

P

=

0

PPin$,

1

P#

1.

Hence 8(P) is the regularrepresentationof $; we may assume that it breaks up into the distinctirreduciblerepresentationsj, of $, each j, appearingf, times where f, is the degree of

3(P)

We then have

,.

f X F.,

=

F,

= j,(P)

0O* 22 The following lemmas 5 and 6 are proved here in a more general form than necessary for our purpose. However, in the form given here, they can also be used in other connections. 23 For this argument, cf. W. Burnside, Proceedings of the London Mathematical Society (2) vol. 1, p. 388-392 (1904). 24 cf. theorem 10 of the paper mentioned in footnote 16. 25 cf. footnote 16.

DEGREE

OF PRIME

GROUPS

ON PERMUTATION

79

Let V be an elementofthe centralizer(S of $3such that the orderv of V is prime to p. Then 3(V) willcommutewith23(P). It followsthat :3(V) is ofthe form

x

([.

whereTMis a matrixof degreef, and I, is the unit matrixof degreef. Then S3(VtP) breaks up completelyinto the matricesT, X FM. If r(,) is the trace

of TA and if 0,(P) is the character of FA(P), we find P(VtP)

=

X,

r)(P)

Since r is of the highestkind, we have (70)

ErW(iA(P)

forP in$,

0

=

P P 1.

where the sum extends over all values of g. Using the Set E 'ri~fM = relationsforthe charactersof $, we derivefrom(70) the equation orthogonality

a~Ti,(i a

()

Z Z ''e(P~~p-A

EE P

M

r)0jO(P)OV

(P)

=E

M

fi A fl = fig (i'fl

This shows that the matrices Tl/f, have the same trace for all values of P. If a, is the 1-representation of $ then T1 is a vth root of unity X where v is the order of V. Hence

tr(T')

=

(,=

fT (i)/pa

=

Tfr(i)

fTxi

The mapping V' -- T' defines a representation of the group {V }. Since its character is identical with the character of the representation V' --> XI,, it follows easily that T' = XAI. Hence 3(V) = XI. This is impossible for V $ 1, because 3 was a (1-1)-representation, and 6 did not contain any invariant elements except 1. Hence the centralizer LEof $3 cannot contain elements of an order prime to p. Consequently, the order of E itself is a power of p. Since G and $3 generate a p-group contained in (, we have (S 5 $3 and this proves lemma 6. Returning to irreducible linear groups (D of degree p whose center consists only of the unit element, it follows from lemma 5 that the order g contains p to the firstpower only. If P is an element of order p, then lemma 6 shows that P} is the centralizer of P. Hence we see

groupsp, the 12. If p is a prime,thenthe transitivepermutation and the irreducible linear groupsof degreep + 1, permutation doublytransitive (*). condition the with center dimensions the {1 satisfy groupsin p THEOREM

THE INSTITUTE FOR ADVANCED STUDY.

VOL. 30, 1944

MA THEMA TICS: R. BRAUER

109

ON THE ARITHMETIC IN A GROUP RING BY RICHARD BRAUER UNIVERSITY

OF TORONTO1

Communicated March24, 1944 1. Every group ? of finite order g determines an associative algebra r, the group ring, over an arbitrary field K. This algebra consists of all linear combinations a =

aiGi

(1)

of the group elements 6? with coefficients a in- K, where equality of these elements, addition and multiplication are defined in the natural manner. The study of the algebra r forms one of the most powerful weapons we have for the investigation of groups of finite order. When the field K is of characteristic 0, the algebra r is semisimple, and the general principles of the theory of algebras can be applied. As is well known, the ordinary theory of group representations and group characters can be obtained in this way. We then assume that the field K is algebraically closed, or at least that K has the property that all irreducible representations of in K are absolutely irreducible. If the latter condition does not hold in the original field K, it will be satisfied in algebraic extension fields of K of finite degree. Without loss of generality, we may restrict ourselves to the case that K is a suitable algebraic number field,2 as we shall assume throughout *this note. There can be little doubt that we are far from knowing all important properties of group characters. In particular, we are interested in further results which connect the group characters directly with properties of the abstract group ?. Any result of this kind means, in the last analysis, a result concerning the structure of the general group of finite order. 2. One approach to our question is to study the arithmetic properties of r after we study the algebraic properties of F. An element a of r is said to be an integer, if all the coefficients ai in (1) are integers of K. The ring J of these integers is not a maximal order,3 and therefore the ordinary theory of ideals does not hold. However, our definition of integer is linked with the group ? in a natural manner, and we would lose this connection, if we replaced J by a maximal order. Actually, a study of most arithmetic properties of a maximal order would not lead us beyond the algebraic properties of r. The arithmetic question in which we are mainly interested4 is the question how a prime ideal p of K (or rather its extension (p) = pJ to J)) behaves in J. This leads to the study of the residue class ring J* = J/(p). But J* can be interpreted as the group ring of ? over the field K* of residue 5

MATHEMATICS:R. BRAUER

110

PROC. N. A. S.

classes of integers of K modulo p. We thus come back to a study of a group ring and of representations of ?, but now with regard to a modular field K*. The investigation of (p) becomes equivalent to the study of the modular representations5 of ?. For every prime ideal divisor 3 of (p), the residue class ring J/$ is a complete matric algebra of a certain degree f over K*. Thus, $ defines a modular irreducible representation ! of ? of degree f, and we have a (1-1)-correspondence between the prime ideal divisors 3kof (p), and the irreducible modular representations 51, $13, 2, ..., a2, ...,

k of ?.

Since J is not a maximal order, we cannot write (p) as a product of powers of the $j. However, we still have a unique representation of (p) as a direct intersection of ideals

ln a n ... n.flt (P)= ifl

(2)

Here, any two of the JY?are relatively prime, and no 9Wfcan be written as an intersection of relatively prime ideals which are different from S?,. The 3i dividing a fixed 93Mform a "block" B,, and according to what we said above, we may also speak of the modular representations ai belonging to the block B.. Every ~i then belongs to exactly one block. Finally, we may associate every ordinary irreducible representation 3j of 6? with exactly one block B.6' Every 3j of the block Br contains only modular constituents ai of B,, and, conversely, every ?i of B7 appears only in 3j of B,. This shows that there exists a connection between the decomposition (2) and the decomposition of r into a direct sum of simple algebras. Denote by p the rational prime number which is divisible by p. If pa is the highest power of p which divides all degrees fi of the ai belonging to the block BT, then p" also is the highest power of p which divides the degrees zj of all ordinary representations 3j of Br.7 We then say that the block B7 is of type a. If pa is the highest power of p dividing the order g of ?, then we call d = a - a the defect of B,. The smaller d, the simpler is the structure of 9mI in (2). If the group characters of ? are known, we can easily obtain the blocks B1, B2, ..., Bt. The main object of this paper is to give a direct characterization of the blocks. For instance, we shall show that the number of blocks of given positive defect d is completely determined by the structure of certain subgroups 9 of ? which are the normalizers 9 of the subgroups i of order pd in ?. 3. The modular group ring J/(p) determines a matrix C of Cartan invariants.8 This C is the direct sum of the matrices C7 of Cartan invariants of the rings J/IJ corresponding to the blocks B7. The degree of C, is equal to the number y, of prime ideals $3 (and of modular representations a1) in the block. The matrix C7 is symmetric, the corresponding

MA THEMA TICS: R. BRA UER

VOL. 30, 1944

111

quadratic form is non-negative, the determinant of CTis a power of p. The coefficients cij are non-negative rational integers, the value of ci depends on the mutual relation of the prime ideals $i and $j. It is easy to determine the elementary divisors of C. Let K1, K2, ..., Kk be the classes of conjugate. elements of 6? which contain p-regular elements, i.e., elements whose order is prime to p. Let gx be the number of elements in Kx so that gx = g/nx where nx is the order of the normalizer of the elements of Kx, and suppose that p divides nx exactly with the exponent px. Then pp', pP2, ..., pPk are the elementary divisors of C.9 Exactly y7 of these elementary divisors must belong to C7, and naturally the question arises in what manner the k elementary divisors of C are distributed into blocks. A partial answer is given by THEOREM

1:

If C, is the Cartan matrix of a block of defect d, then CThas

one elementarydivisor pd while all other elementarydivisors of C7 are powers of p with exponentssmaller than d. As corollaries10we obtain COROLLARY 1: If there exist l, p-regular classes Kx in ? for which the orderof the normalizerof the elementsis divisible by p* but not by p +', then ? possesses at most 1, blocksof defecta-(a = 0, 1, 2, ..., a). COROLLARY 2:

There exist exactly la blocks of maximal defect a, where

g = 0 (modpa),g ; 0 (modpa+l). COROLLARY 3:

only coefficientis 1.

If BTis a blockof defect0, then C7 has the degree1, and its The block consists of exactly one ordinary representation

3 and one modular representation W. Taken as a modular representation,

3 remains irreducibleand coincides with A. The prime ideal 93 belongingto a coincides with 9Jt in (2). In the proof of theorem 1, the following construction can be used. Consider the k columns of the matrix of group characters of 0 which belong to the p-regular classes K1, K2,, ..., K of @. It is possible to select a minor of degree k which is not divisible by p. Exa/ctly y7 of the rows in this minor must belong to ordinary characters ~, of B.. It is then possible to associate y, of the columns with B7, such that each column belongs to exactly one B,, and that the y7 rows and columns associated with B7 form a minor which is not divisible by p.. (This construction may not be unique.) We can then prove THEOREM2:

If the classes Kx, K,, K,

... are associated with the block

Br by the preceding construction,and if p divides the order of the normalizers of the elementsof Ki to the exact exponent pi, then the elementarydivisors of C7 are the powers of p with the exponents px, p , p, .... If ~y is an ordinary character of ?, and if S,(K,) is the value of , for the elements of K,, then it is well known that the numbers cw(K,) = g /,,(K,)/Dg(',,)

(3)

MA THEMATICS: R. BRA UER

112

PROC.N. A. S.

are algebraic integers. The proof of theorems 1 and 2 yields at the same time THEOREM3: A character ,, belongs to a block BT of a defect larger than

or equal to a given numberd, if and only if we havec,(KK) - 0 (modp)for all p-regular classes with p, < d. Two characters~x and ~,, belonging to blocks of defectd appear in the same block,if and only if WX(KV)- co(K^) (mod p)

for all p-regular classes with p, = d. If the Cartan matrix CT of the block B, of defect d has h divisors pd-~, then B7 contains at least h + 1 ordinary characters elementary whose degrees are not divisible by pa-d+l. (By definition, the degrees are COROLLARY 4:

all divisible

by pa-d.)

4. If Kc is a class of conjugate elements of ?, we also denote by Ka As is well known, the Ka then the sum of all the elements in this class. form a basis of the center A of the group ring r. We thus have formulae

K,K6 = Za,o,yKy

(4)

The true significance where the ao.,, are non-negative rational integers. of the numbers cowin (3) is that they form a character of the commutative algebra A. Taking the co,, (mod p), we obtain a modular character of A, or what is the same thing, a character of the center A* of the modular group But characters ,,, of ? belonging to the same block B7 give rise ring r*. to the same modular character of A. Hence we have a (1-1) correspondence between the blocks Br of ? and the characters of the commutative modular algebra A*. For It is easy to prove certain arithmetic properties of the a,,6 in (4). of a elements all if of commutes with element G an instance, subgroup S K, of order pd of ? while no element of K, commutes with all elements of ~, On the basis of this remark and more then a,, must be divisible by p. it is possible to establish connections of a similar statements nature, general between A* = A*(?) and the analogous algebras A*(s), formed by means In this manner we obtain a connection of suitable subgroups 9 of ?. between the characters of A*(() and those of A*(9S), and finally a connection between the blocks of ? and certain blocks of T. Our main results are proved by combining this method with that of section 3. They are contained in the following theorems:

4: Chooseone group b,.of orderpdfrom each class of conjugate THEOREM subgroups of this order of ?, and denote the normalizer of ~, by 9, (a = 1, 2, ..., 1). For the given prime p, all blocks of 9, have at least the defect d. If 9, possesses exactly r, blocks of defect d, then ? contains exactly rl + r2 +

... + ri blocksof defectd.

MATHEMATICS:

VOL. 30, 1944

R. BRA UER

113

To every block B of defect d of ? there corresponds uniquely

THEOREM 5:

a blockB of defectd of one of the R, in theorem4. If cow(K,)in (3) is formed by means of a characterof B while jx(K,) is formed in an analogous manner by means of a character of B, then cw,(K,)

=-Ei

(Ko) (mod p).

(5)

Here, Ka is any class of ?, while K6 ranges over all classes of 91. which lie The ideal p in Ka and whose elements belong to the centralizer T, of ,. denotes a fixed prime ideal divisor of p in the field generatedby the characters of B and B. 5: COROLLARY

If in the notation of theorem 5, the class Ka does not contain

elements of 3,. then Wc(Ka) m 0 (mod p)

(6)

If K, contains elements of T,, and if the order of the normalizer of these elements is not divisible by pd+l, then K, contains only one class Kt which consists of elementsof T, and we have CS,(Ka)-

C,(K#) (mod p).

(7)

The congruences (5), (6) and (7) can be expressed in terms of the characters. We thus obtain the following relations between the characters of ? and those of 9,. COROLLARY6: Let ,, by any character of B, say of degree z, let P. be a character of the corresponding block B of 1,, say of degree w. In the notation

of theorem5, we have {(K,)

n

EZ Wg,.g,

v(KB) (mod ppn) with m = s - a + pa

whereg, is the number of elementsin K,, go is the number of elementsin Ko, p. is the exponent to which p divides na =

g/g,,

and s is the exponent to

which p divides z. In the case of (6), this yields ~,(Ka) = 0 (mod ppm), while in the case of (7) we obtain ,(iK

)--

(K ) (mod pts -- a + d)

wg

wheren is the orderof 9,. Our results do not give any new information about the blocks of highest kind, i.e., of defect 0. We have here M = { }, X,. = , = . For blocks of positive defect d, we may have 9,. = ?, but only when ? has a normal subgroup of an order pd > 1. In any case, every block of defect d of 91,

114

.MATHE.MATICS: R. BRAUER

PROC.

N. A. S.

contains an ordinary representation which associates the unit matrix with the elements of S;J... As a representation of fil../ S;J„ this one is of the highest kind. This shows that the question of finding all blocks of defect d > 0 can be answered by studying groups of smaller order than g. If for every one of these B we know the number of ordinary characters contained in B, we also know the number of blocks of defect 0. By corollary 3, it is the number of classes of @, diminished by the number of ordinary characters which belong to blocks of positive defect. In finding all blocks of @, the following remark is useful. REMARK: In theorems 4 and 5, it is suffi.cient to consider only groups S;J„ which are p-Sylow-subgroups of normalizers of p-regular elements, and for which S;J„ is a maximal normal P-subgroup of If these assumptions are not satisfied, lflo- does not possess blocks of defect d.

m...

1 The work on this note was done while the author was a Fellow of the John Simon Guggenheim Memorial Foundation. 2 Using a theorem of Hasse, Jour. f. d. reine u. angew . .Math., 167, 399-404 (1931), we could even assume that K was obtained from the field of rational numbers by the adjunction of a certain root of unity. 3 For the arithmetic in. maximal orders of algebras, see, for instance, Jacobson, N., "The Theory of Rings," Math. Surveys No. II (1943). 4 For the following cf. Brauer, R., these PROCEEDINGS, 25, 252-258 (1939). 5 For the modular representations of groups, cf. Brauer, R., and Nesbitt, C., University of Toronto Studies,.Math. Ser. No. 4 (1937), and Ann . .Math., 42, 556-590 (1941). I refer to the second of these papers as BN. 6 Cf. BN, §9. 7 BN, §17. 8 BN, § 5. 9 BN, § 16. 1° Corollary 1 is an improvement of theorem 3 of BN. Corollary 2 is identical with theorem 2 of BN. Corollary 3 is identical with theorem 1 of BN.

506

FINITE GROUPS

ON SIMPLE GROUPS OF FINITE ORDER. I RICHARD BRAUER AND HSIO-FU TUAN

1. Introduction. Using the theory of representations of groups we have obtained a number of results for simple groups of certain types of orders. In the present paper, we shall prove the following result: If ® is a (non-cyclic) simple group of order g~pqhg*, where p and q are two primes and where b and g* are positive integers with g*

3, or ®^LF(2, 2m) with p = 2m-\-l, p>3; conversely, these groups satisfy the assumptions. As an application, we determine all simple groups of order prqb1 where p, rt q are primes and where b is a positive integer. The only simple groups of this type are the well known groups of orders 60 and 168. 2. Some known results concerning representations of groups. 1. In this section, some known theorems are given without proof. Most of these results, which are needed in the following, have been obtained in the theory of modular representations of groups. However, all the statements are concerned with the ordinary group characters. 2 2. If © is a group of order g containing k classes Ki, • • • , K^ • • • , Kk of conjugate elements, then there exist exactly k distinct irreducible characters fi(G), • • • , fM(G), • • • , f*(G), where G denotes a variable element of ®. If we restrict G to a subgroup 2ft of order m of ©, then each fM(G) may be considered as a (reducible or irreducible) character of 2W. From the orthogonality relations for the characters of Sft, it follows that

(2.1)

H'MG) ^ 0 (mod m),

where the sum extends over all elements G of 9K. More generally, the same congruence holds, if J* is a linear combination of the f / s with coefficients which are algebraic integers. 3. Let p be a prime number and let p be a prime ideal divisor of p in the algebraic number field generated by all £M(G). Denote by h(G) the number of elements in the class K» containing G. If £*M has degree 0M, the number h(G)^(G)/zll is an algebraic integer. Two characters fM and f v belong to the same p-block, if Presented to the Society, September 17, 1945; received by the editors March 27, 1945. 1 We use the notation of L. E. Dickson, Linear groups, Leipzig, 1901. 2 The fundamental properties of group characters are given in a large number of books. Here we mention only: W. Burnside, The theory of groups of finite order, 2d éd., Cambridge, 1911.

756

757

ON SIMPLE GROUPS OF FINITE ORDER

(2.2)

h(G)UG)/z = h(G)Çv{G)/z (mod p),

for all G in ®. In this manner, the k characters are distributed into a certain number of ^-blocks Bi(p), B2(p), • • • • The first p-block Bi(p) will always be taken as the block containing the 1-character Çi(G) = 1 (for all G). If for all characters J"M of Bff(p) the degree sM of fM is divisible by a power £ a while at least one of the degrees z^ is not divisible by £ a + 1 , then B„(p) is a block of type a. In particular, Bff(p) is of the lowest type if a = 0. An element G is p-regularf if its order is prime to p ; and G is ^-singular in the other case. For every £-block Bff(p) we have 3

(2.3)

H,MP)MQ) = 0,

where fM ranges over all characters of J?ff(/>) and where P is any ^-singular element of © and —1 such that Bi(p) consists of w = (p — l)/t "non-exceptional" characters fi(G), • • • ,Çv>(G)and/"exceptional"charactersÇw+i(G)t • • *,JVM(G). The latter have all the same degree zw+i. To each of these characters Çi(G), there belongs a certain sign ô»= ± 1 such that the following relations hold : (2.4)

Zi ss Ôi (mod p)

(2. S)

tzw+i s dw+i (mod p) ;

(2.6)

for i = 1, 2, • • • , w\

E î A - O

(ô1 =

2l

=l).

( i - 1, 2, . . .

,w).

Moreover, for ^-singular elements P of G, we have (2.7)

UP)

«*
0. Let H be any subgroup of G of an order ph, h > 0, where p is the fixed prime selected above. Denote by C(H) the centralizer of H in G and by W(H) the normalizer of H in G, and consider a subgroup N which satisfies the condition H (H) c N c W(H).

(2)

If K? is the part of K,which lies in S(H), then either K? = 0 if K does not contain any elements of ((H), or K? is a sum of complete classes of N. It can be shown easily that (1) implies KsK -a,,Ta,Ko Consequently, the classes K,, with K?

(mod p).

(3)

0 form the basis of an ideal T*

VOL. 32, 1946

MA THEMA TICS: R. BRA UER

183

of the center A* of the modular group ring r*.4 On the other hand, the K?, 7 0 can be considered as the basis of a subring R* of the center A*(N) of the modular group ring r*(N) of N. Now (3) yields

R* c A*(G)/IT*.

(4)

This relation represents a connection between the group rings of G and of N; it forms the basis of our work. 3. The algebra A*(N) is commutative and splits completely, its irreducible characters Co*are all linear. The character co* of A*(N) induces a character of the subring R*. Because of (4), this character may be interpreted as a character of A*(G)/T* and hence it induces a character w* of A*(G) which vanishes for the elements of T*. If we know how the classes of N are distributed among the classes of G, we can express w* explicitly in terms of Co*. We have

C o*(Kp)

W*(K.)

(mod p)

'

where Kp ranges over all classes of N which belong to Ka. a character Every ordinary character ~, of G determines which is given by c*O(K,)

= g(ag)/naz

(5) co, of A(G)

(6)

where ,, is an element in the class K,, n, is the order of the normalizer of oa,, and z, is the degree of i,. The modular characters cow*of A*(G) are obtained by considering the different w, (mod p). In particular, two characters ~, and bbelong to the same block BT, if they yield the same w*. If is a block of characters of N, there is associated a modular character &* of A*(N) with B,. As described above, this character co* determines a character w* of A(G). Again, this character w* determines a block B. of G. We shall say that B, is the block of G determined by the block B,r of N. It follows from the results of A G R that the defect d, of B, and the defect d, of B7satisfy the inequality

B,

h < da _ d,

(7)

where ph is the order of H. 4. In (2), the group N was left arbitrary to some extent. Choose N now as the normalizer 9 (H) of H. It was shown in A G R that for a given block BT of G, there exist subgroups H and blocks of %I(H) for which the in we holds If consider (7). equality sign conjugate subgroups of G as not essentially different, then H is uniquely determined. We call this = of H the its order is with h dT. H, B,; defect group ph group Again, the block B, of 9(H,) is uniquely determined.

B,

MA THEMA TICS: R. BRA UER

184

PRoc. N. A. S.

to the case of an arbitrary N in (2), we state 1: Let H be a subgroup of order ph of G, let N be a subgroup of THEOREM G satisfying H. (S(H) C N If the block B, of N with the defect 91i(H). the block the defect group IIr, then H C determines G with B, of group H, Returning

N, and H, is conjugate in G to a subgroup of IHr. to the k irreducible characThe k linear characters wi corresponding ters -i of G can be arranged in form of a matrix 02 = (wi(Kj)) of degree k. contains xT ordinary characters If the block fi, then xT rows of Q correto a minor of these x7 rows such Choose A, B,. spond degree x, containing It can then be shown that AT is divisible by p to the least possible power. that it is possible to make this selection of xr columns for each block B13in such a manner that every column appears for one and only one block. for the proof, the theory of algebras This result is by no means trivial; of blocks must be used. and the significance Since the columns of S0correspond to the classes Kj of G, we have associH,C . 5.

B7

ated xr classes Kj with every block B, such that every class is associated of classes for the different The selection with one and only one block. In any case, the number of blocks may be possible in more than one way. the classes associated with classes B, can be shown to be among p-regular in of modular number characters the to B,, and further these y, equal in the sense of associated form a selection classes with B, y, p-regular in Theorem 2. A G R, ? 3, particular So far we assumed that B1, B2, ..., Bt were the blocks of ordinary and to note that the results of modular characters of G. It will be important this section remain valid if every B, is a collection of ordinary and modular of G, such that every ordinary and modular character of G becharacters longs to exactly one B, and that every B, consists of one or several blocks and y, the of G. Again x, denotes the number of ordinary characters in modular characters number of Br. 6. We shall say that a group H of order ph is the defect group of a class of the normalizer of suitable elements of Kj. Kj, if H is a p-Sylow-subgroup is the that This implies ph highest power of p dividing nj in (6); the exWe can now state the following ponent h will be termed the defect of Kj. results THEOREM 2: Let (S)) be a system of subgroups H of orders 1, p, p2, ..., of G such that every subgroup of order ph of G is conjugate to exactly one H in (O). For every H in ( f), find the collection B13( of all blocks B, of 1(H) which determine a given block B, of G, and select a full system of classes Kp of 91(H), which are associated with B (). Suppose that r7(H) of these classes Kp have H classes Kp belong to as their defect group. Different ones of these r,(I) classes thus the classes obtained K,, Ka of G; for the different H in different B' with associated classes a selection of possible (O) form As corollaries,

we have

MA THEMA TICS: R. BRA UER

VOL. 32, 1946

THEOREM3:

185

The number of characters in B. is given by , =

r7(H)

(8)

H

where the sum extends over all H in (O). THEOREM 4: If s,(H) of the r,(H) classes Kp in Theorem 2 are p-regular, then the number of modular characters in B, is given by Y =

H

s(H)

(9)

where H again ranges over all groups of (O). THEOREM 5: If, in (9), H ranges only over those groups have a fixed order ph, the corresponding sum

y(h) = represents the multiplicity C, of the block B,.

s(H),

(H in (,);

of (i)

(H: 1) = ph)

of ph as an elementary

(10)

divisor of the Cartan matrix

It would be conceivable that the numbers r,(H) and s,(H) the special selection of classes of 91(H) associated with B,. this is not so; we have THEOREM 6:

which

depend on However,

The numbers r,(H) and s,(H) in the preceding on the depend only group G, the subgroup H and the block B, of G.

theorems

In order to discuss our results, let us assume for the sake of simplicity that G does not contain any normal subgroup of an order ph > 1. Suppose we know: (a) A complete system of subgroups H of a p-Sylowsubgroup of G, (b) which of the groups H in (a) are conjugate in G; (c) the characters of the normalizers 9(H), H 1, (d) the manner in which the classes of conjugate elements of 91(H) appear in the classes of conjugate elements of G. If H 7 1, then, under our present assumption, 9(H) is a proper subgroup of G. If we know the characters, we can find the modular characters co* of the center A*(91(H)) of the group ring of R9(H), and this gives us the blocks B, of 9(H). Then (5) gives the modular characters w* of A*(G). In this manner, all the characters c* belonging to the different blocks B, of positive defect are obtained. Further, we can determine which B, for a fixed H - 1 belong to B(t), and then find r,(H) and s,(H). This is not sufficient to determine x, and y, completely, since the numbers r,(l) and s,(l) remain undetermined. However, we obtain lower bounds for x, and y,. since Further, any p-singular class Ka has a positive defect, we have r,(l) = s,(l), and hence the excess x, - y, of the number x, of ordinary characters over the number y, of modular characters in B, can be obtained. Finally (10) gives the multiplicity of the elementary divisors different from 1 of C,. This shows that a number of the most important 7.

186

MATHEMATICS:

PROC.N. A. S.

R. P. BOAS, JR.

invariants of the blocks are determined by the information contained in (a), (b), (c), (d). 8. It had been shown in A G R, that if for a subgroup H of order ph in G, the group 91(H)contains g(H) blocks of defect h, then G possesses (H in (P), (H: 1) = ph)

Eq(H), H

(11)

blocks of defect h. It may be remarked that the number q(H) can be determined by means of the group Wi(H)/N = U and its normal subgroup HE(H)/H = V. The characters 0 of V are distributed in classes of characters which are associated with regard to U; two characters 0 and 01 being associated if o(a0) = 0(u-1au)

where a is a variable element of V and u js a fixed element of U. Then it can be shown that q(H) is equal to the number of classes of associated characters 0 of V of defect 0, such that no element u of U exists of order p with regard to the subgroup V for which 0(u-l'ou) = 0Q(). If h > 0, this result requires only the investigation of groups of smaller order than g, in order to obtain q(H) and (11). 9. There does not seem to exist a similar result in the case of blocks of defect 0. As a substitute, we have here the theorem: THEOREM7: The classes of defect 0 in G form the basis of a subalgebra M of the center A* of the modular group ring r* of G. The number of blocks of defect 0 is equal to the rank of Mn for suficiently large n. * Part of the work on this and a following note was done while the author was a Fellow of the John Simon Guggenheim Memorial Foundation. 1"On the Arithmetic in a Group Ring," these PROCEEDINGS, 30, 109-114 (1944). This paper will be quoted as A G R. 2 Brauer, R., Am. Jour. Math., 67, 461-471 (1945). 3 See for instance, Brauer, R., and Nesbitt, C., Ann. Math., 42, 556-590 (1941). 4 We denote the residue class field of the integers of K (mod P) by K* and the group ring of G with regard to K* by r*.

THE RATE

OF GROWTH OF ANALYTIC

FUNCTIONS

BY R. P. BOAS, JR. MATHEMATICAL

REVIEWS,

BROWN

UNIVERSITY

Communicated May 10, 1946

This note presents a theorem of Phragmen-Lindelof type and indicates how it can be applied to establish and generalize results of N. Levinson' on the determination of the rate of growth of an analytic function along a line

MA THEMA TICS: R. BRA UER

VOL. 32, 1946

215

of games. For this see reference 3, pp. 154-155, where another proof of this special case is also referred to. 1 von Neumann, J., "Uber ein okonomisches Gleichungssystem, etc.," Ergebnisse eines

MathematischenKolloquiums,8, 73-83 (1937).

2 Kakutani, S., "A Generalization of Brouwer's Fixed Point Theorem," Duke Math. Jour., 8, 457-459 (1941). 3 von Neumann, J., and Morgenstern, O., "Theory of Games and Economic Behavior," Princeton University Press (1944).

ON BLOCKS OF CHARACTERS OF GROUPS OF FINITE ORDER, II BY RICHARD BRAUER DEPARTMENT

OF

UNIVERSITY

MATHEMATICS,

OF

TORONTO

Communicated July 8, 1946

1.

appeared in these PROCEEDINGS,

The first part of this investigation

June, 1946, p. 182.1 In this note, we shall apply our results to a study of the (generalized) of the arithmetic

numbers2 of a group G of finite order g and decomposition in the group ring r of G.

Let again p be a fixed rational prime number.

Select a full system II

of elements ro = 1, rl, 72. 13s, ... of orders 1, p, p2, ... such that every elein G to exactly one element ment of an order pa of G is conjugate ri of II. Denote by Ni the centralizer of tr in G. A full system 2 of elements of G the different classes'of conjugate elements can be obtained in representing

the following manner:

Let

(), a(i), ...

classes of conjugate elements of Ni. - 0, 7rioi 7rit(i ), ... for i 2.3 If 1, 2, . ., k are are the modular 2p, S?2, ... regular element aoof N1, we

represent the different p-regular

Then S consists of the elements

1 2, .... the ordinary irreducible characters irreducible characters of Ni, then have a formula ,(Jix ) =

of G, and if for every p-

Ed c4 (a)

(1)

where the d}, are algebraic integers, the decomposition numbers, which are independent of C. This formula yields a representation of the matrix Z of the ordinary characters of G as a product of two square matrices D and b Z = DD.

(2)

We have to set a = (i) Z = (i(Tr, ,('))) where / is the row index while every column corresponds to an element c(), i = 0, 1, 2, ...; j = 1, 2, ..., ki, where

ki is the number

of p-regular

classes

of Ni.

Similarly,

in D =

MA THEMA TICS: R. BRA UER

216

PROC.N. A. S.

(diy), the rows correspond to the characters , and the columns to the modular characters yp,of the different N1. Finally, if the rows and columns are

arranged suitably,

0 ( (0

(0)

/((Ho0)) ) .o.

... 0

(3)

))

where, in each partial matrix (s'(o-))) in the main diagonal, the row index is v and the column index is j. The degree of all three niatrices in (2) is equal to the number k of conjugate classes of G, k = ko + kl...,. The square of the determinant of D is a power of p while the determinant of I is relatively prime to p. The formulae (1) show that in order to know the ordinary characters of G, it is sufficient to know the modular characters O of all the Ni including No = G, and the decomposition numbers dx,. the product of the column (i, v) of D with the conjugate complex Actually, c, y, of Ni of the column (i', v') is 0 for i = i' and the Cartan invariant = in of the terms i'.. While this c /, ,, can be expressed for i 4, this does in terms of the ?V. the numbers not enable us to express decomposition

We shall say that for fixed i the elements ritoi) of G belong to the ith section.

3.

In the notation of I, theorem 1, we take H as the group generated

= Ni. The following result can be proved (with by Oi, and M = Q(II) considerable difficulty): THEOREM 1: If the modular character sp of Ni belongs to a block Ba of Ni, then d}, can be different from 0 only for ordinary characters of G which belong to the block BT of G determined by B,. This implies that in each column of D we have

zero except

corresponding to the A, belonging to one block BT of G.

in the rows

It follows that, if

the rows and columns of D are taken in a suitable order, D breaks up comto one block pletely into t matrices Ti, T2, . . ., Tt, each T, corresponding = be a D each det .. Since BT, (r X# 0, 1, 2, ., t). necessarily T,7must of the number characters where of is x, ordinary degree x,, square matrix, of the columns of D here will in general not be the in BT. The arrangement same as that used in (2). ,, of G and the modular characOriginally, only the ordinary characters into blocks B.. It is now natural to ters ?oo,of G itself were distributed

count p'2,i > 0, as a character of Br, if 0 belongs to a block Bl of Ni which B7 in the sense of I. Then B7 consists of x. ordinary characdetermines ters ',, and x, modular characters s'. In our notation, y, of these characters have the upper index i = 0. These are the modular the other o are the modular characters of the groups Ni.

characters

of G,

MA THEMA TICS: R. BRA UER

VOL. 32, 1946

217

As a corollary to theorem 1, we have the following refinement of some of the orthogonality relations for group characters. THEOREM 2: then

If the elements p and orof G belong to different sections of G, E' r (p)W(o) = 0

when in the sum g, ranges over all the characters of G belonging Br.4 4. We state without results which proof the following

to a fixed block are connected

with theorem 1. THEOREM 3: Let BT be a block of G, and D7 its defect group. If no element of Dy is conjugate to ri, then ~, (p) = 0for all characters ~, of BT and all elements of the section of 7ri.5 THEOREM 4: If B7 is a block of defect dT with the defect group D7, there exist blocks Ba of defect dr of Ni which determine B7, if and only if Ti is conIf Ti is conjugate to an invariant jugate in G to an invariant element of D,. element of D,, we can choose the block Ba of defect dT of N1 in such a manner that it determines the block B, of G, and that for every , in B7 there exists a (p in BR such that dy 4= O. Let pa be the exact exponent

to which p divides

g = pg',

(p, g')

g,

1

If po is a prime ideal divisor of p in the field of characters, and if the degree z of g, contains p to the exact exponent a - dy + e, (e g 0), we may even

state in theorem 4 that for a suitable

pf'in B, we have

dy 9 0 (mod pEpo).

(4)

THEOREM 5: If the block B, of N. determines the block B, of G, the defect of B, is at most equal to I, where pl is the order of a maximal p-subgroup of N1 which is conjugate in G to a subgroup of the defect group D7 of B7. If the de1 gree z, of the character ', of B7 is not divisible by p0 - d + where d is the dea character 4yof N1 which belongs to a block of N1 of fect of B,, there exists defect I and for which d7i is not divisible by po. THEOREM 6: If p' is the maximal order of elements of the defect group D7 of B7, then for all ~, in B,, the numbers dty belong to the field of the p'-th roots The characters of unity. , of B7 belong to the field of the (p"g')-th roots of unity. THEOREM 7: If p V 2, and if B, contains y, modular characters of G, then at least y, of the ordinary characters , of B7 are p-rational, that is, they lie in the field of the g'-th roots of unity, (g', p) = 1. In fairly general cases, the exact number of p-rational characters in B7

is equal to y,.

218

MATHEMATICS:

R. BRA UER

PROC.N. A. S.

For p = 2, a result similar to theorem7 can be obtained which is more complicated, and shall not be stated here. The previous results make it possible to prove the following theorem: A block B of defect d contains at most pd(d + 1)/2 ordinary characters. It is probable that the bound pd(d + 1)/2 here can be replaced by pd, but I have been able to prove this stronger result only for d = 0, 1, 2. Theorem 8 implies that if the order g of a group is divisible by the prime number p to the exact exponent a, and if G contains q classes of conjugate elements whose order is prime to p but whose normalizer has an order divisible by p0, then at most qpa(a + 1)/2 of the degrees of ordinary irreducible representations of G are relatively prime to p. 6. As in I, let K be an algebraic number field in which all the simple Denote by p constituents of the semisimple algebra r split completely. a fixed prime ideal divisor of p in K. The ideal (p) generated by p in the ring of integers J of r can be represented as a direct intersection6 5.

THEOREM 8:

(P)=-1, n S2 n... n sl of ideals of J, such that no 9T0 possesses a proper representation as direct There exists a (1 - 1) correspondence between these intersection. of (p) and the blocks B7 of characters of G (for p). "block components" tYrT In particular, the number y, of modular characters of G is equal to the

number of prime ideals $ of J dividing (p). Now, theorem 8 implies THEOREM9: No block component of (p) in J is divisible by more than pa(a + 1)/2 prime ideals of J where pa denotes again the highest power of p dividing g. The y2 coefficients of the Cartan matrix C7 of the block B7 describe, to a certain extent, the mutual relationship between the y, prime ideal divisors Here, C7 $ of 9Tr. They represent interesting arithmetical invariants. We can form the is a symmetric matrix with integral rational coefficients. corresponding quadratic fornl Q. Now our results yield THEOREM10: To given p and given defect d, there exist only a finite number of classes of quadratic forms to which the Cartan form Q of a block of defect d can belong (for an arbitrary group G of finite order). We also quote the following results which can be proved directly without great difficulty. THEOREM11: If the defect of the block B7 is positive, the Cartan form Q does not represent the number 1. (More generally, Q does not represent (integrally) a form of determinant 1. If B. has the defect 0, then C, is of degree 1, and Q is the quadratic form x2

7. It mnay be remarked that blocks of defect 1 can now be discussed The results obtained earlier for the characters of groups rather completely.

PHYSIOLOGY:

VOL. 32, 1946

E. WOLF

219

of an order g = pg', (p, g') = 1 appear as special cases of properties of characters of blocks of defects 0 and 17 Finally, it may be mentioned as a conjecture that it appears probable that for a given p and d, only a finite number of inatrices exist which can occur as Cartan matrices C7of blocks of defect d. 1The first part will be quoted as I. 2 Cf. Brauer, R., Ann. Math., 42, 926-935 (1941). 3 For the results quoted in this section, cf. the paper mentioned in 2. 4 In the case that p belongs to the section of the 1-element, this result has already been obtained in Brauer, R., and Nesbitt, C., University of Toronto Studies, Math. Ser., No. 4, theorem VIII (1937). 5 This generalizes a result obtained in Brauer, R., and Nesbitt, C., Ann. Math. 42, 556-590 (1941) for blocks of defect 0. 6 Cf. Brauer, R., these PROCEEDINGS, 30, 109-114 (1944), in particular, equation (2). 7 Cf. Brauer, R., these PROCEEDINGS, 25, 290-295 (1939), and Ann. Math. 42, 936958 (1941). I take this occasion to mention the following corrections in the first of these papers: In theorem III, the assumption should read n < (2p + 7)/3. The left side of equation (4) should read rpt. + ry,tp. For the results of the last paragraph of section 3, it is necessary to assume that a suitable splitting field is used.

EFFECTS OF EXPOSURE TO ULTRA- VIOLET LIGHT ON HUMAN DARK ADAPTATION* BY ERNST WOLF BIOLOGICAL

LABORATORIES,

HARVARD

UNIVERSITYt

Communicated July 15, 1946

Previously it has been shown that the course of dark adaptation of the eye of the baby chick can be altered by addition of ultra-violet radiation between 290 and 365 mi, to the visible white light of a mercury vapor lamp Exposure to wave-lengths longer than 365 m/u reduring preexposure.l sults in uniform dark adaptation curves, all curves reaching the same final threshold level. The addition of ultra-violet below 365 mi, retards complete adaptation, raising the final threshold considerably above the normal. Extension of the ultra-violet range to about 355 mu causes an increase of 0.3 log unit, to 315 miu an increase of 0.6 log unit, and to 290 m,i an increase of 1.1 log units in the final threshold level. In the baby chick, as in all newly born animals, the absorption by the ocular media is small, therefore a considerable penetration of ultra-violet to For the human eye the ultra-violet transmission the retina is expected. is a function of age,2 depending mainly upon the transparency of the lens;3 it is maximal in infancy and thereafter decreases so that in the adult eye

ANN14ALS OF MATHEMATICS

Vol. 48, No. 2, April, 1947

ON ARTIN'S L-SERIES WITH GENERAL GROUP CHARACTERS BY RICHARD BRAUER (Received

September 3, 1946)

I. INTRODUCTION

1. In a fundamental paper, E. Artin' introduced the general L-Series L(s, x, K/F) of a Galois extensionfieldK ofan algebraicnumberfieldF. Here, x denotesan arbitrarycharacterof the Galois group 3 of K with regardto F. The followingresultsof Artinmay be mentioned: I. If x is a linear combinationEcus of the charactersso,with rational coefficientsc,, then L(s, X,K/F)

= H

L(s, p,, K/F)CP.

II. Let Q be a subfieldof K and & the correspondingsubgroupof 5. If 4t'is a characterof ' and A1* the characterof 5inducedby A', then L(s, A*,K/F) = L(s,

A,&K/s).

III. If the representationof @ belongingto the character x has the kernel 9 and if N is the correspondingsubfieldof K, then L(s, x, K/F)

= L(s, x, N/F)

whereon the rightside x is to be interpretedas a characterof 5/9L IV. If K is an abelian fieldover F and if x is an irreduciblecharacter,then L(s, x, K/F) coincideswith one of the ordinaryL-seriesof the extensionfield K of F. The proofof this fact restson the law of reciprocity. The resultsof Hecke2show in this abelian case that L(s, X, K/F) is a meromorphicfunction which satisfiesa certainfunctionalequation. Further,Artinproved the group theoreticaltheoremthat every characterx is a linear combinationE cep, where the c, are rational numbersand where the sp,are charactersof @ induced by charactersof cyclicsubgroups. In connectionwith I, II, and IV, this yielded at once the result that in the general case L(s, x, K/F) can be continuedanalyticallyover the whole complexplane; indeed, a suitable power L(s, x, K/F)m with an integralrational m is a meromorphicfunction. Moreover,L(s, X, K/F) again satisfiesa functionalequation of the well knowntype. However,since m may be largerthan 1, thisdoes not show that L(s, x, K/F) itselfis a single-valuedfunction. Artinconjectured that thisis, in fact,the case and that L(s, x, K/F) is a productofabelian L-series I

Abhandl., Math. Seminar, Hamburg Univ., 3, 89-108 (1924); 8, 292-306 (1931).

2 Gesells. der Wissens. zu Gottingen, Nachrichten 1917, 299-318; Math. Zeitschr. 1,

357-376 (1918); 6, 11-51 (1920).

See also E. Landau, Math. Zeitschr. 2, 52-154 (1918).

502

ARTIN'S L-SERIES WITH GENERAL GROUP CHARACTERS

503

with integralrational exponents. Using the method above and the factIII we can derivethisat once fromthe followinggrouptheoreticalstatement:' THEOREM 1. If (5 is a groupoffiniteorderg, everycharacterx of 5 can be of characters expressedas a linear combinationwithintegralrationalcoefficients A*, such thateveryw*is a characterof (D inducedbya linearcharacterw of a subgroupof 5. This conjectureof Artinwillbe provedin the presentpaper and it will thereby be shown that L(s, x, K/F) is meromorphic. A second, strongerconjecture fromthe 1-character, remainsopen. If x is a simple characterof 5, different thenArtinsurmisesthat L(s, X, K/F) is an integralfunction. 2. Instead of provingTheorem 1 directly,we shall derive it in Section III froma related Theorem 2. It seems expedientto indicate brieflythe connection between both theorems. Let ***, 2, 9. Denote the classes of conjugate elementsof 5 by P be the fieldof the 2gthroots of unity.4 It is not difficultto see (cf. 7, 8) that Theorem 1 will be proved when we can show the followingstatement:If a congruence k

(1)

E

i=1

(mod qt)

ci~(Si)-?0

ci modulo a power qt of a primeideal q of P, and with q-integral5coefficients in P, holds forevery characterw*,then the correspondingcongruence k

(2)

a, c xG2) i=1

0

(mod qt)

holds forevery characterx of 5. Using resultsconcerningthe modular characters6we can split the sum (2) into a numberof partial sums, each of which must be congruentto 0 (mod qt) if (2) holds. These partial sums are of the 3 This form of the conjecture was mentioned to me by Artin in a conversation which was the starting point of the present investigation. A similar, somewhat stronger, conjecture was given in H. HASSE, Bericht uiberneuere Untersuchungenund Probleme aus der Theorie der algebraischen Zahlkorper Part II, p. 160 (Jahresber., Deutsche Math. Ver. Erganzungsband 6, 1930) with the same purpose. However, in the form given by Hasse, the conjecture is not correct. 4 The choice of the field P is arbitrary to a large degree. What is required is that P is an algebraic number field which contains the characters of 3 and of the subgroups of (. Later, it will be convenient if every rational prime factor of g is the square of an ideal of P. 5 A q-integer is a quotient a/l of two integers a, is of P such that the denominator iB is prime to q. 6 The theory of modular characters will not be used in the following. However, we mention the fact here, because it allows us to see the reason for the procedure followed in this paper. The connection between (2) and (3) can also be obtained as a corollary to our Theorem 2 below.

504

RICHARD BRAUER

followingtype. Let q be the rational prime divisible by q and let A be a qregular7elementof 3. Then (2) forall x impliesthat (mod q')

E'c, x(f,) -0

(3)

where the sum ranges over all those classes fst of 5 whichcontainelementsG with A as theirq-regularfactor.8 We shall denote the systemof these classes by e(A). Since it is easy to derive (2) from(3), we will have to show that (1) forall charactersc&o*implies (3) forevery characterx of 5 and all q-regularelements A of . of the normalizer9(A) of A and let t be the Let e be a q-Sylow-subgroup subgroupof 5 generatedby A and C. If ,t1is an irreduciblecharacterof &, it can be seen withoutgreat difficulty (see 9 below) that the character4,* of 5 induced by 1 can also be induced by a linear characterw of a suitable subgroupof ,. If therefore(1) holds forall charactersw*,the corresponding congruencewith A,*instead of c* will be true. It turnsout that these latter congruencesare sufficient to prove (3). In fact,we state THEOREM 2. Let A be a q-regularelement of (5 and let& be thegroupgenerated by A and a q-SylowsubgroupC of thenormalizerJi(A) of A in 5. If a congruence (4)

E

ci *(ft)

0

(mod q')

withq-integral coefficients ci holdsfor all characters4A*of ( whichare inducedby

irreduciblecharactersi/ of A, thenfor everycharacter X of 3,

(5)

'cjX(S;)

(mod q')

whereStj in (5) rangesoverthesystem25(A) of thoseclasses of 5 whichcontain withA as theirq-regular elements factor. The congruences(4) and (5) can be expressedin terms of elementsof &, the charactersof & and the coefficients describingthe relations between the charactersof 5 and of A. Theorem2 will be proved in Section II. In Section III, the proofsketched above, that Theorem2 impliesTheorem 1, will be givenin detail. II. PROOF OF THEOREM 2 3. Let q again be a rational prime number. If A is a q-regularelementof of the normalizer the group @ of finiteorderg, and ife is a q-Sylow-subgroup 91(A), the groupt = A, e } generatedby A and C is a directproduct __

-

{AI X .

7 That is, an elementwhoseorderis primeto the primenumberq.

8 Every elementG can be writtenuniquelyin the formQR whereQ has an order 2 1 whichis a powerof q whileR is a q-regularelementcommuting withQ. We thensay that R is the q-regularfactorof G.

ARTN'S L-SERIES WITH GENERAL GROUP CHARACTERS

505

An irreduciblecharacter4, of D is thereforethe product r4 of an irreducible characterr of {A I and an irreduciblecharactert of 0 in the followingsense: If the arbitraryelementH of o has the formA'Q with Q in A, then 6(H) = O(A'Q) = r(At4(Q) of ( can be definedby The induced character01A*

(G)=

(6)

(R GRp yo

E

whereR, ranges over a completeresiduesystemof ? modulo !, = Carp. Let again q be a primeideal divisor of q in the fieldP of the 2gthroots of unity. Assume that a congruence(4) with q-integralcoefficients ci in P holds for all irreduciblecharactersy6of A. Then (4) will still be true for all linear combinations4,6of the irreduciblecharactersof t with q-integralcoefficients, provided that 4/*is still definedby (6). We want to choose the expression L' such that 4,t*vanishesforall classes ftiof conjugate elementswhichdo not contain an elementG with A as its q-regularfactor; that is, 0&*(t) = 0 for all Let ri, 2 * classes S, whichdo not belongto the systeme(A). a be the be the different of {A l and let 6i, t2 6X*Xm characters irreducible different irreduciblecharactersof D. Set (for,u = 1, 2, * , m) (7)

4,,(H) = 4,I,(A'Q) =

a

E

a-1

Da(A) ra(A)A(Q).

This is a linear combinationof the irreduciblecharacters ra(AVt~6(Q) of t, with q-integralcoefficients ra(A). The orthogonalityrelations for the characters of {A} show that #VI(A'Q)= 0, if A id A'. It followsfrom (6) that *,(G) = 0, if G is not conjugatein 5 to an elementAQ, Q ink. Hence, ^,6*Gti) = 0, if S1idoes not belongto the systemC(A). for4,in (4), If we substitute 1,6 Let us it will be sufficient to let ftirange over the classes of the system6(A). , tRh. choose the notation such that 6(A) consistsof the classes 91, 92, Then (4) reads h

(8)

(mod qt)

c,,*(Gj)

E

, 1

(I = 1, 2, **, im).

4. We have to prove (5) only forthe case that the characterx is one of the

irreducible characters si,

p2,

***,

the congruences (*)

h j-1

Cic o((1)

pk

of @. In otherwords,we haveto prove 0

(mod q')

9 If the elementRGR- 1 does not belongto 0, we have to set #(RpGRT1) = 0. For the propertiesof induced characters,see G. FROBENIUS, Sitzungsber.,Akad. der Wissens. Berlin, 1898,501-515. Compare also A. SPEISER, Theorie der Gruppen von endlicher Ordnung,3rd ed. Berlin,Springer,1937,?64.

506

BRAUER

RICHARD

for K = 1, 2, * k,I. Now the character,6* of 5 is a linear combinationof It followsthat the lefthand side of (8) is the corresponding the charactersp linear combinationof the left sides of the congruences(*). Our next task is to studymorecloselythe connectionbetweenA, and the C . * of 05 induced by the characterDad of !i is a linear The character( combinationof the p,, say

ra 'i

(Pai) =

(9)

rapg, ran 2 0. Now (7) gives with integralrational coefficients

(10)

=

(I a

?a(A)

= rangm)* a

Pa(A)

E

X

rapg' Xp.

Let soledenote the characterp, conjugate complex to p, @ic ,

At=

so that 1', 2', ***, k' formsa permutationof 1, 2, Write (10) in the form (11)

=

Ad

**,

k and, of course,(K')'

=

K.

SWK,10

where (12)

W

=

E

a

?(A)

rag.

Accordingto a theoremof Frobeniuson induced characters,the coefficients

ra,, in (9) appear also in the relations which express sp(H) (for an element

H = A'Q of I) in termsof the irreduciblecharactersrat6 of I:

so(A(Q)=

rareKra(A')t1(Q). aml E

For v = 1, this gives (cf. (12))

sox(AQ) =E

m

jL-1

Replace

(13)

K

tA4(Q).

by Kt and take the conjugate complex. Thus

(Pv(AQ)= E wgjta(Q).

The class Mj forj = 1, 2, ***, h containsan elementAQU) whereQ(J)belongs to thenormalizer91(A)and theorderof Q(j)is a powerofq. By Sylow's theorem, Q(j) is conjugate in 9Z(A) to an elementQj of I. Then AQ(j) is conjugate to AQj and hence p i containsan elementAQ, with Q,in Z, (j = 1, 2, ***, h). 10It is only for formal reasons that we prefer to denote the coefficientsby W'tt instead of us;, .

507

ARTIN'S L-SERIES WITH GENERAL GROUP CHARACTERS

The relations(8), (11) forthe classes j, (j = 1, 2, * **, h) and (13) are the basis of our work. We introducematrixnotationand set

r

(14)

= 0, then g itself is an irreducible character while in the other case - e is an irreducible character.8 In order to find all irreducible characters of (M, we have to find all solutions of the Diophantine equation -

(7)

Y Xpxumpu

I

in rational integers xp. The coefficients mpa of the quadratic form on the left side can be found, if the ,p* are known. Only solutions xp are to be used for which Yxpwp*(R) > 0. There are exactly k distinct expressions e.xpwp* formed by means of such solutions xp. These kcexpressions are the k irreducible characters of (M. , Rk of THEOREM 4. Suppose that the number Ic of classes Al, R2,* conjugate elements of (Mand the number gi of elements in !R are 7known. Suppose that a complete systermof elementary subgroutps$ of (Mis given, (subgroups conjugate in (Mmay be considered as not essentially different). , 21 of Assume further that for each $ the numberI of classes Q1, 22, . conjugate elements of $ and the number hj of elements of ?j is 7known,that it is known to wvhichclass S, the elements of 2j belong and that the values w (2j) of the linear characters w of $ are 7known. Then the irreducible characters of (Mare completely determined.

As already remarked in the introduction, the construction of all linear characters ta(2j) of $ requires only the knowledge of all normal subgroups I0 with cyclic factor group of the elementary group Sg and the information to which particular coset (mod R)o the class 2j belongs. Since the characters o1*, )25 *, *wr* are, in general, linearly dependent, the equation (7) has, in general, infinitely many solutions. However, it can be seen without difficulty that the solutions can be found in a finite number of steps. 8 This method has been used by I. Schur in order to find the characters of special groups.

714

RICHARD BRAUER.

4. An analogue for modular characters. Let p now be a fixed prime number. We prove THEOREM5. If D is the character of an indecomposable constituent of the modular regular representation of (M(mod p), then 1 can be written in the form E aa(wa

=

where the au are rational integers and where the wu* are characters of {M induced by linear characters wa of elementary subgroups $u of orders prime to p. Proof. We first observe that 1 may be considered as an ordinary (reducible or irreducible) character of (Mwhich vanishes for the p-singular classes of O.'9 All characters wu* in Theorem 5 vanish for the same classes. As in the proof of the theorem on induced characters (see 1), it is sufficient to show that if a congruence (8)

~

0

cito (9t)

(mod qt)

modulo a prime ideal power qt' of a suitable algebraic number field has q-integral coefficients c, and holds for all characters wu* in Theorem 5, then the corresponding congruence (9)

, ci,@D(!A) O= (mod ql)

holds for 1. It is sufficient to restrict the, summation to p-regular classes a. If q does not divide p, it follows at once from Theorem 2 of the paper quoted in 1 that (8) implies (9). It remains to treat the case that q is a prime ideal divisor of p. Let A be an element of At and let t, 4%* , denote the linear characters of the cyclic group {Al}; Set -f-1

+

(AV) = E Tq, (A))t (AV). c=o

Then (10)

a

+ (AP)_

v

AP-A

The induced expression is +*(G) (1/a):q+(RGR-1) where R ranges over all elements of (M. Now, (10) yields =

(G)

n(A), 0,5

G in

i

G not in Ri

I Cf. R. Brauer and C. Nesbitt, Annals of Mathematics, vol. 42 (1942), pp. 556-590, in particular, equation (9) and the argument in ? 14.

715

APPLICATIONS OF INDUCED CHARACTERS.

glg, is the order of the ilormalizer of A in (X. Substituting

where n (A)

=

this for

in (8), we find

Wc*

c,n(A)

Now

(modqt).

zO

(Aj) is divisible by the highest power of q dividing n(A). cj(D()

~O

Hence

(mod qt).

This implies (9), and Theorem 5 is proved. The linear combinations of the characters w1o, (J28, * S* with integral rational coefficientsform a module Q. If elements of Q are linearly dependent, there exists a linear relation with integral rational coefficients. This is seen at once when the elements of Q are expressed by the irreducible characters of (M. are linearly Let i1, '2, q,W be a* basis of Q. Then i1, *2< * *iw numbers. independent in the field of all In particular, the characters I1, 25,* * . of the distinct indecomposable constituents of the modular regular representation of (Mbelong to Q and they are linearly independent. Every element of Q vanishes for all p-singular classes Ai of (Mand can, therefore, be expressed by the qi with integral rational coefficients.10 Hence the I?D also form a basis of Q. This shows that the number w of basis elements of Q is equal to the number of distinct ID, that is, to the number of p-regular classes Ri in (X. Further, the q' and the Ii are connected by a unimodular linear transformation with integral rational coefficients, W

.

(11)

.

.

+=

b,ip?.

. 5fw of Q when the wu* are known, it seems that the J?j themselves cannot always be determined on the basis of this information. It follows from (11) and the orthogonality relations for modular group characters that

While it is of coursepossibleto determinea basis +1,

(12)

( qa,8= (1/g) Egiha i

)(S +f (As)-

t25

.

.

bapCpubou p,

10If a linear combination t with integral rational coefficients of the ordinary of Q3vanishes for all p-singular elements of 6, then a concharacters XI, X21 -* with complex sideration of ranks shows that t can be written in the form t coefficients hi. The orthogonality relations for modular group characters yield of * * are the modular irreducible characters hi= (l/g) > (R-1) (1(R) where 01,02,( and where R ranges over all p-regular elements of (. Each ?, (R) can be written as a linear combination of the Xj (R) with integral rational coefficients. Substituting this expression for p, (R), we easily see that the h, are rational integers.

716

RICHARD BRAUER.

where the cpa are the Cartan invariants of (M. The matrix with the coefficients is equal to BCB' where B= (bafi), C = (C,af). The corresponding qa: quadratic form is equivalent to the form with the matrix C. This yields 6. If the characters w)T in Theorem 5 are known, a quadratic be form can found which is equivalent to the form whose matrix is the Cartan matrix of ( for p. THEOREM

Consider an element

t

of D, ==

Ixuif.

It follows from (12) that (13)

(1/g)

, gj(9

=)T

)

, xpxcJqpc7.

If the expression (13) is equal to 1, then + t is an irreducible ordinary character of (. Since t vanishes for all p-singular classes of (M,its degree is divisible by the highest power pa of p which divides g.11 Conversely, if $ is an irreducible ordinary character of (M whose degree is divisible by pa, then t vanishes for p-singular classes and belongs, therefore, to Q. If we set =

EXutfiu,

the coefficients X(Jgive a solution of E

xpxJqpu

1.

We thus have 7. Let pa be the highest power of p which divides the order g of (. The number of ordinary irreducible representations of ( whose degree is divisible by pa is equal to the number of representations of 1 by the , xw and quadratic form in Theorem 6. (We count XIc,x2* xI,- x2, - xw as the same representation.) THEOREM

The number determined in Theorem 7 can also be characterized as the

number of blocks of defect 0 of ( (for p). To some extent, Theorem 7 fills a gap left in the investigation of the blocks of a given group.12 UNIVERSITY OF TORONTO.

See the paper quoted in 9. R. Brauer, Proceedings of the Nationat Academy of Sciences, vol. 30 (1944), pp. 109-114, vol. 32 (1946), pp. 182-186 and 215-219. 12

TRANSACTIONS OF THE ROYAL SOCIETY OF CANADA VOLUME XLI

: SERIES III

:

MAY, 1947

SECTION THREE

«< «< «< «< «< ((( «< ((( «< «< «< «< ((( ((( «< «< «< «< «Nix(APi)

whereNi is definedin (14). Take two vectorsof X, Xwiththe componentsxi and u withthe components us , and definethe innerproduct(g, u) by (S.

u) =

Es xtivf

The innerproductof two vectors of Y is definedin an analogous manner.It followsfromthe orthogonalityrelationsforthe charactersof (Mand of M that

(22) (23)

(x(AP), x(APj)) = n(AP,) 4is, (#(Pt)), t(P5'0))

-

q()

'Aj

CHARACTERS

OF FINITE

GROUPS

OF

363

ORDER

whereqj? as above denotesthe orderof the normalizerof P(X) in I. Because of the orthogonality(23), the h vectors 4(PiX)) are linearlyindependent.This impliesthat the h - m vectors3X) are linearlyindependent.Further,(20) and (23) show that w(pS(p )), )s))

(24) (25)

=

Pi,

= Np~p,

(p, Ib)

and it followsfrom(19) and (24) that bj is orthogonalto all vectorsgM* Since the m vectors bi are orthogonaland not 0, they are linearlyindependent.We now see that the m vectorsbj togetherwith the h - m vectorsg(x)forma basis of X. we can set If Zo is the subspace ofX spannedby the j (26)

(mod Z0)

ei=-?1ci&

wherethe ci, are complexnumbersin K(E). Since each a, is orthogonalto Zo, (26) and (25) yield (es, Up) = cipNp.Hence Uphas the ith componentaipNp Then (25) implies Formingthe determinantforp, ar= 1, 2, * on determinants, we find

, m and usinga

wellknowntheorem

where A ranges over all minorsof degree m of the (h X m)-matrix(ci,). In particular,therewill exista minorA such that

v(A^-II; No) < 0.

(27)

We may assume that A appears in the firstm rowsof (cip). Since Zo belongsto the kernelof T, it followsfrom(26) that ej and have the same image. Thus (18) and (21) yield (Wli,

WV*

EK

i

=

W)-

,

=

y

CiA

ci;A,,&

ciryN x(APA).

Hence, by (22) W;

I,,

Cj

Nyn(AP,,).

Formthe determinantoftheseexpressionsfori, j = 1, 2, matrix(wj) is denotedby W0, thisyields

,

m. If the (k X m)-

det (WoWo)= det (Ai)H,, N,,H,, (N,,n(AP,)). On account of (27) and lemma 2, we find v (det WoWo) < 0. of Wo are algebraicintegers,the equality sign must hold. Since the coefficients The matrixWo is of the type occurringin Lemma 1 and hence we have proved Lemma 1.

364

RICHARD

BRAUER

4. Results related to Theorem 1 We firstprove Theorem 2 formulatedin the Introduction.The necessityof the conditionsis trivial.On the otherhand, if 0(G)

?k1aixi(G)

=

is a generalizedcharacter,we have (1/g)Ea I 0(G) 12= ?k

ai.

from Now condition(III) of Theorem2 showsthat onlyone of the ai is different 0 and that thisai has the value +1. Thus, O(G) = ?xi(G). If 0(l) > 0, the + sign must apply and O(G) is an irreduciblecharacterXi(G) as was to be shown. The conditions(II) and (IV) of Theorem2 can be replacedby the REMARKS. one conditionthat, foreveryelementarysubgroup@?,the restrictionof0(G) to to require e is a (reducibleor irreducible)characterof Le.Also, it is sufficient (II) or our new conditiononlyformaximalelementarysubgroupsC?of 5. Further,only one groupfromeach class of conjugatesuch groups @ has to be considered. As an applicationof Theorem1, we now prove the followingTheoremon induced character(see footnote'). in the 3. Everycharacterx of a group(Moffiniteordercan be written THEOREM form

x=

(28)

cip

characterof an elementary subgroupLi of (M,where whereeach hi is an irreducible of 5 inducedby4if, and wheretheci are rationalintegers. 4,* is thecharacter charactersofall elementarysubgroups PROOF. Let sPirangeoverall irreducible of(M.Suppose we have n such characters,i = 1, 2, * , n. If so*is the character of (Minducedby Ais, we have formulas

(29)

AX

=

(i = 1, 2,

bijxj

...

,n)

to prove (28) in the case that It is sufficient with integralrational coefficients. x = Xi is an irreduciblecharacterof (M.We wish to show that a formula(28) forx = Xj withintegralci can be obtainedby solvingthe equations (29) forthe Xi . This will be establishedby showingthat if p is a rationalprime,the n congruences (30) (i = 1, 2, **

(mod p)

1bijzj _ 0 ,

n) fork rationalintegersz1, **

k,Zk have

only the trivial solu-

tion.2

Indeed, supposethat (30) holds forsome p and k rationalintegersz1, Set

(31) 2

...

, Zk .

O(G) = (1/p)Fki zjxi(G).

This impliesthat n > k and that the kthdeterminantdivisorof the matrix(bis) is 1.

CHARACTERS

OF GROUPS

OF FINITE

ORDER

360

It is clear thatO(G) is a class functionon (M.Set Si =

bjzj,. (1/p)E1,_j

By (30), si is a rationalinteger.Further,by (31) and (29) (32)

(1/9)EG0(G)6*(7Gr) = (1/p),jbjzj

= Si.

The charactersi* of (Minduced by the character /ijof a subgroup(s of order es can be definedby ij*(G)= (11ei)Ex .i(XGX-') whereX rangesover 3 and wherewe set V/i (Y) = 0 for Y q Es . If this is used in (32), we find Si=

(gei)-lG

Ex 0(G)4i(XG-'X-1).

Set hereXGX-1 = H, C = X-'HX. Then X and H rangeindependentlyover (M. Since O(G) is a class function,this yields Si = (gei)-1Ex

?y O(H)oi(H-)

= (1/ei)jH 0(H)0i(H-).

Here, H can be restrictedto C~i, since xVi(H)= 0 forH (33)

j;

si = (1/ei)jH o(H)4i(H-1),

(H e Ci).

Consideronly values i forwhich Hi is a given elementarysubgroupe of (M. Then in (33) 41ican be an arbitraryirreduciblecharacterof A. The restriction of 0 to e is a class functionon Le.Since the si are rationalintegers,(33) shows that the restrictionof 0 to Lfis a generalizedcharacterof A. In fact O(H) = JEi ssif(H)

(forH E A).

We have thus shownthat O(G) satisfiesconditions(I) and (II) of Theorem1. Hence O(G) is a generalizedcharacterof (M.This implies that the coefficients zj/p in (31) are integers.Hence the solutionzj of (30) is the trivial solution; zj _ 0 (mod p) forall j. This concludesthe proofof Theorem3. We next prove a lemma concerningelementarysubgroups. LEMMA 3. Let e = 21 X v be an elementary subgroupwhere$ is a p-group for someprimep and 21a cyclicgroupof an ordera primeto p. If st is an irreS o ; $, of e, and duciblecharacterof A, thereexistsa subgroup o = 21 X Do, 9 t1 of e inducedby4io. a linearcharacter' i/oof L&suchthat is thecharacter PROOF.The degree of every irreduciblecharacterV/of e is a power p8 of p, since 4 is the product of irreduciblecharactersof the cyclic group 21and the p-group$. If s = 0, the lemma is trivial; we may take C& = @, 4/o= 41.We may assume that the lemma is proved in all cases wheres has a smallervalue. relations Let X be a linear characterof A. It followsfromthe orthogonality that 4', containsX with the same multiplicitym with which 4' appears in 4AX. 3

By a linearcharacter,we always mean a characterof degree1.

366

BRAUER

RICHARD

Since 4A'X and i1 have the same degree,m < 1. Hence ",6 containsX withthe multiplicity1 or 0; we have the firstcase, if /'X= 4t.The linearcharactersX satisfyingthis conditionforma multiplicativegroupA. If we set ,V = Ex + Lhi, (X e A), (34) all the charactersAsi on the righthave degreeswhichare properpowersof p. On comparingdegrees,we see that the orderof A is divisibleby p. Hence A contains an elementXiwhose orderis p. Thus the values X1(E), E e A,,are the pth roots of unity.If j is the kernelof Xi, then Gi1is a normalsubgroupof index p of A. For A e Xf,the value X1(A) is an ath root of unity. Hence X1(A) = 1, K a Y. Hence L has the form i = ?f X $1 where $1 is a subgroup of $. If E is restrictedto tSj, then X1(E) becomes the principalcharacterof L.,. Since the principalcharacterXoof e appears among the X in (34), it follows that Ai(E1)k(Ei) as characterof , containsthe principalcharacterof El with a multiplicityhigherthan 1. This impliesthat ql(E1) is reducible. Let 4 be an irreduciblecharacterof Y occuringin #&(E1). By the Frobenius reciprocitytheorem,the character0* of e inducedby 4 containss. The degrees Dgo and Dgt of4 and ^, are powersof p. Since Dgq < Dg4t,we have p Dgq < Dg4,.Now, 0* has the degreep Dgq ? Dgai. It followsthat 0* = A. Since Dgo < Dgp, the lemmais trueforthe irreduciblecharacter4 of the elementarygroup @,. Hence thereexists a subgroupL& = ?1 X $o and a linear characterqto of o such that 4,oinduces the character4 of j . Then 4toinduces the character0* -= , of (Yand this showsthat Lemma 3 holds for4,too. CombiningLemma 3 and Theorem3, we find 4. Everycharacterof 65 is a linearcombination THEOREM withintegralrational coefficients of charactersof 65 induced by linear charactersof elementary subgroupsof 65. We now show THEOREM 5. The followingconditionsare necessaryand sufficient in orderthat a function0 defined on 65belongto thecharacter ringof65withregardtothefieldK. (I) 0 is a classfunction. (II*K) If X is a linearcharacterof an elementary subgroupe of ordere of 65,the number (E e (Y) 0(E)X(E ') (Il/e) is an integerof K. PROOF. If 0 is a class function,we can express0 in the form(1) wherethe coefficients as are given by (2). It followsfromTheorem 4 that the ai will be integersof K, if and onlyif the sum S = (1/g) EsG (G)X*(T1)

EE

is an integerof K foreverycharacterX*of 65induced by a linearcharacterX of an elementarysubgroupL of 6. An argumentsimilarto that used in connection with(32) above yields

S = (I/e) Now, Theorem5 is evident.

ER e(E)X(E'1)

(E e @).

CHARACTERS

OF GROUPS

OF FINITE

ORDER

367

In the same mannerin which Theorem 2 was deduced fromTheorem 1, we can obtain a criterionforirreduciblecharactersfromTheorem5. Thus, we have THEOREM 6. Thefunctione(G) is an irreducible character ofthegroup(Moffinite order,if and onlyif thefollowingconditionsare satisfied (I) 9(G) is a classfunction. (II*) If e is an elementary subgroupof (5 and X a linearcharacterof I, if e is the orderof @, then

(35)

(1/e)J:e O(E)X(E 1)

(E e (9),

is a rationalinteger.

(III)

(1/g)JoeI(G)

12

=

1,

(Ge-().

o (1) >_O.

(IV)

We can write the condition(II*) in a different form.Consider a fixedelementarygroup (Y = 2[ X 3. Every linear characterX of (Y can be interpreted as a linear characterof the factor group L/L' where A' is the commutator group of L&= 2[ X A. Clearly T' is equal to the commutatorgroup 2$' of 93. Let Y range over the elementsof Y/0. Each Y is then a residue class of e modulo C'. If we denote the rationalinteger(35) by z(X), then z(X)

=

(/e)

y2(j2EeyO(E))X(Y-').

Multiply this with X(X) whereX is a fixedelementof Y/I' and add over all linearcharactersX of /0'. If e' is the orderof A', this yields (36)

AEeX0(E)

=

e'l:

z(X)X(X).

Conversely,the condition(II*) will hold for a fixedelementarygroup A, if foreveryX e /', we have formulas(36) whereX ranges over the linear characters of the abelian group C/&' and where the z(X) are rational integersdepending on X but not on X. 5. Linear charactersof ( In the case of a linear character,Theorem 2 can be stated in the following simplerform: THEOREM 7. The followingconditionsare necessaryand sufficient in orderthat 0(G) be a linear characterof (i. (I ') O(G) is a class functionwhichis not identically0. (II') The equationO(AB) = O(A)O(B) holds (a) if A and B are twocommuting elements ofrelatively primeordersof0; (b) if A and B are elements ofa subgroupof 5 of prime power order. PROOF. The necessityof the conditionsis clear. Conversely,assume that the conditionshold. It followsfrom(IJJb) that the restrictionof e to a p-group$ containedin 3 is a linearcharacterof A3.Then (JL'a) showsthat the restriction of 0 to an elementarysubgroupe of 5 is a linearcharacterof A. Since everyG

368

RICHARD

BRAUER

of _ 1 ofp. Ofcourse,both A and B are powersof G. Then A willbe calledthe p-regularfactorofG. We shall also use the factthatif0 is an elementof XR suchthat9(G) and 0(A) lie in Z, then 0(G) E @(A)

(12)

(modp).3

This can be seenas follows.If we replace0 by thecyclicgroupgenerated by G, then0 willstilllie in the corresponding characterringand henceit willsuffice to prove(12) in thecase that(Mis cyclic.Then the irreducible characters xi of = = If B has the order e are linearand G = AB implies q Xi(G) xi(A) xi(B). pt. thisimpliesXi(G) = xi(A)'. Now 0 has theform8 = Ei aixi withai e R. On we obtain raisingthisequationto thepth powerr timesin succession, 0(G)q-Hi

aixi(G)Y= Hi a2xi(A)' 3(A)'

(modpR).

By Fermat'stheorem, 8(G), O(A)' X 0(A) (mod p) and thisyields O(G)' (12).4 if 2Xis a cyclicgroup We shallsay thata group p = S X e is p-elementary, of orderprimeto p whilee is a p-group.

LEMMA3. If everyp-elementary subgroup1gof ( is containedin a group of thefamily I 1, thenthereexistsan element7)of VR 8uchthat,for everyG e 0, 7)(G) e Z and ha

(modp). v(G) -1 PROOF. Let A rangeover a full system{A of representatives forthose elements.In order classesofconjugateelementsof? whichconsistofp-regular (13)

3The

use of this well known congruence can be avoided in the following; cf. footnote 5. 4 Since R has a Z-basis, we have pR n z = pZ. Hence two elements of Z which are congruent (mod pR) are indeed congruent (mod p).

6

RICHARD BRAUER AND JOHN TATE

to constructelements71A of VR forevery to proveLemma 3, it will suffice properties: A e {A withthefollowing (I) lA(G) e Z foreveryGe @, (II) ,lA(G) = 0, if the p-regularfactorof G is not conjugate to A, 1 (modp), ifthep-regular factorofG is conjugateto A. (III) ,A(G) Indeed,ifsuch 71A are known,we can take tj in Lemma3 as thesumofthese aA forall A in {A). For everyG e (M,thereexistsexactlyone A in {A I which factorofG. Henceexactlyone 1A(G)is congruent is conjugateto thep-regular to 1 (modp) and all othersvanish. elementA, let at be thecyclic In orderto construct an rAfora fixedp-regular I5(A) of groupgeneratedby A, let e be a p-Sylowsubgroupofthe centralizer = 2[ X A3.Then p is p-elementary and henceis contained A, and set 'h = in some t e {H. I. Now Lemma2 can be applied.This yieldsan element4 of VRsuchthat+(G) e Z foreveryG in @, +6(G)= 0 if G is not conjugateto an elementofAZ, and +(A) = ((E(A):Z). Now (E(A):3) is primeto p, sincee is a p-Sylowsubgroupof (F(A).Hencewe can finda z EZ suchthatz(((A): ) 5 1 (I). (modp). If we set X1,= z4, thenqa e V8 and lAhas therequiredproperty If thep-regular factorofan elementG of@ is notconjugateto A, thenG is not conjugateto an elementofAZBand hence4) and lAvanishat G. Thus 1A has 1 (mod p) and now (12) showsthat A has property(II). We have nA(A) theproofoftheLemma.5 (III). This finishes property theorem. 6. We can nowprovethefollowing in somegroup of @ is contained THEOREM 3. If every subgroup p-elementary = = . e with 1, thengo Vz Sx ofthefamily1a1I and if g pmgo (go, p) If c is PROOF. Let v have the same significanceas in Lemma 3. Set q = pm. = cP 1 mod p9+1* 1, 2, .), then a rational integerwith c 2 1 mod pi, (3

Hence (13) implies

(

(14)

m

(mod ptm)

1

forall G in (. Since VRis an ideal and hence a ring,qq e VRi. Set qq = 1 + e0 . Then (14) shows that Theorem2 can be applied to e = goe0whichis obviously

VRa class function.It followsthat g9oo E VRand hencego = gego Now Lemma1 yieldsg0e Vz as we wishedto show. subgroupof @ is containedin somegroup COROLLARY. If everyelementary then ta f {goaI, -

Uz = X()

= Vz.

Indeed,Theorem3 can be appliedforeveryprimep. Since Vz containsa rationalintegerprimeto p foreveryprimenumberp, we have 1 e Vz. Since Vz is an ideal of Uz, Vz = X(()

= Uz.

The equationX(@) = Vz yieldsTheoremA and the equationX(@) = Uz yieldsTheoremB statedabovein 1. 5 Instead of using (12), we can use the resultstated in footnote2 and Sylow's theorems to showthat +(G) ((S(A):5B) mod p, if the p-regularfactorof G is conjugate to A. This impliesthat 1Ahas the property(III).

ON THE

CHARACTERS

OF FINITE

GROUPS

7

If R is an arbitrary integraldomaincontaining Z, thenVR_ Vz. Underthe assumptionof the Corollary,1 e Vz 9 VR and hence UR = Xi(@) = VR. HARVARD UNIVERSITY BIBLIOGRAPHY

On Artin's L-series with generalgroup characters,Ann. of Math., 48 (1947), pp. 502-514. of thecharactersof groupsoffiniteorder,Ann. of Math., 2. R. BRAUER, A characterization 57 (1953),pp. 357-377. einer endlichenGruppe, des Charakterringes 3. P. RoQvETTE,Arithmetische Untersuchung J. Reine Angew.Math., 190 (1952),pp. 148-168. 1. IT.

BRAUER,

Richard Brauer: Collected Papers Volume II Finite Groups

Mathematicians of Our Time Gian-Carlo Rota, series editor Richard Brauer: Collected Papers Volume I Theory of Algebras, and Finite Groups edited by Paul Fong and Warren ]. Wong [17]

Richard Brauer: Collected Papers Volume II Finite Groups edited by Paul Fong and Warren J. Wong [18] Richard Brauer: Collected Papers Volume III Finite Groups, Lie Groups, Number Theory, Polynomials and Equations; Geometry, and Biography edited by Paul Fong and Warren ]. Wong [19] Paul Erdös: The Art of Counting edited by Joel Spencer [5] Einar Hille: Classical Analysis and Functional Analysis Selected Papers of Einar Hille edited by Robert R. Kallman [11] Mark Kac: Probability, Number Theory, and Statistical Physics Selected Papers edited by K. Baclawski and M. D. Donsker [14]

Charles Loewner: Theory of Continuous Groups notes by Harley Flanders and Murray H. Protter [l] Percy Alexander MacMahon: Collected Papers Volume I Combinatorics edited by George E. Andrews [13] George Polya: Collected Papers Volume I Singularities of Analytic Functions edited by R. P. Boas [7] Goerge Polya: Collected Papers Volume II Location of Zeros edited by R. P. Boas [8]

Collected Papers of Hans Rademacher Volume I edited by Emil Grosswald [3] Collected Papers of Hans Rademacher Volume II edited by Emil Grosswald [4] Stanislaw Ulam: Selected Works Volume I Sets, Numbers, and Universes edited by W. A. Bayer, J. Mycielski, and G.-C. Rota [9] Norbert Wiener: Collected Works Volume I Mathematical Philosophy and Foundations; Potential Theory; Brownian Movement, Wiener Integrals, Ergodic and Chaos Theories; Turbulence and Statistical Mechanics edited by P. Masani [10] Norbert Wiener: Collected Works Volume II Generalized Harmonie Analysis and Tauberian Theory; Classical Harmonie and Complex Analysis edited by P. Masani [15] Oscar Zariski: Collected Papers Volume I Foundations of Algebraic Geometry and Resolution of Singularities edited by H. Hironaka and D. Mumford [2] Oscar Zariski: Collected Papers Volume II Holomorphic Functions and Linear Systems edited by M. Artin and D. Mumford [6] Oscar Zariski: Collected Papers Volume III Topology of Curves and Surfaces, and Special Topics in the Theory of Algebraic Varieties edited by M. Artin and B. Mazur [12] Oscar Zariski: Collected Papers Volume IV Equisingularity on Algebraic Varieties edited by ]. Lipman and B. Teissier [16] Note: Series number appears in brackets.

Richard Brauer

Collected Papers Volume II Finite Groups

Edited by Paul Fong and Warren J. Wong

The MIT Press Cambridge, Massachusetts, and London, England

Publisher's note: Richard Brauer left an annotated set of his publications. Following his wishes, these annotations have been incorporated by hand into these reprints. Publication of this volume was made possible in part by a grant from the International Business Machines Corporation. Copyright© 1980 by the Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. This book was printed and bound by The Murray Printing Company in the United States of America. Library of Congress Cataloging in Publication Data Brauer, Richard, 1901Collected papers. (Mathematicians of our time; v. 1 7-19) Text in English and German. Bibl1.ography: p. CONTENTS: v. 1. Theory of algebras and finite groups.-v. 2. Finite groups.v. 3. Finite groups, Lie groups, number theory, polynomials and equations; geometry, and biography. 1. Mathematics-Collected works. I. Fong, Paul, 1933- II. Wong, Warren J. III. Series. 80-17622 510 QA3.B83 ISBN 0-262-02135-8 (v. 1) 0-262-02148-X (v. 2) 0-262-02149-8 (v. 3) 0-262-02157-9 (complete set)

1901-1977

Contents (Bracketed numbers are from the Bibliography in Volume I)

Volume II Finite Groups (Continued) [64] On groups of even order (with K. A. Fowler)

3

[65] Zur Darstellungstheorie der Gruppen endlicher Ordnung

22

[67] Number theoretical investigations on groups of finite order

61

[68] On the structure of groups of finite order

69

[70] A characterization of the one-dimensional unimodular projective groups over finite fields (with M. Suzuki and G. E. Wall)

78

[71] On a problem of E. Artin (with W. F. Reynolds)

106

[72] On the number of irreducible characters of finite groups in a given block (with W. Feit)

114

[73] Zur Darstellungstheorie der Gruppen endlicher Ordnung II

119

[74] On finite groups of even order whose 2-Sylow group is a quaternion group (with M. Suzuki)

141

[76] On blocks of representations of finite groups

144

[77] Investigation on groups of even order, I

14 7

[78] On finite groups with an abelian Sylow group (with H. S. Leonard)

150

[79] On groups of even order with an abelian 2-Sylow subgroup

165

[80] On some conjectures concerning finite simple groups

171

[81] On finite groups and their characters

177

[82] Representations of finite groups

183

[83] On quotient groups of finite groups

226

[84] A note on theorems of Burnside and Blichfeldt

239

[85] Some applications of the theory of blocks of characters of finite groups. I

243

[86] Some applications of the theory of blocks of characters of finite groups. II

259

[89] On finite Desarguesian planes. I

287

[90] On finite Desarguesian planes. II

. 294

[91] Investigation on groups of even order, II

322

[92] Some applications of the theory of blocks of characters of finite groups, III

328

[93] A characterization of the Mathieu group M 12 (with P. Fong)

359

[94] An analogue of Jordan's theorem in characteristic p (with W. Feit)

389

[95] Some results on finite groups whose order contains a prime to the first power

402

[97] On simple groups of order 5 · 3a · 2b

421

[98] Über endliche lineare Gruppen von Primzahlgrad

471

[99] On a theorem of Burnside

495

[100] On blocks and sections in finite groups. I

499

[101] On pseudo groups

521

[102] On blocks and sections in finite groups, II

531

[104] On a theorem of Frobenius

562

[105] Defect groups in the theory of representations of finite groups

566

viii

CONTENTS

Finite Groups (Cöntinued)

ANNALS

OF MATHEMATICS

Vol. 62, No. 3, November,1955 Printedin U.S.A.

ON GROUPS OF EVEN ORDER By

RICHARD

BRAUER

AND

K. A.

FOWLER

(ReceivedSeptember 14,1954) I. Introduction 1. Let 65 be a group of finiteorderg. We prove firstthat if g > 2 is even, then thereexists a propersubgroupof orderv > Vg. The proofis quite elementarybut the methodcannotbe applied if g is odd, thoughit seemsprobable that a similarstatementholds in that case too. Indeed, if 65 is soluble of an orderg > 1, g not a prime,it is veryeasy to see that 65 has a propersubgroup of orderv ? Vg. It is a well known unproved conjecturetha& all groups of odd orderare soluble. We shall use the terminvolutionfora group elementof order2. If m is the total numberof involutionsof 65 and if we set n = g/m,the same method showsthat 65containsa normalsubgroupV distinctfrom 5 such that the index ofV is either2 or is less than [n(n + 2)/2]! (where[x]denotesthe largestinteger not exceedingthe real numberx). If J is an involutionin 65 and if n(J) is the orderof its normalizer91(J)in 5, then n < n(J). It then followsthat there exist only a finitenumberof simple groupsin which the normalizerof an involutionis isomorphicto a givengroup. 2. The followingterminology will be useful.An elementG of a group 65 will be said to be real in 65if G and G-' are conjugatein 65.1 Real elementsdifferent fromthe identity1 occuronlyin groupsof even order.With G, everyconjugate elementis real. We may therefore speak of real and non-realclasses of conjugate elementsin 65. from1 in 65. We introducea Let 65* denote the set of elementsdifferent "distance"d(G,H) forany two elementsG, H of ,5. If G = H, set d(G, H) = 0. If G # H and if there exists a chain Go, G1, *... GI of elements of 65#with Go = G. G. = H such that Gi-1and Gi commute,let d(G, H) denotethe length 1 of the shortestsuch chain connectingG and H. In particular,d(G, H) = 1 if and onlyif G and H commuteand G # H. If theredoes not exista chain connectingG and H, set d(G, H) = ocxIt is clear that this distance has all the usual propertiesexceptthat it can be infinite. If 9 is a subset of 65#and G E 65, we definethe distanced(G, DI) of G from DI to be the minimumof the distancesd(G,H) forH e W. 3. In Section III, it is shown that if the group 6 contains more than one class of involutions,then any two involutionshave distance at most 3. This impliesthat if the two elementsG and H both have normalizersof even order 1 An element of(Dhas a realvalueforG. G is realin (Difand onlyifeverycharacter NotethatifG is notrealin 5,it maybe realin groupscontaining (Das a subgroup.

565

566

RICHARD

BRAUER

AND

K. A. FOWLER

in a groupof thiskind,thend(G, H) < 5. The values 3 and 5 givenherecannot be replacedby smallervalues. from1. If G 4. Section IV deals with propertiesof real elementsG different has distance at least 4 fromthe set 9) of involutions,this distance is infinite. Actually,the normalizerg = 9(G) of G in (Mis an abelian groupwhichis the normalizerof each of its elementsH $ 1. This impliesthat d(H, L) = X if H is an elementof &, H # 1, and L is an elementof 65 not in SA.SubgroupsS& in groupsofeven order.They have a number of thistypeoccurfairlyfrequently of interestingproperties.In particular,the orderh of SDis relativelyprimeto everyelementof t into its its index.There existinvolutionsJ whichtransform inverse.If n(J) is the orderofthe normalizer9%(J)of J in 6, thenh < n(J) + 1, unless & is a normalsubgroupof 65; in the lattercase, 65 "splits" into g and a subgroupQ3 of 9(J). There exist infinitelymany simple groups 65 each of whichcontainsa subgroupg of the type here discussedwithh = n(J) + 1. Our resultsconcerningreal elementsG # 1 of distance at most 3 fromthe set 9 of involutionsare ratherfragmentary. It can happen that the distance fromG to 9 is actually equal to 3. In this connection,we show that, under certainconditions,some ofthe Sylow subgroupsof 65are abelian. It is, of course, veryeasy to constructgroupsin whichno Sylow subgroupis abelian. However, a largenumberof "interesting"groupsseem to possess some abelian Sylow subgroups.Perhaps,in view of this,our resultdeservesconsideration. 5. The last sectiondeals with propertiesof the charactersof groups of even order. If n has the same significanceas in 1, there exists an irreduciblereal ofa degreeless thann. On thebasis ofthisremark, character,notthe 1-character, one can studythe cases wheren is small. If the resultsof C. Jordanand H. F. Blichfeldton lineargroupsof a givendegreecould be improvedmaterially,this would make it possibleto improvethe resultsmentionedabove in 1. If p is a primedividingg with the exact exponenta, it may be that (Mdoes not possess irreduciblecharactersof defect0 for p, that is, characterswhose degreesare divisibleby pa. On the otherhand, many "interesting"groups do conditionsfor the have such characters.In Section V we give some sufficient existenceof charactersof defect0. Finally,if (Mcontainsa subgroup& of the type discussedin 4, ratherdetailed informationconcerningthe values of the irreduciblecharactersof @ for the the characters elementsof lb can be given.This is of greathelp in constructing of (M. 6. NOTATION. The normalizerof an elementG of (Mwill be denoted by 9(G) and its orderby n(G). The set of elementsX of 65 which transformG into G or G-1,that is, forwhich X-1GX

=

G or X-1GX=

G-

ON GROUPS

OF EVEN

567

ORDER

formsa subgroup91*(G).If G is non-realor ifG is an involution,%*(G) = 9(G). If G is real and of ordergreaterthan 2, 9*(G) has order2n(G). The classes of conjugateelementsof 65will be denotedby go, S1, ** ,-, Here, go will be the class containing1. Then the classes containinginvolutions are taken, say these are the classes St1, -*. , gR. Next we take the otherreal classes and finallythe non-realclasses. Usually, Gi will denote a representative elementforSi . If ni = n(Gi), thenRi consistsof g/njelements. The groupringof 65formedover the fieldof rationalnumberswill be denoted by r. With each Ri we associate an elementKi of r. Here, Ki is the sum of the , Kk_1forma g/nielementsof ,i . As is well known,the elementsKo, K1, basis forthe centerA of r, and hencewe have formulae KiKj = E,=o

(0)

aijAKA.

Here, the aij, are non-negativerationalintegers. If g is a subgroupof 65,the index of ' in 6 will be denotedby (65:b). II. Existence of large subgroups 7. Let 9Mbe the set of involutionsof the group 65 of even order.Then 91Iis ,Sr. Set the unionof R1, R2,

+ Kr. M = K, + K2 + Then M is that elementof A whichis the sum of all the involutionsin 65. It followsfrom(0) that we have formulae (1)

M

(2)

2

=

ciKi . Ei=o

Clearly,the coefficient ci is equal to the numberof orderedpairs (X, Y) of involutionsX, Y such that (3)

(X,Ye 9).

XY=G,

We show of65whichtransform LEMMA(2A). If G2 5 1, thenci is thenumberofinvolutions 1 wherev, is thenumberof thenci = viGi into G7'. If Gi is an involution, in 9(Gi). Finally,forGi = 1, ci = m. involutions PROOF.If X, Y satisfythe conditions(3), then G-=

Y-'X-1 = YX = X-'(XY)X

= X'1GjX.

Conversely,if X-'GiX = G.' and X E 9, then the elementY = XGi satisfies the equation y2 = 1. Hence the conditions(3) are satisfiedif Y # 1. We have Y = 1 if and onlyif Gi = X, and then G, itselfis an involution.All the statementsof the lemma are now readilyobtained. (2B). If Gi is non-real,ci = 0. For any Gi, ci _ n(Gi). If Gi is COROLLARY an involution, ci _ n(G) - 2. PROOF.If Gi is not real, the lemmashows that ci = 0. If Gi is real, thereare Gi into GU1,and henceci ? n(Gi) = ni. exactlyn(Gi) elementswhichtransform Finally,ifGi is an involution,vi < n - 1 and hence ci _ ni - 2.

568

RICHARD

BRAUER

K. A. FOWLER

AND

8. Countingthe numberof groupelementsoccurringon both sides of (2), we obtain mI= =

icig/ni

wherem is the numberof involutionsin (M.We now apply (2A) and (2B). For each real G2, cig/ni _ g. If k1is the numberof real classes,we obtain

(4)

m2

_

m+

Et-j

(v

- 1)g/ni+ (k - r- 1)g

wherethe last termoriginatesfromthe ki of orderlargerthan 2. The numberof involutionsis given by

r - 1 real classes Mi of elements

-

m = Et=l gini.

(5) If we set

v = Max{vi, v2, ..., vr} we obtain (4*)

m2
4/g. PROOF. If we set m = g/n,then (2C) yields g ? hn2+ (v - h)n

hn(n - 1) + vn.

ON GROUPS

OF EVEN

569

ORDER

One of the groups91(Gi) with 1 < i ? r containsv involutionsand hence v < ni- 1 forsome such i. On the otherhand, (5) yields

n

(5*)

n7 + n-' +

-+

7-

and hencen < ni . It now followsthat g < hn,(ni - 1) + (n, - 1)ni

=

(h + 1)ni(n

-

1).

If ni < h, then g < h3.Now h is the orderof a subgroup9Z(H) and, since H is conjugateto H-' # H in 5, we have 91(H) ; M.Hence the theoremholds in thiscase. If h < ni, we have g < n' . If ni # g, the theoremis true with !3 = 9(Gi). It remainsto deal with the case ni = g. Then O5 contains an invariant involutionGi. We may assume that the theoremhas been proved for groups of even orderless than g. If g/2 is even and g $ 4, we may apply the theorem to 05/{ Gi}. It followsthat (M/{ Gi contains a subgroupQ* 5 (M/{ Gi of an orderv* > 9g/2.Since we may set T3* = T3/{MG}whereT3 is a subgroupof = orderv 2v*of 6, we have g > v > -A4g> 4/g Again the theoremholds. It is trivialforg = 4. Finally,if g/2 is odd, it is well known2that (Mcontainsa subgroupof order Hence the theoremholds in all cases. g/2and g/2 > -A/q. thegroupQ3in (2D) can be (2E). If (Mdoes notcontaininvariantinvolutions, chosenas thenormalizerof a suitablereal elementof (M. 9. We use a methodvery similarto that in 8 in orderto obtain a slightly strongerresult. Some of the classes l, M2 X * may consist of invariantinvolutions. Assumethat the notationis chosensuch that these are the classes f, withi 1, 2, *.., r'. If thereis no invariantinvolution,set r' = 0. Since we have n = g, 9(Gi) = 5 i = 1, 2, **,r', we have ci = vi- 1 =m-1 for for1 < i < r'. Now (4) becomes (ni m2 < m + r'(m - 1) + Zrr+l The formula(5) can be writtenin the form m = r' +

-

Zt=T'+l

2)g/ni + (k-

r - 1)g.

g/n

and we obtain m2 < m + r'(m-

1) + (r - r')g -2(n

-

r') + (k#- r

1)g

and hence {7\a!b nt_

i 1at1i I1{Z __ 2E

2 For instance, this is( a simple consequence of a Theorem of Burnside. See [3], p. 133.

570

BRAUER

RICHARD

AND

K. A. FOWLER

Let u denote the minimalindex of all the subgroupsof 6 whichare distinct ,r, we have from i,and assume that u > 2. Since n, < g fori = r' + 1, u < g/niforthesei, and (5*) yields (r - r')u < m - r'. For i = r + 1, r + 2, * , ki- 1, let 91*(Gi)denotethe groupof order2ni introducedin 6. Since 9*(Gi) has a subgroup9(Gi) of index 2 and we assumed u > 2, we must have 9*(Gi) P-? 6. It followsthat u cannot exceed the index g/2niof 9*(Gi) in 65.Now (6) implies 2u(kj

-

r - 1) < g - 1-m.

If we combinethe last two inequalitieswith (7), we obtain m2
1, that is, u < n(n + 1)/2. Suppose that r' > 1 and assume that u ? n(n + 2)/2, u # 2. Then (8) yields (r' - 1)n > gn/2u and hence

g < 2u(r' - 1). The r' invariantinvolutionstogetherwith 1 forma subgroup(Yof orderr' + 1 from65/c has an index of the centerof 6. Since any subgroupof 65/c different at least u, the orderof such a subgroupis not greaterthan g/u(r' + 1), and g/u(r' + 1) < g/u(r' - 1) < 2. It followsthat 6/c is certainlycyclic.Hence 65is generatedby G and at most one additionalelement.Since G lies in the centerof 6, it followsthat65is abelian. But then 6 has a subgroupof index 2, whereaswe assumed u > 2. Thus the assumptionsu > 2, u > n(n + 2)/2 lead to a contradiction.This proves the theorem: THEOREM (2F). Let 6 be a group of even orderg whichcontainsexactlym involutions. If n = g/m,then65 containsa subgroupof index u such thateither u = 2 or 1 < u < n(n + 2)/2. If 6 containsat mostone invariantinvolution,the numbern(n + 2)/2 can be replacedbyn(n + 1)/2.

ON GROUPS

OF EVEN

ORDER

571

We now are able to obtain a slightimprovement of (2D). We state: COROLLARY (2G). If (5 is a groupof evenorderg > 2, thereexistsa subgroup Soforderv > V2g - 1/3. 8#P PROOF. If 0 contains a subgroup of index 2, the statementis true, since > g/2 V2g forg ? 4. Assumethat 5 does not have subgroupsof index 2. Suppose firstthat 5 does not contain invariantinvolutions.If Q3is a subgroupof 5 of maximalorderv < g, Theorem(2F) showsthat g/v< n(n + 1)/2. It followsfrom(5*) that n < n1, and sinceni is the orderofa subgroupR (G1) X 0, we have n1 < v. Thus 2g < v2(v+ 1). One easily sees that this impliesv > V2g - 1/3. The case in which 5 containsinvariantinvolutionscan be treatedin a manner similarto that used in the proofof (2D). THEOREM(2H). If 5 is a groupofevenorderg whichcontainsm involutions and if n = g/m,thenthereexistsa normalsubgroupV X (5 of 5 such that $/SVis isomorphicto a subgroupof the symmetric group on u letterswith u = 2 or u < n(n + 2)/2. In particular,((: ) = 2 or ((: V) < [n(n + 2)/2]! If 5 contains at most one invariantinvolution,the numbern(n + 2)/2 can be replaced by n(n + 1)/2. PROOF. If Q3is the subgroup mentionedin (2F), there correspondsto Q3a permutationrepresentation of 5 of degreeu. We obtain (2H) by takingfor V the kernelof this representation. COROLLARY (2I). If 5 is a simplegroupof evenorderg > 2 whichcontainsm involutions and if n = g/m,then g < [n(n + 1)/2]! If J is an involution of (5, thenn in theinequalitycan be replacedbyn(J). There existonlya finitenumberofsimplegroupsin whichthenormalizerofan involution to a givengroup. is isomorphic PROOF. The firststatementfollowsat once from(2H), since a simple group ofeven orderg > 2 cannotcontainan invariantinvolution.The secondstatement thenfollowsfrom(5*), whichimpliesn < n(J). 10. For a later application,we mentionstill anotherresultwhichis obtained by the methodused above. THEOREM (2J). If 5 is a groupof evenorderg whichcontainsexactlym involutions,thenthenumberk1of real classesof 5 satisfiestheinequality kPROOF.

1 ? m(m t 1)/g.

In (4), vi can be replacedby ni - 1. Then In2
(r - 1)n,'.

In particular,we have nr > V(r - 1)g. This showsthat if 5 does not contain invariantinvolutionsand if r ? 2, then the groupMBin (2D) can be chosenas the normalizerof an involution.For r > 3, we obtain an improvementof (2G). we state introducedin the introduction, 12. Usingthe terminology thenany two THEOREM (3D). If 0 containsmorethanone class of involutions, involutions of (Mhavedistanceat most3. classes, it follows PROOF. If the two involutionsX and Z belong to different from(3A) that d(X, Z) < 2. Suppose then that X, Z belongto the same class Ri. It followsfrom(3A) that some elementX1 of fi must commutewith an involutionY not in ti. AfterreplacingX1 and Y by conjugates,we may assume X equals X1 . Since Z and Y belongto different classes, we have d(Z, Y) < 2. On the otherhand, d(X, Y) = 1. Hence d(X, Z) < 3. thenany two COROLLARY (3E). If (- containsmorethanone class ofinvolutions, elements G1and G2withevenn(G1),n(G2) havedistanceat most5. Indeed, if n(Gi) is even, thereexistsan involutionXi whichcommuteswith Gi. Hence d(G1, X1) < 1,

d(X1, X2) < 3,

d(X2, G2) < 1.

It followsthat d(G1, G2) < 5. There existgroupswithmorethan one class of involutionsin which REMARK. involutionsof distance3 occur. For instance,let (Mbe the symmetricgroup on p letters,wherep is a primeand p _ 5. If G is a cycle of lengthp, thereexists an involutionX whichtransforms G intoG-'. Then Y = XG also is an involution. If an elementZ commuteswithboth X and Y, then Z would commutewith G and henceZ would be a powerof G. The onlypowerof G whichcommuteswith an involutionis 1. Hence Z = 1 and this showsthat d(X, Y) > 2. A similarargumentcan be used to prove (3F). If (Mis a groupof evenorderwhichcontainsa real elementG such that from1 in %(G), then(M containsinvolutions n(H) is odd for everyH different whichhavedistancegreaterthan2. One can also show by examplesthat the number5 in (3E) cannotbe replaced by a smallervalue. IV. The set of real elements 13. We prove LEMMA (4A). If G is a real elementof thegroup(Mof evenorderand if n(G) G intoG-'. All involutions J whichtransforms is odd,thenthereexistsan involution in into the number and ofsuchinvolutions G-' are conjugate (, G whichtransform is equal to theindexof91(G) n W9(J)in 91(G). PROOF. Since n(G) is odd, G is not an involution.Then the group9*(G) has

574

RICHARD

BRAUER

AND

K. A. FOWLER

even order2n(G) and therefore it containsan involutionJ. Since n(G) is odd, J G into G-1. cannottransform G into G. Hence J transforms = If X is any involutionsuch that X-'GX G-', one sees as in the proofof (3A) that X is conjugateto J in 9*(G). The numberof elementsin the class of J in 9*(G) is equal to the index of9*(G) n 9(J) in 9*(G), and everyelement in this class is an involutionwhich transformsG into G-'. Since J does not belong to the subgroup9(G) of index 2 of 9*(G), it followsthat 9(G) n 9(J) has index 2 in 9*(G) n 9(J). Hence the index of9(G) n 9(J) in 9(G) is equal to the index of9*(G) n 9(J) in 9*(G). This completesthe proof. The followingtheoremis essentiallya restatementofa resultdue to Burnside, [2],pp. 229, 230. THEOREM (4B). Let & be a subgroupof 65.If thereexistsan involutionJ in the from1, thenS& normalizer of S&,and if J commutes withno elementof S& different is abelianofoddorderand J transforms everyelement of& intoits inverse. PROOF. Every elementof S& can be writtenin the formJH-'JH forH e A. Hence J transforms every elementof S& into its inverse.This implies that S cannot containan involution.For H, K e KH = J(I-'AK-')J = (JH-I'J)(JK-1J)

=

HK,

and hence 1g is abelian. As an immediateconsequenceof (4B), we have of 65 whichis transformed COROLLARY (4C). AssumethatG # 1 is an element into its inverseby theinvolutionJ. If d(G, J) > 3, then9(G) is abelian of odd orderand J transforms eachelement of9(G) intoits inverse. We now prove THEOREM(4D). Assumethatd(G, J) _ 4 in (4C). Then9(G) is thenormalizer ofeachofits elements different from1; and d(H, Z) = oo forH e 9(G), Z o 91(G), Hs 1. PROOF. It followsfrom(4C) that 91(G) is abelian. If H e 9(G), then91(G) ; 9(H). For H $ 1, we have d(H, J) ? 3 and hence (4C) can be applied to H instead of G. Since 9(H) is abelian and G e 9(H), we have 9(H) ; 91(G) and hence9(G) = 91(H). It is now clear that any elementH z 1 of9(G) has distanceat most 1 from from1 of9(G) and distance oc fromeveryelementnot everyelementdifferent in 9(G). G $ 1 has distanceat least4 fromtheset9)1 COROLLARY (4E). If a real element thend(G, DI) = oc and 9(G) is thenormalizerof each H $ 1 in of involutions, 9(G). Indeed, (4A) showsthat thereexistsan involutionJ such that J-'GJ = G-'. Then (4C) and (4D) apply. Since n(G) is odd, all involutionslie outside9(G). 14. We considera subgroupSg of an arbitrarygroup05 of finiteorderg such from1. Our results that Hi is the normalizerof each of its elementsdifferent will apply to the subgroup9(G) in (4D).

ON GROUPS OF EVEN ORDER

575

It is clear that S& will be abelian. If p is a primedividingthe orderh of 9, thereexistsan elementP of orderp in S&.Let $ be a p-Sylowgroupof (Mwhich containsP, and let PO 5 1 be an elementof the centerof $3.Then Po e 9(P) = S&and hence91(Po) = A. Since $ C 9(Po) = 9, it followsthat p is primeto the indexg/hof A. Let W denote the normalizerof & in 65. Since & is a normalsubgroupof9 and since h is relativelyprimeto (J: ), thereexists a subgroupQ such that ([3], p. 125) A=S

n,

tn9

={1}.

If G has orderw, then 9 has orderhw. In 91,the h - 1 elementsH z 1 of & have normalizersof orderh. Hence they are distributedinto (h - 1)7w classes of conjugateelementseach consistingof w elements.In particular,w divides h - 1. Actually,if pa is the highestpower of a primep dividingh, then w divides pa - 1. This is seen by consideringa Sylowgroupof A~. If an elementA of 65 transforms an elementH z 1 of & into an elementK of SA,we have A-1J(H)A = 9(K), that is, A-'SA = & and hence A lies in 91. No two distinctconjugates of & can have an intersectiondifferent from1, sinceeach conjugateof St is the normalizerof each of its elementsdifferent from 1. Now the argumentsleading to Sylow's theoremshow that the number of conjugatesis congruentto 1 modulo h. If we denote this numberby 1 + Nh, whereN is a rationalinteger,theng/(hw)= 1 + Nh and henceg = wh(l + Nh). We have proved THEOREM (4F). If 5 is a groupoffiniteorderg and & is a subgroupsuch that it is thenormalizerof each of its elements different from1, then& is abelian and its orderh is relatively primeto its indexg/h.We can set (9)

g = hw(1 + Nh);

h-

1 = wt,

wheret, w, and N are rationalintegers, t > 0 w > 1, N _ 0. The normalizer 9 of Sghas orderhwand thereexistsa subgroupG oforderw suchthat (10)

% = " ';

'gn G=

I{1}. Each elementof !t different from1 is conjugatein (, to exactlyw elementsof 1 and any twoofthesew elements are conjugatein 9. 15. We now take for St the group91(G) in (4D). Since J maps each H eon its inverse,we have J e 91. AfterreplacingG by a conjugate group in 1, we may assume J e G. Let H be a fixedelementof 't different from1. Since 91(H) = Et, only the elementsof tJ will transformH into H-'. On the other hand, for W e G, the elementW-'JW transformseach elementof St into its inverse.Hence we have W-1JW = HoJ withHo E 'D. Since J, W e U, it follows that Ho e Gi3.Now (10) shows that Ho = 1 and hence that W-'JW = J. We now have

576

RICHARD

BRAUER

AND

K. A. FOWLER

different THEOREM (4G). If (Mis a groupof evenorderg and G is a real element from1 whichhas distanceat least4 (and hencedistanceso ) fromthesetofinvolutions, 9(G). If J is an involutionwhichtransforms G theresultsof (4F) apply to = T into G1' and henceeveryelementof S& into its inverse,we may chooseQ in (10) 91(J). The numberw is even. and i33 S suchthatJ e i33 16. Let H e ?a, H - 1. Our resultsshow that thereexistexactlyh involutions H into H-'. It then followsfrom(2A) and its proofthat there whichtransform existexactlyh orderedpairs (X, Y) ofinvolutionswiththe productH; moreover, H into H-1. Then, by if (X, Y) is any such pair, then both X and Y transform (4A), X and Y belong to the same class of (Mas J does. Denote this class by Al . We now see that thereare exactlyh orderedpairs ofelementsof M1withthe productH. From (0) we have Etoal1~ (11) K1 = If H e Rx, then alix is equal to the numberof orderedpairs of elementsof .-t withthe productH. Hence alix = h. The numberof elementsin .W is g/ni. It followseasily that a110= g/n, Counting the numberof group elementsoccurringon both sides of (11), we obtain

(12)

(g/n1)2

=

g/n, + 2:k-1 anlig/n.

X For the class RxA,we have aix = h, nx = h, and thus the termin (12) fori t classes SIxwhich contain elementsof S is g. There are exactly (h - 1)7w from1. different It may happen that there are several non-conjugategroups of the same type as & whose elementsare transformedinto their inverses by the same involutionJ. Let us denote these groups by Hi (i = 1, 2, ... , s) and let tj forAs as t had forSA.Then the elementsof It different have the same significance from1 will contributeti termsg to the sum in (12) and differentS must conterms.Hence tributedifferent g2/n'> g/n1+

gE=,- ti

g ? ni +

n2E

tj.

We have shown (4H). Suppose that J is an involutionof (5 and that thereexist elements intoits inversebyJ, each ofwhich X1, X2, * *', X8 each of whichis transformed has distanceat least4 fromJ, and whichare suchthatno elementoftheclass ofXi withXi fori F j. If theelements from1 of Sk =9=(Xi) belong different commutes classesin (M,then tot; different (13)

g ? n(J) + n( J)2ZJ

tiE

17. Again let St be the group mentionedin (4G), and considerthe h sets H9MwithH e A. If two such sets H19i and H29JwithH1, H2 E a~, H1 ? H2,

ON GROUPS

OF EVEN

577

ORDER

have an elementD in common,there exist involutionsJ1 and J2 such that D = H1J1= H2J2, Hi'H2 = J1J2 . Then, as we have seen in the proofof (2A), J1 transforms HL1H2 into its inverse.Hence J1 e S9J, H1J1 e S?J, and hence n S H29J1 !J. Conversely,if H and Ho are any two elementsof &, then HAS J-'H'HoJ = H-'H. It followsthat (H-'HoJ)2 = 1. Since J does not lie ill !t, H-'HoJ e W1. This showsthat ! J is containedin each of the sets HR. Thus, = n H2W H19 )J. If m again denotes the numberof involutionsin (M,then each HE contains the h elementsof !9J and m - h elementswhichdo not appear in any of the othersets HAm. The numberof elementsin the union of the sets HW is thereforeh + h(m - h). No elementof St can appear in a set HE since & has odd orderand cannotcontainan involution.Since we have g elementsin 3, at most g - h distinctelementscan lie in the union of the sets H9R. Hence h + h(m This yieldsmh - g


g/m n, then < h(h -2)n

(14)

18. Set n(J) = ni . Since w divides n1(cf. (4G)), we can set n1 = wz, where z is a positive rational integer.On the other hand, ni divides g/h. Indeed, if thiswerenot so, therewouldexistan elementP in 9(J) ofa primeorderdividing h. Some conjugatePI of P thenbelongsto S? and hence n(Pi) = h. This implies that n(P) = h. However, since J e 9(P), n(P) must be even and we have a contradiction.Thus n1 divides g/h. Since n1 = wz, it followsfrom (9) that z divides 1 + Nh. Set 1 + Nh = zx, wherex is a positiverationalinteger.Then g = hwxz;

= wt,

h-1

1 + Nh = xz,

ni = wz.

Suppose that h > n1. It then followsfrom(5*) that h > n and hence (14) holds. Since n(h - ni) < ni(h -n), < h(h -2)nl

(16)

Set v = h - n1> 0. We thenhave xv = Ah

-

xnl

= A

-w

-xwz

It followsfrom(15) and (16) that x _ (h (17)

x-wN

= 1;

-2)/v.

+

h(x

-

wN).

Thus, we must have

xv= -w + h.

RICHARD

578

BRAUER

AND

K. A. FOWLER

Since wz = ni = h - v, the equation (15) becomes

g = hni(h - w)/(h - ni) = h(h - v)(h - w)/v.

(18)

On the otherhand, (13) showsthat ni

+ nt


w, z > 1. If we had w). the lefthand side of (20) would be positive, h ? (n2 + n1 w2)/(ni Hence impossible. whichis -

-

h
n1, it follows 1 w), (mod + 1 ni By (15), h = = In the lattercase, (19) becomes + w. 1 + h or + nH 1 ni that h (1 + nit)(1 + w) < h(1 + ni) = (tw + 1)(1 + ni), and thennit + w < wt + ni, ni(t - 1) < w(t - 1). Since ni > w, we must have t = 1. Then h - 1 = w, and since h = 1 + ni + w, we obtain ni = 0, a contradiction.It followsthat we must have h = 1 + ni . This yieldsthe result (4J). Let 6i be a groupof evenorderg whichcontainsa real element THEOREM Let J be O different from1 and withdistanceat least4 fromthesetof involutions. G intoG-1. If thegroup9(G) is notnormalin 5, whichtransforms any involution thenits orderh is at mostn(J) + 1. The case h = n(J) + 1 occursonlyif g9 in (4G). h(h - 1)(h - w), wherew is theorderof thegroupQ mentioned many groupsin whichthe case h = n(J) + 1 occurs. There exist infinitely If 6 is the groupLF(2, 2a) of orderg = (2a + 1)2a(2a - 1), thereexists only one class of involutions,r = 1, n = ni = 2a, and thereexistsa subgroup& of 2a + 1. Here, w = 2, t = (h - 1)/2 = 2a1. the type herediscussedwithh 19. We concludethis sectionwith a fewsimpleremarks: (4K). If 9P is a setconsistingof moinvolutionsof 65,any subgroupV of order into from1 whichis transformed 1 > g/(mo+ 1) containsan elementLo different its inversebysomeelementJ of 9Vo. PROOF.If V containsan elementJ of 9), we may take Lo = J and we have J-1LoJ = L-1. Assumethen that V and 9o are disjoint.

ON GROUPS

OF EVEN

ORDER

579

If two sets Li~o and L29I0, withLi , L2 e V,L1 - L2, are not disjoint,there existinvolutionsJ1, J2 e 9o such that L1Ji = L2J . Then L71L2 = J1J2and the elementLo = LIl' L2 E 3 into its inverse. ,J1transforms If the 1sets LaoM,L e V,are pairwisedisjoint,theirunioncontainslmodistinct elements.Since no elementof V appears, we have lrn0< g - 1 and hence 1 < g,/(mO+ 1).

As a special case, we note (4L). If 65 containsm involutions,any subgroupe of order1 > g/(m + 1) containsreal elements different from1. In particular,if n = g/m,any subgroupof order1 > n containsreal elementsdifferent from1. The following exampleshowsthatthisresultcannotbe improvedsubstantially. If 65 = LF(2, q) and q is a primepower with q _ -1 (mod 4), the subgroups of orderq do not contain real elementsdifferent from1. On the otherhand, m = q(q - 1)/2 and g/(m + 1) < q + 1. As anotherconsequenceof (4K), we note that ifthe group65in (4G) contains r > 2 classes ofinvolutions.Q , gr and if J -t, then 9

h 0. Since k-1 f2 =0 -::9, Li~o

it followsthat 1 + (k1 - I)f2 < g. Now (2J) showsthat m(m,+ 1)f2< g(g - 1). This yields thereexists (5A). If 65 is a groupof evenorderg whichcontainsm involutions, a real character, notthe1-character, of a degreef suchthat

f ? -\Ig(g- 1)/(m2+ M). In particular,if n = g/m,thenf < n. Thus ifn < 2, that is, if at least half of the elementsof (Mare involutions,65 has a real characterof degree 1 whichis not the 1-character.This impliesthat (Mhas a normalsubgroupof index 2. One can also show that the elementsin 65 of odd orderforma normalsubgroupof (5. Using the knowngroupsof degree 2, one can obtain then(Mhas (5B). If thegroup(5 ofevenorderg containsat leastg/3 involutions, a normalsubgroup65osuchthat6/6o eitheris cyclicoforder2 or 3 or is theicosahedralgroupoforder60. If (5A) is combinedwithJordan'sand Blichfeldt'sTheoremson lineargroups of given degrees,resultssimilarto (2H) can be obtained.However,our present knowledgein this matterdoes not enable us to improvethe resultsgivenin II. 21. We denoteby Xpithe value of the characterX, forthe class fi . It is well knownthat to everycharacterXpof(5 therecorrespondsa characterwpofdegree 1 of the centerA of the group algebra r. The values of wpforthe elementsof the basis Ko, , Kk-1 of A are givenby wop(Ki)- gXPi It followsfrom(0) that Wp(Ki)wp(Kj)=

Z.

aij,,wp(K,),

and hence

gnj n, xpiXpj= fp aijmxp1ng * -1 -1

-

If we multiplyboth membersof the last equation by pxfora fixedvalue of X, add over all p, and apply the orthogonalityrelationsfor group characters,we obtain the well knownformulae (21)

airsA=

gn7'n 7'ZEPxpixpRPXfP7-

ON GROUPS

OF EVEN

581

ORDER

As above, we denote the classes containinginvolutionsby Qll, Q12X. It followsfrom(1), (0), and (2) that

cx= Ad >-

(22)

aix.

If we take into account that Xpiis real fori

(23)

CA=

gZ j=, ni

=

1, 2,

,

r, we obtain

n-j-1 EP xpiXpjXpfp-

We now show (5C). Suppose thatG is a real elementof thegroup65 of evenorderg for which G intoG-1 (cf. 4A). If p whichtransforms n(G) is odd and thatJ is an involution is a primewhichdividesn(G) withtheexactexponentv butwhichdoes notdivide n(J), then65possessesa p-blockB of defectd ? v. We may chooseB suchthatit for2. containscharacters of positivedefect all involutionstransforming G into r1' that follows from (4A) PROOF. It lie in one class, say in M,. If G belongssay to Rx, then (2A) and (4A) show that is a divisornx/sofnx,,wheres is the orderof91(G) n 91(J). cx = alixand that con Then, fori, j = 1, (21) becomes 2 xxEPX21PXfp-. nolfi=

gs

P

Since p and n1are relativelyprime,theremust exist a value of p such that -1 2 x1 2 Kx gnx XPLPxfp7 =Xplwp(Kx)

is not divisibleby a primeideal divisorp of p in the fieldof characters.Then (24)

Xpl 9 0 (mod p),

wp(Kx) 9 0 (mod 0).

The second conditionimplies that xp belongs to a p-block of defectd ? v. Indeed, if Xpbelongsto a p-blockB of defectd, thereexistsa characterx, in B withg/(pdf,) primeto p. Now wp(Kx)- w(K\) (modp). Since wp(Kx) = gxpx/(nxfp)

is primeto p, nxmustbe divisibleby pd, that is, v > d. The firstcondition(24) impliesXpi # 0, and hence Xp must be of positivedefectfor2. (5D). If, in (5C), n(G) is primeto p, thenthereexistsa character COROLLARY for2. of 65whichis ofdefect0 forp and of positivedefect 22. There is a second case in whichwe can prove that 65 possessescharacters of defect0. and thatforsomeodd primep thereexists (5E). Suppose thatJ is an involution a primepowergroup$o oforderpc > 1 suchthatno elementof $o different from1 is mappedon its inverseby any conjugateof J. If p dividesn(J) withtheexact characterxp whosedegreeis divisibleby exponentv, thenthereexistsan irreducible for 2. In particular,if 93ocan be takenas a pC-^V and whichis of positivedefect p-Sylowgroupof 1 and if v = 0, thenxpis ofdefect0 forp. PROOF. If J belongs to RI, then the proof of (2A) shows that under our

582

RICHARD

AND

BRAUER

K. A. FOWLER

assumptionsala- 0 foreach class Rx with X > 0 whichcontainselementsof . For P e $o , set 13o (25)

A(P)

=

>3PXP(J)2XP(P)fP

It followsfrom(21) that 4/'(P)= 0 foreveryelementP # 1 in $ . For P = 1, we see fromthe orthogonality relationsthat AV(1)= Ep xp(J)2 = n(J).

of 93o,it followsthat If sodenotesthe characterof the regularrepresentation

'P

(26)

n(J)/pcp.

On the otherhand, we can expressthe restriction of X. to 9Soin termsof the irreduciblecharactersof o13,in orderto obtain an expressionof 4l(P) in terms ofthe irreduciblecharactersof 93o. If the restriction of xpcontainsthe 1-character of $3 with the multiplicityb., then comparisonof the multiplicityof this 1-characterin (25) and (26) yields Exp(J)2bp/fp = n(J)/pC

Since the righthand side containsp withthe exponentc - v in the denominator,and since all x,(J) and bpare rationalintegers,theremust exist a value p such that pCv dividesfpand xp(J) - 0. This yieldsthe statement. An immediateconsequenceof (5E) is and letp be an odd primedividingg. If no element (5F). Let J be an involution intoits inverseby J, and if p dividesn(J) uith theexact of orderp is transformed character exponentv and g withtheexactexponenta, thenthereexistsan irreducible whosedegreeis divisiblebyp a-p and whichis ofpositivedefect for2. 23. We conclude the paper with some remarksconcerningthe case that 3 containsa subgroup?& whichsatisfiesthe assumptionsof (4F). It is immaterial here whetherthe orderg is even or odd, but we mentionthese remarkshere since they can be applied in the case of (4G). If the notationis the same as in (4F), thereexist t charactersG1, * , a of degreew of the groupS. Then (M ,'t all of the same degreez such that possessest irreduciblecharacters'P,

'Pj(H) = y + 6Qj(H) forH e &, H $ 1. Here, -yand 5 are independentofj and H, -yis a rational integer,and a = 11. For elementsG of (Mwhichare not conjugateto such an elementH, we have '1(G)

=

'2(G)

..

= 'P6(G).

If xj are the othercharacters,not of this "exceptional" kind,then xj(H) has a fixedrationalintegralvalue aj forall H e a, H $ 1. The degreefj ofXj satisfies the congruence (mod h), a,

ON GROUPS

OF EVEN

583

ORDER

whilethe degreez of the 4,j satisfies (mod h).

z _ Py+ bw

These resultsare a special case of a more generalresultwhichwas obtained originallyas an application of a theoremon characters[1]. A direct simple proofusing induced characterswas given by M. Suzuki. Using the orthogonalityrelations for group characters,one obtains the followingadditionalrelations: (t _ 1)'y2+ (Qy

2

)2+

Za

= w+

l,

teyz+ Efjaj

-bz.

HARVARD UNIVERSITY UNIVERSITY OF ARIZONA REFERENCES

[1] R.

A characterization of groupsoffiniteorder,Ann. of Math., of thecharacters 57 (1953),pp. 357-377. [2] W. BURNSIDE, Theory of Groupsof Finite Order,Cambridge,1897. Leipzig,1937. [3] H. ZASSENHAUS, Lehrbuchder Gruppentheorie, BRAUER,

~3RAU~R, 1~. Math. Zeitsehr. Bd. 63, S. 406---444 (1956)

Zur DarsteUungstheorie der Gruppen endlicher Ordnung Dem Andenken meines verehrten Lehrers ISSAI ScHtm gewidmet Von

RICHARD BRAUER w 1. Einleitung In seinen Vorlesungen fiber Gruppentheorie stellte I. Schur es gelegentlich als Wfinschenswert dar, die Darstellungen der Gruppen endlicher Ordnung yon einem mehr zahlentheoretischen Standpunkt aus zu untersuchen. Die folgenden Ausffihrungen shad ein Versuch, ein solches Programm in Angrfff zu nehmen. Es sei @ eine Gruppe endlicher Ordnung g, und / ' = F ( @ , /2) sei die zugeh6rige Gruppenalgebra fiber einem gegebenen K6rper/2. Das allgemeine Element v o n / ' hat dann die Form

~=Za~G,

(t.t)

GE~

wo die Koeffizienten a~ E1emente aus/2 sin& W~hlen wir/2 als algebraisehen Zahlk6rper endliehen Grades, und ist o der Ring der ganzen algebraisehen Zahlen in/2, so bfldet die Menge der Elemente (t.1) mit Koeffizienten a o aus D eine Ordnung J in/,1). Man kann dann die Zahlentheorie in J untersueh'en. Dabei ist zu beaehten, dab J" ffir g > t nieht Maximalordnung ist, so dab die Ergebnisse yon Speiser, Artin und ttasse nieht ohne weiteres angewendet werden k6nnen. Zwar ist /~ halbeinfaeh, und daher kann man J in eine Maximalordnung einbetten, doch gehen dabei gerade die yon unserem Standpunkt interessantesten arithmetischen Eigenschaften verloren. Als erste Frage haben wir die Untersuchung der Primideale ~3 yon J. 3). Der Durchschnitt yon ~ mit o ist ein Primideal p yon o, und dann ist J ) ~ _ ~ J . Die ~ enthaltenden Ideale yon J entsprechen eineindeutig den Idealen des Restklassenringes J* = J/PJi Die Restklassenabbildung (mod ~J) bildet den Teilring o yon J auf einen K6rper/2* ab, der init o/~0identifiziert werden darf. Man sieht dann, dab man J* als die Gruppenalgebra F(@, Q*) yon @ fiber/2* auffassen kann; (t .2)

J*

----

F(~,/2*);

/2* = o/~,.

1) Einige Begriffe und Resultate der Algebrentheorie, die im folgenden verwendet. werden, sind irn A n h a n g (w 14) zusammengestellt. ~) Vgl. w 14, (a) und (c). Un• einem Ideal wird stets ein zzveiseitiges Ideal verstanden. Ferner wird yon einem Ideal einer Ordnung J vorausgesetzt, dal3 es yon Null verschiedene Elemente yon 0 enthiilt.

Zur Darstellungstheorie der Gruppen endhcher Ordnung

407

Ein p enthaLtendes Ideal ~ von J ist dan n und mtr dann Primideal von J, wenn sein Bild ~* in J* Primideat von J* ist. Aus der Strukturtheorie der Algebren folgt, dab dies dann und'mlr dann der Fall ist, welm ~* maximal in J* ist, d.h. wenn J*/~3* eme einfache Algebra ist. Es seien ~1, t~ . . . . . ~z die wesentlich verschiedenen irreduziblen Darstellungen yon ~ durch lineare Transformationen mit Koeffizienten aus g2*. Jedes ~i kann zu einer Darstellung v o n / ' ( ~ , Q*) erweitert werden. Ist ~* der Kern yon ~ in diesem Sinn, so sieht man leicht, dab ~*, ~* . . . . . ~ gerade die verschiedenen Primideale yon/~(@, ~*) sind. Das ~* entsprechende Ideal ~i yon J besteht aus denjenigen Elementen 0~ von I, ffir deren RestMasse ~* (rood ~J) die Gleichung ~i(e*) = 0 gilt. Die so erhaltenen Ideale ~ , ~z . . . . . ~ sind dann die stimtlichen Primidealteiter yon p J in J. Es ist also die Untersuchung der zu einem festen Primideal 1~yon ~ geh6rigen Primideale ~ von J tiquivalent mit der Untersuchung der irreduziblen Darstellungen yon @ in f2*. Mit derartigen ,,modularen" Darstellungen von endlichen Gruppen haben sich C. J. Nesbitt und der Verfasser ill einigen friiheren Arbeiten befagt~). In der vorliegenden Arbeit treten die modularen Gruppencharaktere nur als Hilfsmittel auf. Alles, was hier benStigt ist, ist mit kurzen Beweisskizzen in w3 zusammengestellt. Will man die Struktur des Ideals ~ J von J weiter verfolgen, so liegt es nahe, die Restklassenalgebra J* -=F(| Q*) als direkte Summe yon direkt unzerlegbaren Idealen 91" darzustellen;

(t.3)

- 91, 9 91" r

| 91t.

Setzt man dann ~ *,,,=

Y,

i~l, 2,..., t

und ist ~ (t.4)

9 1,",

% *. =

n

i=l, 2, ..., t

das ~B* entsprechende Ideal yon J, soist p l = ~1~ ~B~r~... r~~ .

Die Ideale ~i Und ~i sind fiir i ~/" teilerfremd; (~), ~Bj)= J ; keins der Ideale ~Bi ltiBt sich nicht-trivial als Durehschnitt teilerfremder Ideale darstelIen. Die Darstellungen (1.3) und (t.4) sind eindeutig bestimmt. Wir nennen ~B1, ~B2. . . . . ~Bt die Blockideale yon p J i n J . Da ~BI~ ... ~Bt~PJ ist, refit jeder Primidealteiler ~ yon pJ'genau eins der Blockideale N,. Ist also B, die Menge der Primidealtefler yon ~B~, so sind die l Primideale ~3~, ~32 . . . . . ~ auf t ,,Bl6cke" B 1, B~, .:., B~ verteilt. Nach dem oben Gesagten k6nnen wir dann auch yon einer u der l modularen DarsteUungen ~1, ~z . . . . . ~z auf die t B16cke sprechen. W i e in w4 genauer auseinandergesetzt werden wird, fiihrt dies auch zu einer Verteilung der gew6hnlichen irreduziblen Darstellungen yon I~ anf die t B16cke. 3) Man vgl. R. BRAUER and C. N~SBITT, University of Toronto Studies, Math. Series No. 4, 1937; Annals of Math. (2) 42, 556, P a t t i .

408

RICHARD BRAUER :

Diese Bl6cke yon DarsteUungen sgllen in der vorliegenden Arbeit untersucht werden4). Kennt man die Charaktere yon @, so kann m a n dara us weitgehende Information fiber die BIScke und die zugeh6rigen Blockideale ablesen. Dies wird in w4 und w5 ausgeffihrt. Ist p die durch p teflbare rationale Primzahl, und enth~lt g die Primzahl p genan in der Poten z ~ba,' so verstehen :wir lmter dem De/ekteines Blocks B die kleinste ganze Zahl d,~ ffir die pa-~ die Grade aller Darstellungen in B teflt (w6). Das Hauptziel dieser Arbeit ist es, Zusammenh~ge zwischen gruppentheoretischen Eigenschaften Yon @ und Eigenschaften der BlScke aufzudecken. Daraus ergeben sich dann .neue Eigenschaften der Charaktere yon @. ~)iese Aufgabe wird ill w7 in Angriff genommen. Auf Grund einer einfachen gruppentheoretischen Bemerkung ergibt sich ein Zusammenhang zwischen den B16cken gewisser Untergruppeh von i@ und denB15cken van @. Wir k6nnen dann in w8 jedem Block B von@ yore Defekt d e~e Untergruppe ~ yon @ yon der Ordnung pd, seine De/ektgruppe, zuordnen. Mit ~ ist auch jede konjugierte Gruppe Defektgruppe yon B; davon abgesehen ist ~ eindeuti~ dutch B bestimmt. Ist ~ eine beliebige Untergruppe yon @, deren Ordnung eine Potenz #~ yon p ist und ist ~ der N0rmalisator yon ~ in @, so gibt es eine eineindeutige Zuordnung zwischen den B15cken B yon ~ vom Defekt d und den B16cken B yon ~ mit der Defektgruppe ~ (wt0). Die Behandlung yon ~ l~13t sich dann welter auf Untersuchung yon 9~]~ zurtickftthren. Ffir die Werte der Charaktere Xi in B erh~lt man Kongraenzen nach Potenzen yon p, vgl. (t2A). Interessiert man sich z.B. ffir die Bestimmung der Anzahl der B16cke B, von einem gegebenen Defekt d, so sieht man jetzt, dab man diese Frage im Fall eines positiven d vollst~ndig auf die Untersuchung yon Gruppen yon ldeinerer Ordnung als g zurtickftihren kann. Im Fall d-----0 versagen diese Methoden, da bier 9~=@, ~>= {1} ist. Als Ersatzleiten wirim wt3 ein andersartiges Resultat ffir die Anzahl der B16cke vom Defekt 0 her. Die Untersuchung soll in einer weiteren Mitteilung fortgesetzt werden, w~hrend Anwendungen an anderer SteUe behandelt werden sollen. w2. Bezeichnungen Im folgenden ist @ stets eine endliche Gruppe der Ordnung g. Ist ~/eine beliebige Teilmenge yon @, so bezeichnen wit mit 3(9/) den Normalisator yon 9/, d.h. die Untergruppe von @, deren Elemente X die Gleichung X ~ = 9/X erfiiUen. Ferner sei ~ (~) der Zen~ralisator yon ~, das ist die Untergruppe von @, deren Elemente mit jedem Element yon ~ vertauschbar sin& Besteht 9/aus einem einzigen Element A, so schreiben wir,~(A) (=~(A)) ftir ~R(9/). Die Ordnung yon ~R(A) werde mit n(A) bezeichnet. 4) 13bet einen Tell der Ergebnisse berichten ohneBeweis drei,Noten, Proc. Nat. Acad. Sci. ]O, 109-~-114 (1944); 32, t82--186, 2t5--2t9 (t946).

Zur DarsteUungstheorie der Gruppen endlicher Ordnung

409

Es seien im folgenden ~1, ~ , ..., ~ stets die Klassen konjugierter Elemente yon @. Ist also G~E ~ , so besteht ~ aus gin (G~) Elementen. Die Anzahl k der Klassen ~ ist dann a u c h die Anzahl der irreduziblen Charaktere yon @, die mit Z1, •2 . . . . . Z~ bezeichnet werden sollen. Es sei. x~ der Grad yon X~; xi=x~(1). Ist 3 ein beliebiger K6rper, so kann man die Gruppenalgebra/'(@, 3 ) yon @ fiber 3 bilden. Dies i st eine assoziative Algebra der Dimension g fiber 3, deren g Basiselemente so gew~hlt werden k6nnen, dab sie bei Multiplikation eine zu (~ isomorphe Gruppe bilden. Man kann. dann die Basiselemente mit d e n Elementen yon @ identifizieren, und dann ist das allgemeine Element y o n / ' ( ( ~ , 3 ) v o n d e r F0rrn (t.t) mit K o e f f i z i e n t e n a a E 3 . Offenbar ist das Einheitselement yon @ auch Einheitselement von /'((~, 3). Mit K~ werde stets das Element yon T'((~, 3 ) bezeichnet, das die Summe der Elemente der Klasse ~ , ist,

(2.t)

K ~ = v C. GE~

Bekanntlich bilden die k Elemente K1, K~ . . . . . K~ eine Basis des Zentrums Z(@, 3) der Gruppenalgebra. Wegen dieses Zusammenhangs nennen wir Z(~3, 3 ) die Klassenalgebra von ~ fiber 3. D i e Multiplikation der Basiselemente ist durch Formeln (2.2)

K ~ K a = Z a~a~Kr

gegeben. Hier is t a~a v eine nicht-negative ganze rationale Zahl, die v o n d e r Wahl des K6rpers unabh~ngig ist. Ist Gv ein Element von ~ , so ist n~mlich a~a ~ die Anzahl der geordneten Paare (X, Y) von @, die die Bedingungen XY

= G~,

XE~,,

YE~ a

erfiillen. Hat ~ Pnmzahlcharakterlstlk p, so kann man natfirlieh a~,or (rood p) nehmen. Ist ~ der K6rper der komplexen Zahlen, und ist ~i die zum irreduziblen Charakter Zi yon ~5 geh6rige Darstellung yon @, so kann man ~i als irreduzible Darstellung von/'(~5, 3~ auffassen. Ist ~' ein beliebiges Element yon Z(@, 3), so ist ~(~) nach dem Sehurschen Lemma ein skalares Vielfaches a~ (~)E der Identit~t E; o~(~) E 3 . Als Funktion yon ~ stellt dann ~oi einen linearen Charakter der Klassenalgebra Z(@, 3) dar: Natfirlieh ist ~0~ vollst~ndig dutch die Werte o~ (K,) ffir ~ = 1, 2 . . . . . k bestimmt. Diese kann man in bekannter Weise durch Bereehnung der Spur yon ~ ( K , ) bestimmen. Ist G~ ein Element von ~.,, so wird es oft bequem sein, auch o~.(G,) flir ~o~(K~) zu sehreiben; man hat dann (2.3)

,oi(K,) ---- ~oi(Ga) --

g zi(G~) n (G~,) x i '"

(x i = Xi(t)).

Da 9r Z~. . . . . ~ orthogonal und daher linear unabh~ingig sind, sind die k linearen Charaktere o~, ~o~. . . . . co~ von Z((~, 3 ) voneinander verschieden. Da Mathematische Zeitschrift. Bd. 63

28

410

RICHARD BRAUER :

Z(@, ~) die Dimension k hati sind dies s~imtliche irreduziblen DarsteUungen der KlaSsenalgebra. Man sieht auch jetzt noch leicht ein, dab Z(@, ~) halbeinfach ist. Jedes x~(G~) ist eine Summe von g-ten Einheitswurzeln. Ersetzt man also dell K6rper der komplexen Zahlen ~ durCh einen die g-ten Einheitswurze!n enthaltenden Teilk6rper $2, so zeigt (2.3), dab alle ~o~(K~) in Q hegen. Daher besitzt auch Z(@,/2) genau k verschiedene lineare Charaktere, die wir ebenfalls mit co1, o~2, ..,, cok bezeichnen. Im folgenden ist/2 stets ein die g-ten Einheitswurzeln enthaltender algebraischer Zahlk6rper endlichen Grades. Wie in w1 sei o der Ring der ganzen atgebraischen Zahlen in $2, und essei ~ ein festes Primideal yon o und p die durch ~0 teilbare rationale Primzahl. Die Restklasse eines Elements ~ von o (mod p) wird stets mit x* bezeichnet werden; es sei (2.4)

$2* =

~/~

der Restklassenk6rper. Ferner sei v stets die zu ~ geh6rige exponentielle Bewertung yon D, die wir so normieren, dab (2.5)

r (p) = t

ist. Gelegentlich w i r d die Bezeichnung 0~ fiir den Ring der fiir p ganzen Zahlen von $2 und 9. ftir das Piimideal yon o. verwendet werden. Der Restklassenk6rper 0,/9~ darf mit $2* identifiziert werden. Ist G ein Element yon | so schreiben wit zur Abktirzung v (G) ftir v (n (G)) und nennen diese Zahl den De/ekt yon G. Die p-Sylowgruppen des Normalisators 92(G) haben dann also die Ordnung p./c). Geh6rt G~ zur Klasse ~ , so setzen wir noch (2.6)

v ( ~ ) = v(G~) =v(n(G~)),

(GaE~e);

wir nennen v (~) den De/ekt der Klasse ~ . Ein Element G yon ~ heiBe p-regulgr, wenn die Ordnung yon G zu p teilerfremd ist. Da das gleiche dann auch fiir alle konjugierten Elemente gilt, so k6nnen wir yon p-reguliiren Klassen konjugiertey Elemente spreehen. w 3. Eigenschaften yon modularen Charakteren

Wir geben in diesem Abschnitt eine Schilderung der Theorie der modularen Gruppendarstellungen, soweit sie im folgenden ben6tigt wird. (3 A) Es sei 1" eine Algebra yon endlicher Dimension i~ber einem algebraisch abgeschlossenen Kb'rper ~ yon Primzahleharakteristik p. Es sei S der yon den Elementen o~fl--fo~ mit c~, fl E l " au[gespannte lineare Teilraum yon l'," es sei T die Menge der Elemente ~ von _F, /i~r die es eine Potenz q yon p gibt, derari daft 0q zu S geh6rt. Dann ist T ein linearer Teilraum von l'. Ferner ist die Anzahl der wesentlich verschiedenen irreduziblen Darstellungen yon l':gleich der Dimension des Restklassenraumes F/T.

Zur Darstellungstheorie der Gruppen endlicher Ordnung

411

Beweis, Es seien 0 und a zwei Elemente v o n / ' ; man bilde (0 + a) p durch Ausmultiplizieren der Klammern. Zu jedem von ~P und a p verschiedenen Glied ~geh6ren p Glieder, die aus i b m durch zykllsche Permutation der Faktoren entstehen. Offenbar sind diese p Glieder untereinander kongruent (rood S). Daher ergibt sich

(3.t)

(e + a) p ~ ~P + a~ ( m o d S).

Silid ~, fl Elemente yon F u n d wendung von (3A), dab

setzt man y=fl(o~fl) p-l, so folgt unter Ver-

(~fl " f l ~ ) P - - (~fl)~ -- (fl~)~ = ~7 - - ? ~ -------0

(mod S)

ist. Jetzt sieht man, dab die p-te Potenz jedes Elementes yon S wieder zu S geh6rt. Unter Verwendung von (3A) folgt nun leicht, dab die i n (3A) deftnierte Menge T wirklich ein linearer Tefiraum yon / ' ist, und ferner dab T__) S ist. Ist F einfach, also eine volle Matrizenalgebra fiber •, so besteht S, wie man leicht sieht, gerade aus den Matrizen von der Spur Null. Ist ~/ein Idempotent mit yon Null verschiedener Spur, so geh6rt ~/ nicht zu S und auch nicht zu T. Da nun 1~/S die Dimension 1 hat u n d / ' > T ~ S ist, so folgt hier, daB T / T die Dimension t hat. Im allgemeinen Fall folgt aus der Nilpotenz des Radikals N v o n / ' , daB N zu T geh6rt, t3esitzt nun / ' genau l wesentlfch verschiedene irreduzible Darstellungen, so ist I?/N direkte Summe von l Idealen F/, die einfache Algebren sind. Hat T/ die entsprechende Bedeutung ftir F/ wie T fiir _P, so folgt jetzt leicht, dab die Dimension yon I ' / T gleich der Summe der Dimensionen der F//Ti ist. Nach dem oben Gesagten ist aber jede der letzteren Dimensionen t, und daher hat I ' / T die Dimension l wie behauptet war. (3B) Die Anzahl der wesentlich verschiedenen, irreduziblen Darstellungen einer endlichen Gruppe g3 in einem algebraisch abgeschlossenen K6rper E yon Primzahlcharakteristik p ist gleich der Anzah! der p-reguldren Klassen kon]ugierter Elemente in | Beweis. Wir nehmen/'----/'(gfi, 3 ) und verwenden dieselben Bezeichnungen wie im Beweis von (3 A). Ist G ein beliebiges Element von @, so k6nnen wir G = R P setzen, wo R und P Potenzen von G sind und w o R p-regul/~r ist, w/ihrend die Ordnung von P eine Potenz q yon p ist. Aus (3A) mit g = R P , a = - R ergibt sich (R P -- R)# -----R# P# -- R p (roodS).

Diese Beziehung erhebe man wieder in die p-~ Potenz und beachte, dab wie oben bemerkt, die p-te Potenz eines Elementes yon S wieder zu S geh6rt. F/ihrt man so fort, so findet man schlieBlich (R P -- R)q ~ Rq P q -- R q = R q -- R q = 0

(mod S).

Dies heiBt aber, dab R P -- R E T ist, also G ~ R (mod T). 28*

412

RICHARD BRAUER :

Jetzt folgt, dab (rood T) das allgemeine Element (t.t) yon ri@, ~ ) einem Element derselben Form kongruent ist, in dean nur die p-regul~ren Elemente yon ~ roll Null verschiedene Koeffizienten haben. Sind ferner G und G1 zwei konjugierte Elemente yon ~, so ist G ~ G1 (rood S) und also auch (rood T). Daher kann man setzen (3.2)

~z ~ ~, aRR

(mod T),

R

wo R ein Vertretersystem ftir die p-regul/iren Klassen durchl~uft. Diese Elemente R bilden ein (rood T) linear unabh~ngiges System yon Elementen v o n / ' . Ist n/imlieh 0r in (3.2) ein Element yon T, so gibt es eine Potenz q von p, derart dab 0r E S ist. Da die'Elemente R p-reguNr sind, kann man q So w/ihlen, dab fiir jedes R die Gleiehung R q ~ R gilt. Erhebt man nun 0r in (3.2) in die q-te Poten z, so erg.ibt sich mit Hilfe yon wieclerholter Anwendung yon (3.1), dab ~, a~ R ~ 0 (mod S) R

ist. Nun sieht man abet leicht, dab S aus den Elementen (t.t) besteht, fiir die (mod p),fiir jede Klasse Ri die S u m m e der Koeffizienten der Elemente G in Ri verschwindet. Da alle R in (3.2) zu verschiedenen K1assen geh6ren, sind (rood p) alle a~ und daher alle a R in (3.2) Null, wie zu zeigen war. Es bilden also die Elemente R in (3.2) eine Basis f/Jr ! ' (mod T). Daher ist die Dimension l yon F I T gleich der Anzahl der p-regul/iren Klassen Ri, und nun liefert (3 A) die Behauptung. Wir beweisen weiter den ffir uns wiehtigen Hilfssatz (3 C) Sind ~ , ~2 . . . . . ~l die l p-reguliiren Klassen yon ~, und ist Gj ein Repriisentant /r ~j, so kann man l (gew6hnliche) irreduzible Charaktere Z1, Z~ . . . . . Z, yon ~ so ausw~hlen, daft (3.3)

Det (z,(Gi)) @ 0

(modp)

ist, wobei in der Determinante der Zeilenindex i und der Spaltenindex f die Werte t, 2 . . . . . l durchlau/en. Beweis. W~ire (3 C) nicht richtig, so k6nnte man l ganze Zahlen u 1, u 2. . . . . u, in f2 so w/ihlen, dab ffir jeden irreduziblen Charakter Zi die K0ngruenz gilt l

Y u, z~(~,) - o

(rood p)

und etwa %, ~ 0 (rood p) ist. Ist 6) eine beliebige Linearverbindung der Zi mit ganzen Koeffizienten in s so ist dann l

(3-4)

Z ui 6) (Gi) ~ 0 i=1

(mod p).

Es sei nun N = {Gin} die yon G,~ erzeugte zyklische Gruppe, b sei die Ordnung und ~1, ~2. . . . . ~b seien die b irreduziblen Charaktere von ~. Dann w/ihle

Zur Darstellungstheorie der Gruppe n endlicher Ordnung

418

man eine p-Sylowgruppe ~ yon ~(Gm) und setze ~ = { ~ , ~ } = ~ X ~ , Ist B Q mit B 6 ~, Q 6 ~ ein variables Element yon ~, so ist ein Charakter vqi v0n ~ dnrch Oi (B Q) = ~i(B) definiert. Es sei (9i der von z$i induzierte Charakter yon @. Setzt man b

O ='Y, ~i (Gin)O~, i=1

so ist O eine Linearverbindung yon Charakteren yon | mit ganzen Koeffizienten aus /2. Aus der Definition der induzierten Charaktere ergibt sich b

(e},)

XE~/=1

wo ~gi(XGiX-1) gleich Null zu setzen ist, falls XGjX -1 nicht zu ~ geh6rt. Ist XGjX-16 @und i < i ~ l , so ist XGiX-1 p-regul~ir und also ein Element B yon ~. D a n n ist zgi(XGjX-X) =el(B). Aus den Orthogonalit~itsrelationen ftir die Charaktere yon ~3 ergibt sich jetzt, dab der Wert der inneren Sumrne in der letzten Formel 0 oder b ist, je nachdem ob B#G,,, oder B=G,,, ist. Das letztere ist nur ffir G i = G,~, d.h. ffir ~'--m m6glich. Jetzt folgt leicht, dab O ( G / ) = 0 ffir ? ' # m u n d O(G,~):b(~(G,~):~) ist. Wegen der Art, in der wit ~ konstruierten, ist dieser Wert O (G,~) eine zu p teilerfremde ganze Zahl. Setzen wir dies in (3.4) ein, so erhalten wir einen Widerspruch, da u ~ 0 (rood P) war. Damit ist (3 C) bewiesen. Da der K6rper s nach Voraussetzung die g-ten Einheitswurzeln enth~ilt, kann man die irreduzible Darstellung ~ yon @ mit dem Charakter Zi i n / 2 schreiben, i = t, 2 . . . . . kS). E b e n s o k a n n man die Koeffizienten der l absolut irreduziblen modularen Darstellungen ~1, ~z . . . . . ~z v o n @ im Restklassenk6rper Q* w&hlen. Ist 0, der Ring der flit P ganzen Zahlen in/2, so darf man ferner annehmen, dab ~i(G) ffir jedes G6(~ Koeffizienten in o~ hat~). Die Restklassenabbildung ffihrt dann ~i in eine Darstellung Y* m i t Koeffizienten in 0,/p,=/2* fiber. T r i t t die irreduzible modulare Darstellung ~ j in ~i mit der Vielfachheit dii auf, so nennen wir die n icht-negativen ganzen rationalen Zahlen dij die Zerlegungszahlen yon (~; i = t, 2 . . . . . k;/" = ~, 2 . . . . . l. Wir definieren jetzt den Charakter 9 einer modularen Darstellung ~ mit Koeffizienten in/2*. Dabei beschr~nken wir uns auf p-regul~re Elemente G yon q6. Hat G die Ordnung h, so sind die charakteristischen Wu§ e*, e* von ~(G) s~mtlich h-te Einheitswurzeln. Da h zu p teilerfremd ist, geht s) Vgl. etwa R. BRAUER, Amer. J. Math. 69, 709 (t947). Ersetzt man gegebenfalls .(2 durch einen Erweiterungsk6rper endlichen Grades, .so kann m a n die Verwendung dieser Tatsache vermeiden. s) Ygl. w 14, (d) und (1413). Dort wird a u c h gezeigt, dab die im folgenden definierten Zahlen dii ungeAndert bleiben, wenn man ~i dutch eine ~hnliche Darstellung mit Koeffizienten in 0v ersetzt.

414

I~CHA~DBRKUER:

jedes e* aus einer eindeutig bestimmten h-ten Einheitswurzel ei ill/2 dutch Restklassenabbildung heIvor. D a n n setzen wir 9(G)~. ~.e~. Es ist also 9 eine ffir p,regul/ire Elemente G E @ definierte Funktion mit Werten in /2. Natfirlich h/tngt ~0(G) nur von der Klasse konjugierter Elemente ~ ab, zu der G geh6rt. (3D) Haben die irreduziblen modularen Darstdlungen ~1, ~2 . . . . . ~z die Charaktere 91, 9~ . . . . . 9l, sind dq die Zertegungszahlen, so ist fi& p-regul~re Elemente G l

(3.5)

x~(G) = Y, d~j ~j(c). i=x

In der Tat stimmt ftir p-regul~ires G der Weft des Charakters von 2" mit dem Wert xi(G) des Charakters Zi von ~ fiberein, und da ~i genau d,i-mal in ~?* auftfitt, so erh~ilt man sofort (3.5). Setzt man die Werte (3.5) in (3.3) ein, so erhiilt man (3E) Die aus den Werten der l irreduziblen modularen Charaktere q~i [i~r die Repriisentanten G1, G,, .,., Gz der 1 p-regul~ren Klassen yon @ gebildete Determlnante ist nicht dutch p teilbar ; (3.6)

Det (r

g~ 0

(m6dp).

Sind [erner ~1, Z* . . . . . Zl wie in (3 C) gewiihlt, so ist (3.7)

Det ( d q ) . 0

(modp);

(i,] = 1, 2 . . . . . l).

Wir setzen noch k

(3.8)

c . = Y, do~d~i= c.,

(r

= t, 2 ..... 0;

0=1

(3.9)

O, = X di,zi,

(i = 1, 2 . . . . . l).

i=l

Die clt sind nicht-negative ganze rationale Zahlen, die sog. Cartanschen Invarianten yon @ (oder eigentlich von /'(@, /2*)). Es ertibrigt sich hier, ihre algebrentheoretische B e d e u t u n g zu diskutieren. Die #i w gew6hnliche Charaktere von @, die in der modularen Theorie eine besondere Rolle spielen, worauf wir ebenfalls nicht n/iher einzugehen brauchen. Aus (3.9) und (3.5) ergibt sich unter Verwendung yon (3.8) die Beziehung I

(310)

#i(G) = ~ cq~oi(G) i=1

(G p-regullir).

Die Orthogonalit/itsbeziehungen ftir die Charaktere yon @ besagen, dab ~ ~. zI(G) z~(H) = n (G) oder

= 0,

je nachdem ob die Elemente G und H yon @ konjugiert sind oder nicht. W~ahlt man ffir H eins der p-regul/iren Elemente G1, G, . . . . . Gz, sagen wit

gur DarsteUungstheorie der Gruppen endlicher Ordnung

415

H=G~, und drtickt man x~(G~) nach (3.5) durch 9j(G=).aus, so fo!gt mit Hilfe yon (3.9), dab 1

~. #i(G) 9i(G~) = n(G) oder i=z

(3At)

-----0

ist, ie nachdem ob G und G~ konjugiert sind oder nicht. Ftir p-singul~res G tritt der zweite Fall ein. Aus dem Nichtverschwinden der Determinante .(3.6)" ergibt sich dann, dab # i ( G ) = 0 sein mug. Fiir p-reguliires G folgt ebens 0 ans (3.6), dab #i(G) durch die h6chste Potenz yon p teilbar ist, die in n(G) aufgeht. Wegen ~bl.(G)= #i(G-1 ), n(G)=n(G -1) haben wit (3F) Die in (3.9) de/inierten Charaktere q~j verschwinden fiir alle p-singul~ren E l e ~ i i r p-regulgres~G !St v ( q~j(G)) ~ v (G). Ist 6 p-regul~ir, So kann man #i(G) in (3At) nach (3A0) durch %(G) ausdrticken. Schreibt man fiir G~ wieder H, so erh~ilt mall (3 G) Fiir p-regulgre Elemente G und H yon @ gilt l

(3.t2)

l

~.#i(G) q~i(H) = ~.ciig~(G) gi(H)=n(G) /,~=l

I"=I

oder

0,

~e nachdem ob G und H kon~ugiert sind oder nicht. w 4. B16cke y o n C h a r a k t e r e n

Wit sehicken unserer Untersuchung einige Bemerkungen tiber die Klassenalgebra Z = Z ( ~ , Q ) voraus. Wie bereits in w2 bemerkt wurde, besitzt Z die k versehiedenen linearen Charaktere col, co2. . . . . co~ und da Z die Dimension k hat, sind dies die s~mtlichen irreduziblen Darstellungen yon Z. Offenbar ist der Durehsehnitt J ~ Z : J0 eine Ordnung yon Z, die die o-Basis K 1, K s . . . . . K k besitzt; der Restklassenring fo/PJo daft mit Z * : Z((~,Q*) identifiziert werden. Anwendung der Restklassenabbildung auf o~1, o~, ..., ~ liefert dann k lineare Charaktere o~, co'2, .-., co~ yon Z*, die aber im aUgemeinen nicht alle verschieden sind. Jeder Homomorphismus W der Algebra Z* fiber Q* auf einen Erweiterungsk6rper von Q* ist ein linearer Charakter von Z* und stimmt mit einem der o~+ * tiberein ~). Wie in (1.3) betrachten wit die Zerlegung der modularen Gruppenalgebra /'((~, Q*) in direkt unzerlegbare Ideale (4.t)

F(@, aQ*) =. ~* +9~* G . . . O9~*.

Setzt man dementsprechend (4.2)

t = *7* + ~7" + " " + .7",

('7" E 2I*),

so sind die *7* orthogonale Idempotente des Zentrums Z ~ = Z ( ~ , $ 2 *) yon /'(~, ~*). Das Ideal *7*Z* yon Z* ist direkt unzerlegbar und man hat

z* = ~* z* | n* z* | ~) Vgl. w14; (14A) und (i4C).

| n* z*.

416

mC~A~O BRAUER :

Als kommutative direkt unzerlegbare Algebra ist ~* Z* primer. Ist also ~* das Radikal yon ~ Z*, so ist ~* Z*/~, ein K6rper, der als Erweitenmgsk6rper /2* von/2* betrachtet werden kann. Man erweitere den natfirlichen Homomorphismus von ~/**Z* auf/2** zu einem Homomorphismus y~, yon Z* auf/2**, indem man ~ , ( ~ ) = 0 fiir ~ E ~* Z* mit a ~ z vorschreibt. Nach dem oben Gesagten erh~lt man so einen linearen Charakter ~v, von Z*; jeder lineare Charakter yon Z* stimmt mit einem dieser ~0, fiberein. Man hat ~0,(~?**)=t. Wie in wl sei ~8.* die Summe der 91" mit a4= v. Da die Ideale !8.* von /'((~,Q*) die Bilder der Ideale !3, yon F(@,/'2) in (t.4) sind, so haben wit das Resultat (4A) Hat das Ideal p ] in J die Blockideale ~1, ~3s. . . . . ~3~ und biidet der nat~rliche Homomorphismus yon J au[ J / p J = F ( @ , [2*) das Ideal ~ au[ das Ideal ~3" yon I'(@,/2*) ab, so gibt es lineare Charaktere ~ yon Z(| ~Q*), derart daft ~.(~*) = 1 ist; z = 1, 2, ..., t. Die linearen Charaktere ~ , ~)~. . . . . ~ sind die s~mtlichen linearen Charaktere der Algebra Z (@, T2*). . Ist wie immer p ein festes Primideal von Q, so ordnen wir jedem Blockideal ~3. von p ] einen,,Block" B. zu. Darunter werden wit eine gewisse Menge yon gew6hnlichen und modularen irreduziblen Darstellungen von verstehen. An Stelle von Darstellungen werden wir auch von den zu B. geh6rigen gew6hn~ichen und modularen Charakteren von ~ sprechen. Den linearen Charakter ~. yon Z in (4A) nennen wir ~en dem Block B. zugeordneten linearen Charakter der modularen Klassenatgebra Z* =Z(@,/2*), Wit rechnen eine gew6hnliche irreduzible Darstellung ~ von@ und ihren Charakter Z~ zum Block B., wenn fiir den Z~nach (2.3) entsprechenden linearen Charakter o~ von Z die Gleichung gilt (4.3)

co* = ~v,.

Aus dem oben Gesagten ergibt sich (4B) Jeder irreduzible Charakter Z~ yon ~ geh6rt zu einem und nur einem Block B, und ieder Block enth~lt Charaktere Zi. Zwei Charaktere Z~ und Zi geh6ren dann und nur dann zum selben Block, wenn [i~r alle G C ~3 die Kongruenz gilt (4.4) oJi(G) =---coi(G) (modp). Wir z~ihlen eine modulare irreduzible Darstellung ~i und ihren Charakter ~j zum Block B,, wenI~ der Kern ~* yon ~i das Ideal ~8" enthiilt. (4C) Jede irreduzible modulate D"arstellung ~j geh6rt zu einem und nur eine~n Block B, und ieder Block B, enthiilt derartige ~j. Geh6rt die gew6hnliche irreduzible Darstellung ~ zu B,, so gehb'ren alle in ~* als Bestandteil au[tretenden ~i zu B~. Beweis. Der erste Tell yon (4 C) ergibt sich unmittelbar aus dem in w1 Gesagten. Geh6rt ~i zu B,, so gilt (4.3). Wegen ~,(n*) = r ist ~o*~(-,,/,j----t.*' Ais Element yon Z* hat ~* die Form ~ b~ K~ mit b~C/2". W~hlt man b,

Zur Darstellungstheorie der Gruppen endlicher Ordnung

417

in o derart dab die Restklassenabbildung von 0 auf/2* gerade ba in b~ fiberffihrt und setzt man ~ , = ~ . baKe, so ist ~, ein Element yon J o = Z • J , das durch die Restklassenabbildung vdn J gerade in 7" fibergeht. Dann ' ist ~i(~,) =coi(~,) E, also Ei*(~**)=c~ t*/**'E*J = E * . Tritt nun US als Bestandteil in ~* auf, so ist ~j(~*)=E*. Also ist ~*~ ~* u'iad daher ist ~ , ffit aq=* nicht in ~3j enthalten. Dann muB ~3,(=~j sein, und daher gehSrt ~i zum Block B,, wie zu zeigen war. Aus (4C) ersieht man, dab die Zerlegungszahl dii verschwinden muB, wenn ~ und ~j zu verschiedenen B16cken geh6ren. Daraus folgt unter Verwendung von (3.8), dab cii= 0 ist, wenn ~ und ~i zu verschiedenen B16cken geh6ren. Dies Resultat litBt sich folgendermaBen aussprechen (4D) Bildet man aus den Zerlegungszahlen d~i und den Cartanschen Invarianten cij Matrizen D und C, so sind bei geeigneter Anordnung der Indizes diese Matrizen direkte Summen yon den den einzelnen Blacken B~ entsprechenden Teilmatrizen D, bzw. C,. Besteht B, aus k, gew6hnlichen Darstellungei1 ~ und aus l, modularen Darstellungen ~j, so hat D, gerade kT Zeilen und t~ Spalten, Die Matrix C, ist quadratisch Vom Grade l,. Aus (3.8) ergibt sich (4.5)

C, = D',D,,

wo D~ die Transponierte yon D~ ist. Beschr~nkt man sich ausschlieBlich auf die Betrachtung der gew6hnlichen Darstellungen, so h~ngt die Verteilung auf B16cke aUein v o n d e r rationalen Primzahl p und nicht von dem Prknidealteiler p von p ab. Beim Beweis dieser Behauptung dart man annehmen, daB/2 normal fiber dem K6rper Q der rationalen Zahlen ist. Jeder Primidealteiler p' yon p in/2 geht aus dem festen Primideal p dutch Anwendung eines Elements T der Galoisschen Gruppe von/2 fiber Q hervor. Ist nun e eine primitive g-te Einheitswurzel, so ffihrt T dann e in eine Potenz es mit einem zu g teileffremden Exponenten s fiber. Da ;r eine Summe von g-ten Einheitswurzeln ist, wird xi(G) in xi(G s) transformiert, also Oi(G) in oi(G~). Wendet man T auf (4.4) an und ersetzt G* dutch G, so sieht man, dab (.4.4) ffir p' an Stelle yon p gilt. Daraus ergibt sich die Behauptung. Kennt mall die gew6hnlichen irreduziblen Charaktere X~ v0n @," so kann man ffir jedes Primideal sofort die Anzahl t der B15cke B, bestimmen, d:h. also feststellen, Wieviel Blockideale ~, in der Zerlegung (t.4)yon p J a.uf:, treten. Die Anzahl der in ~, aufgehenden Primideale ~i yon f i s t di e Anzahl l , der modularen DarsteUungen {~. in B,. Sind die modularen Charaktere 9,i bekanr~t, so kann man leicht l, bestimmen. Die unten beim Beweis yon (5A) v~rWendete Methode zeigt, dab man l, auch finden kami wenn man nur die X~ kennt. Da unter unseren Voraussetzungen d er Restklassenfing" J/?~] ein voller Matrizenring fiber/2* ist, ist seine Struktur dutch seine Gradzahl/i bestimmt. E s ist'aber b der Grad der zu ~j gehSrigen modularen Darstellung ~i, als~ Da sich die Cartanschen Invarianten c~)aus

418

RICHARD BRAUER.

den modularen Charakteren berechnen lassenS), so erh~lt man auch gewisse Information beziiglich des Restklassenrings J]~3~. Kennt man die modularen Charaktere 9i, so kann man auch ein System yon Kongruenzen fiir die Koeffizient~n a G des Elements ~ in (t.t) aufstellen, die die notwendigen und hinreichenden Bedingungen darsteUen, d a b ~ zu dem Primideal ~j geh6rt. Es sei 9*(G) die Spur yon ~i(G) fiir G E @. Setzt man wie gelegentlich in w3 G = R P, wo R und P Potenzen yon G:sind, und wo R p-regulAr ist, wAhrend ctie Ordnung yon P eine Potenz von p ist, so sieht man leicht, dab 9* (G) -----9* (R) ist, und dab 9*(R) aus dem Werte 9~(R)des Charakters 9~ yon ~i fiir das p-regulAre Element R dutch Restklassenabbildung hervorgeht. Dann gilt (4E) Das zur irreduziblen modularen Dar~tellung ~i yon @ gehb'rige Primideal ~ yon f besteht aus denienigen Elementen ~=~.aGG yon J, aGE o,/iir die die G!eichungen (4.6)

Y, a8 9~*(G/-/) = 0

]i~r alle H E ~ gelten. Hier bezeichnet a8 das aus a~E o dutch Restklassenabbild~ng ( m o d p) erhaltene Element yon ~* = o/p. Beweis. Man sieht leicht, dab die Elemente ~ E J, fiir die Gln. (4.6) gelten, ein Ideal ~ yon J bilden; ~ =( ~ ( J . Daher ist entweder J = ~R oder f = ~,. WAre J = ~ , so miiBte die Spur yon 3i(~*) ftir alle ~* E J* verschwinden, was nicht der Fall ist. Daher ist ~R= ~i, wie behauptet war. Der Beweis gilt auch dann noch, wenn Q ein beliebiger algebraischer Zahlk6rper ist, also nicht notwendig die g-ten Einheitswurzeln enth~ilt. In der Tat ist dann Q* ein endlicher K6rper, also vollkommen, und daher enthalt J* Elemente ~*, denen in der irreduziblen Darsfellung ~i yon J* lineare Transformationen ~i(~*) mit yon Null verschiedener Spur entsprechen. w5. Zuordnung yon p-regul/iren Klassen zu den Bl6cken

EnthAlt der Block BT wie zuvor lT modulate Charaktere und i s t l die Anzahl der p-regul~ren Klassen yon @, so ist nach (3 B)

(5.t,)

l=l, +l~+

... + l , .

Wir wollen jetzt die l p~regul~ren Klassen so den B16cken B, zuordnen, dab dem Block B, gerade l~ Klassen entsprechen, z = t, 2, ..., t. Dabei w~klen wit die Bezeichnungen so, dab dieselben Indizes j bei den ~0i E BT und bei den B T zugeordneten Klassen auftreten. Genauer wollen wir zeigen s) S i e h t m a n die Z a h l e n n (G) tiir p - r e g u l g r e s G als b e k a n n t a n , so s i e h t m a n dies a u s (3A2) u n d (3.6).. D e u t l i e h e r n o c h e r g i b t sich diese B e h a u p t u n g a u s d e n FormeL,x ~iig=F, gi(R) ~i(R), wo R iiber alle p-regulAren E I e m e n t e li~uft u n d wo (y/f) die zu R C~-(cii ) i n v e r s e M a t r i x C -x bezeichnet. Fiir d e n Beweis der F o r m e l n sei a u f die in a) z i t i e r t e n A r b e i t e n verwiesen.

Zur Darstellungstheorie der Gruppen endlicher Ordnung

419

(5 A) Es sei S t dig Menge der l TIndizes j, fi~r die % zu B, geh6rt, z = t, 2 . . . . ,, I, Man kann die gew6hnlichen Charaktere Xi yon @ und die Klassen ~ so numerieren, daft alle Klassen ~ ;nit i E S~ p-reguiSr sind, daft die Charaktere Xi mit i E St zu B T geh6ren und daft fiir jedes z die/olgenden Beziehungen gelten (5.2) (5.3) (5.4)

Det (x,(Gj)) • 0 Det (q~i(Gi)) • 0 Det (de]) , 0

(modp), (modp), (modp),

j (i, jE ST),

wobei in allen drd Determinanten der Zeilenindex i und der Spaltenindex j iiber die IT Werte in S , lau[en. Beweis. Nach (3 C) kann man l Charaktere Xl so auswiihlen, dab (5.5)

Al = Det (x,(Gi)) ~_ 0

(modp)

ist, wobei hier i und j die Werte t, 2 . . . . . l durchlanfen und G~, G 2. . . . . G1 Vertreter ffir die l p-reguliiren Klassen ~1, ff~. . . . . ffl sin& Entwickelt man L1 nach dem Laplaceschen Satz, so kann man zl in der Form schreiben

A = ~ ::i::/ICG, I/1r. 2 .../1~o, 1 Q

wobei A~ ) eine Unterdeterminante ist, in der die zu BT geh6rigen unter den Charakteren Xi auftreten. Es gebe etwa m, derartige Charaktere. Ist m , = 0, so ist A~Q}= t zu setzen. Da in A gerade I Charaktere X~ anftreten, hat man

(5.6)

l .= m 1 + m2 + " " + m,.

Wegen (5.5) kann man ein Glied "-x' ACQ)ACQ)... -'~. A~e) so answ~hlen, d a b (5.7)

A(Q)/,,fl,~) A~~) ~ 0 -a~ .,. z.a I

(mod p)

ist. Driickt man die in A(,a) auftretenden Charaktere nach (3.5) dutch modulate Charaktere 9/aus, so treten nut modulate Charaktere ~oiEB T auf, vgl. (4D). WAre nun der Grad m, yon z]~) gr6Ber als die Anzahl 1T der modularen Charaktere in B,, so wiirde A~) verschwinden, was im Widerspruch zu (5.7) steht. Daher ist m T ~ l T ftir jedes T. Vergleich yon (5.t) mad (5.6) zeigt jetzt, dab m T-- l, ist, ,-----t, 2 . . . . , t. 9 )fmdert man jetzt die. Numerierung der Charaktere Z~ und der Klassen ~j so ab, dab ftir jedes , in der Determinante A~)"gerade die lT Charaktere ~ mit i E S, und die lT Klassen ~i mit j E S, auftreten, so folgt (5.2) ans (5.7). Drtickt man wie oben die Xl in dieser Determinante dutch die ~0i aus, s o sieht man, dab die Determinante in (5.2) das Produkt der Determinanten in (5-3) und (5.4) ist. Daher sind auch diese beiden Determinanten nicht durch p tefibar. Da die dq ganze rationale Zahlen sind, kann man in (5.4) den Modul p dutch p ersetzen, und damit ist (5 A) bewiesen. Die Bedingungen von (5A) k6nnen im aUgemeinen auf mehrere Weisen erftillt werden.

4~0

RICHARDBRAUER:

Nach (3A2). ist

, c=a ,,(o=) ,;(ca) = n (c.) wobei 6 = a = 0 oder I zu setzen ist, je nachdem ob a=~fl oder a = f l ist; a, f l = t, 2 . . . . . l. ist ~ die Matrix (9I(Gi)), C die Matrix (c,;) und N die Matrix (n(Gi)8~i), (i, ~ = t, 2 . . . . , l), so kann man die Gleichung folgendermal3en in Matrizenform schreiben (5.8)

~' C ~ = N .

Arbeitet man im Ring o, der fiir p ganzen Zahlen yon/2, so ist die Deterrninante yon 9 nach (3.6) eine Einheit. Daher zeigt (5.8), dab die Elementartefler yon C die in n (Gx), n (G~). . . . . n (Gt) auftretenden Potenzen einer lokalen Primzahl yon o, sin& Statt dessen kann man dann auch die Potenzen (5.9)

p,{o0, p, Io,). . . . . p, Io,I

nehmen. Da C ganze rationale Koeffizienten hat, sind dies auch die Elementarteller yon C im Ring I~ der fiir p ganzen, rational'en Zahlen. Tats~chlich ist digDeterminante yon C eine Potenz yon pg), und daher sind die Zahlen (5.9) die Elementarteiler yon C im Ring der ganzen rationalen Zahlen. Es wird aber im folgenden geniigen, im Ring Ip zu operieren. Wit zeigen (5 B ) D i e Elementarteiler der zum Block B~ geh&igen Teilmatrix C, yon C sind die.Zahlen p,IG~) mit i E S~.

Hat C, die Elementarteiler ex, e2, ..., e~, so zeigen wir zun~chst, dab bei geeigneter Anordhung der i in S, die Ungteichungen ei ~ #~ Icjl gelten. Wit k6nnen die ej als Potenze 0 yon p w~hlen und setzen dementsprechend ej = p'~. Ist H , eine Diagonalmatrix, die in der Hauptdiagonale die Elementarteiler ei enthiflt, so gibt es zwei in I~ unimodulare Matrizen U = (u~i) und V = (v~i), (i, i E S~), vom Grade l, derart, dab Beweis.

U C V-X= H, ist. Nach (3A0) und (4D) haben wit ffir /~-regul~res G

= Zc,

&).

i 6 s,:

Man setze (5. t 2)

~, (C) = 2; ui; ~,' (C) ; iEs t

~ (C) = g vii ~ (C) ;

(r ~ S,).

9 iES t

Wegen (5,iO) nimmt dann (5.1t) di~e Form an ~j(a) = ~; ~i(C) = p',~;(a).

(iC &).

g) Vgl. R. BRAUER,Ann. of Math. 42, 53~61 (194t) oder Ann. of Math. $7,'357~377 (t953), Satz 13.

Zur Darstellungstheorie der Gruppenendlicher Ordnung

421

Jetzt zeigt (3 F), dab flit a lle p-reguliiren G und fiir alle i E S, die Unglei-. chungen gelten (5.t 3)

(G)) >

- *r

Aus (5.12) erh~It man

I)et (~,(~;)) = Det Iv, j) Det (9,(Cj)) i

(i, i ~ ST)

Da V unimodular ist~ ist Det (vii) eine Eiltheit von I#. Aus (5.3) folgt daher,d a b die Determinante auf der linken Seite eine Einheit von o, ist. Nimmt man also die lT Elemente Gi mit/" E ST in geeigneter Anordnung, so' dam man annehmen, d a b

ist. Dann ist v(~i(Gi))=O, mid jetzt zeigt (5.t3), dab ,i>~(Gi) ist, ~ e wir es zeigen, wollten. Diese Ungle.ichungen gelten fti~ jeden Block B,. Da C die direkte Summe der Matrizen CT ist, ist die Menge der Elementarteiler yon C die Vereinigungsmenge der Mengen tier Elementarteiler tier C,. W(irde auch nur in einem einzigen Fall in ei21,(Gf) das Ungleichheitszeichen gelten, so k6nnten wit nicht in (5.9) die Elementarteiler yon C haben. Also gilt stets ei=v(Gi) und damit i~ (SB) vollstiindig bewiesen. Unsere Methoae liefert auch noch den Hilfssatz (5C) Ist p" der gr6flte Elementarteiler yon C,, so gel~,n f~r ~ q~E B,, aUe XiE B, und aIle ~-regulaven Elemente G yon @ die Beziekungen ,,

>

-

m;

,, ( z , ( G ) )

>,,(c)

-

m.

In der Tat folgt aus (5A3), dab v(~i(G))~ v(G)--m ist, i E S , . Da V unimodular war, kann man nach den zweiten Gln. (5A2) q~i als Linearverbind u n g der ~i mit Koeffizienten in o, darsteUen. Daraus folgt die erste B ehauptung (5.t4). Die zweite Behauptung folgt aus der ersten unter Beachtung von (3.5) und (4D). w 6. Der Defekt eines Blockes Es sei jetzt ib~ die hSchste in g aufgehende Potenz der Primzahl p, also in unseren Bezeichnungen a = v (g). Wit definieren den Defekt d eines Blockes BT als die kleinste ganze rationMe Zahl d, ftir die die Grade x~ aller Charaktere ~,E BT durch p~-a teilbar sind. Man kann dann also x,=pa-ax~ mit ganzem rationalem x'~ setzen; es gibt Zi E B,, fiir die x~ zu p teilerfremd ist. Wir haben bei der Definition des Defekts die gew6hnlichen Charaktere yon B T verwendet. Wir h/itten ebensogut die modularen Charaktere nehmen k6nnen. In der Tat gilt (6A) Hat B, den De]ekt d, so sind die Grade ]i aller modularen irreduziblen Charaktere q~ in B T s~mtlich dutch p~,a, abet nicht alle durch p~,a+i teilbar. Beweis. Man bestimme die ganze rationale Zahl ~ so, dab die Grade ]~ der 9iEBT alle durch p~-~, aber nicht alle durch p "-~+i teilbar sind. Aus

422

RICHARDBR,~UER:

9(3.5)ergibt sich ffir G = t unter Berficksichtigung yon (4D), dab p~-~ ein Teller yon p,-d ist. Andererseits folgt aus (5.4) und (3.5), dab ffir p-regul~re Elemente die 9i C B~ als Linearverbindungen von l~ geeigneten Charakteren Zi aus B~ mit rationa~en, ffir p ganzen, K0effizienten darstellen kann. Ftir G = t zeigt dies, dab p~-d ein Teiler von pa-~ ist. Daher mul3 d-----6 sein, wie behauptet war. Es seien jetzt Determinanten A~ I, ~ACQI,..-, A~~i wie in (5.7) ausgew~hlt. Verwenden wit ,dieselben Bezeichnungen wie in w 5, so haben wir also (6.t)

A~~ = Det (~i(Cj)) • 0

(mod p),

wo i und ]" die l, Werte in S, durchlaufen. Man w~thle eine feste Spalte, etwa die r-te Spalte und multipliziere die Elemente dieser Spalte mit gin (G~). Nach (2.3) ist g Z~(Gr)/n(G,)=xioJi(G,). Da x~ durch pa-d teilbar ist und ~oi(G,) ganz algebraisch ist; so ist g A~~ durch p=-d teilbar. ~Aus (6..1) folgt dann (6.2)

v (G,) ~ d.

Da die Xl mit i E S, alle zum Block B~ geh6ren, so sind nach (4.4) die l, Zahlen oJi(G,) ftir die l, Werte i E~S, aUe untereinander (mod p) kongruent. Daher kann das Gleichheitszeichen in (6.2) h6chstens, dann gelten, wenn o~i(G,) ~ 0 (mod p) ftir i C S~ ist. Nehmen war nun an, dab ffir alle r E S~ das Ungleichheitszeichen in (6.2) gilt. Aus (5 B) folgt dann, dab alle Elementartefler von C kleiner als p~ sind. Dann ist m ~ d - - t in (5.14) und ftir G = I ergibt sich, dab ffir die Grade aller Zi E B, die Ungleichung v (x,) > a - - d gilt. Dies widerspricht der Definition des Defekts d. Wir k6nnen also r E S, so w~ihlen, dab (6.3) v(G,) = d , e~,.(G,)~-0 (modp) ist. W~ihlt man, was nach Definition yon d m6glich ist, den Charakter Zi E B, zun&chst so, dab v ( x i ) = a - - d ist, so hat man (,o,(ol)

-

( gz, IGI I =

+ ,,(z,(o))

-

,,(c) -

d --

Insbesondere ist o~(G)-------0(m0d p), falls v ( G ) < d ist. Wegen (4.4) gilt dies ffir alle i E S,. Sagen wir wie in w2, dab ein Element. G den Defekt Vo hat, wenn v ( G ) = % ist, so liefert (6.3) das folgende Resultat (6B) Ein irreduzibler Charakter Zi yon | geh6rt dann und nur dann zu einem Block yore De/ekt d, wenn /i~r alle Elr G pon@ yon einem De/ekt kleiner als d die Kongruenz wi(G ) =--0 (rood p) gilt, wghrend es p-reguliire -Elemente G yore De/ekt d gibt, /iir die a~i(G)~O (mod p) ist. (Wie immer ist w~ der zu Zi geh6rige Charakter der Klassenalgebra Z(@, Q).) Es sei jetzt B, ein Block, ffir den l ~ 2 ist und es seien r und s zwei verschiedene'zu S, geh6rige Indizes. Multipliziert man dann in (6A) die r-te Spalte mit gin(G,), die s-te Spalte mit gin(G,) und drfickt man in beiden

Zur Darstellungstheorie der Gruppen endlicher Ordnung

423

Spalten wieder die Xi nach (2.3) durch die wi aus, so erhält inan eine Formel _g_ _g__ LJ ... = J ... , n(G,) n(G:,)

X;W;(G,), ... , X;Wi(G.), ...

J,

wo auf der rechten Seite die i-te Zeile der Determinante angegeben ist, und wo in den nicht hingeschriebenen Spalten ganze algebraische Koeffizienten, nämlich Werte X;(G;) stehen. Da alle XiEB-r Grade xi haben, die durch pa-d teilbar sind, so kann man aus der r-ten und der s-ten Spalte je den Faktor herausziehen. Ist dies geschehen, und wendet man auf die so entstehende Determinante die Restklassenabbildung (mod \:)) an, so werden nach (4.4) zwei Spalten proportional; man erhält also 0. Nach (6.1) hat man also a-v(G,) +a-v(G.) > 2(a-d), d.h.

pa-d

(6.4)

v (G,)

+ v (G,) < 2d.

Ist G, wie in (6.3) gewählt, so hat man v(G,) (% (A)) +

(t) >

(no(W)) +

FaBt man all dies zusammen, so ergibt sich, dab ftir die Summe 27' der zu einem gegebenen W geh6rigen Glieder O(W') die Beziehung grit

v(27~) >

v(G)+

v ( w ) - - v'(no(W))

-59 (0(A))

> v(a)

+

v(t).

Da 272 eine Summe yon Teilsummen 27~ ist, ist

v (27,) >

(G) +

Wir k6nnen nun 02.2) in der Form schreiben

•i(G) ~

71Txi(wO(G)

--

272)

(modp.P~(vl-~Cg/")) 9

Jetzt sieht man, dab man hier das Glied 273 weglassen dad. Wit haben damit d a s folgende Resultat b e w i e s e n 0 2 A ) Es sei ~ eine Untergruppe yon @, deren Ordnung eine Potenz ~a der Primzahl p ist; es sei ~: = ~ (~)), ~ = ~ (~)2 M a n bestimme ein volles System yon Charakteren 01, O~. . . . yon ~,7~/~) yore De#kt O, derartig daft Oi in ~)/~) eine Tr~gheitsgruppe ~d~) hat,/i~r die der Index ( ~ d ~ ) zu p teiler/remd ist. I n diesem S y s t e m / i n d e man ein vollst~ndiges Teilsystem ~9"" 1.

J

I:~!.:., c; vanishes for

all

6. Congruences for the degrees of the characters of @

If y; is any character of ~( T), it follows from the orthogonality relations that y;(l)

+ (h/2)1/;(B)

+(h/2)1/;(BH) + Lw,, show that 2 This yieldsf

+

I:w,,.1 (ci(ff)

= 2r (mod 2h). fi

1

= 0.

Similarly, for j = 1, 2

+ öih + Öf I:w,,.1 1 = 0

Here, I:H,,,.1 l = h - 1, and hence f; last 2 congruences (28) is analogous. (11.H)

+ e;(ff))

= Öj

(mod 2h). (mod 2h).

The proof of the

We have

+ 02 x2(X)

= rxw (X),

for elements X of@ which are not conjugate to element H' -c;c 1.

In particular,

(29) This follows from (ao - a1)x(X) = 0, (as+l - a1)x(X) = 0 in conjunction with the values of the coefficients of the columns ao - U1 and Us+1 - a1 obtained above (x 1 was the unit character, ö1 = 1).

7. The dass relation Let ~1 , ~2 , • • • , ~k denote the classes of conjugate elements where we choose that notation such that 1 t ~1, T t ~2, Hit ~2+J for j = 1, 2, · · · , s (cf. (11.A)), and where ~1, ~2, • • · , ~t (t ~ 2 + s) are the classes consisting of the "real" elements G of @, i.e., the elements G which are conjugate to their reciprocals a-1• W e work in the group algebra r of @ over the field of rational numbers or rather in the center Z of r. lf Ki denotes the sum of the elements in ~i, then K1, K2, · · · , Kk form a basis of Z. In particular, we have an equation (30)

90

FINITE GROUPS

ONE-DIMENSIONAL UNIMODULAR PROJECTIVE GROUPS

731

Here, ci denotes the number of ordered pairs (X, Y) of elements of &' 2 such that XY is equal to a fixed element Gi e Sri . Since X, Y have order 2, the equation XY = Gi implies a-;1 = YX = y- 1Gi X. Conversely, if X has order 2, and if XGi x-1 = a-;1, then for Y = XGi , we have XY = Gi , and Y2 = XGi XGi = 0-;1Gi = 1. Thus Y has order 2, except when Y = l, Gi = X. This latter case arises only if Gi has order 2, i.e., if j = 2. Thus (11.I) If Gi is a representative of Sri, then for f ~ 2 the number Cj in (30) denotes the number of elements X of order 2 which satisfy the equation

x- 1aj x = a-;1. For j = 2, Cj is one less than the number of X of order 2 which satisfy the corresponding equation. If j = 1, c1 is simply the number of elements of order 2, i.e., the number of elements of &'2. By (11.A), c1 = g/2h. For j = 2, we may take Gi = T, and c2 1 is the number of elements of order 2 which commute with T. lt follows from (11.A) that c2 1 = h l, c2 = h. For 3 ;;;; j ;;;; 2 s, we may choose Gi = Hi-2 • Since

+

+

X.-1Gj X = G;1

+

implies

+

XE in(IHj-ZI) = CJ,(T),

( cf. (1.C) ), the number Cj denotes the number of elements of order 2 of the dihedral group CJ,(T) which transform Hi- 2 into its reciprocal (Hi-z)-1 • Hence Cj = h for j = 3, · · · , 2 s. If 2 s < .i ~ t, the elements Gi and G11 are conjugate in @. Hence there exist exactly c(Gi) elements X in @ such that x- 1Gi X = a-;1. Then X 2 commutes with Gi . If the order of X 2 contained a prime factor of r+ 1v = 2h, by (11.A), X 2 would belong to one of the classes .R'1 , &'2, · · · , .R'2+s . Moreover, for X 2 ~ 1, CJ,(X2) would consist only of elements which lie in the same s 2 classes. This is impossible for j > s 2 since Gi e CJ,(X2 ). Hence X 2 = 1. Thus we have exactly c(Gi) elements X of order 2 for which x- 1Gi X = G-;1 and Cj = c(Gi) for 2 s < j ;;;; t. Finally, for j > t, the elements Gi and G-;1 are not conjugate, and Cj = 0. If we count the number of elements of @ appearing as summands on both sides of (30), we have

+

+

+

+

+

(g/ c( T) )2 = Substituting the values of

Cj

L~=l ci(g/c( Gi )) .

just found, this yields s+2

2

4gh 2 = 2gh

t

+ 2gh h + ~ _hg h + J=s+a .L 1=8

This yields g2/4h2 = g/2h

+ g/2 + g(t

whence (31)

91

REPRINT OF [70]

C

(Gg ·) c(G_;). J

- 2),

732

R. BRAUER, M. SUZUKI, AND G. E. WALL

8. The degrees of the irreducible characters of @

lt will be necessary to separate the cases r = l and r = -1. The case r = l. lt follows from (29) that Ö2 = 1 since rf = f > 0. As shown by (29), the character x2 must be real, since x:2 cannot be equal to any of the characters except x2. If !2 = 1, then @ would have a linear character x2 ~ x1 , x: = x1 . Since the multiplicative group of linear characters is isomorphic with @/@', it would follow that @/@' has even örder. This is impossible, if@ does not have anormal subgroup of index 2. Hence h ~ 1, and (28) shows that !2 ~ 2h + 1. Now (29) shows that f ~ 2h + 2. Interchanging xa and X4 if necessary, we see from (29) that we may assume ö3 = 1. By (28), Ja ~ h l, f4 ~ h Ö4, J; ~ 2h for J° ~ 5. Thus

+

(32)

f ~ 2h

+

2, !1 = 1, !2 ~ 2h

+

f4 ~ h

Ja

1,

+

+

~ h

+

l,

Ö4 , f; ~ 2h for J° ~ 5.

Since we have s = h/2 - 1 characters xli>, we shall have k - s - 4 = k - h/2 - 3 characters x; with J° ~ 5. Now, the order g is the sum of the squares of the degrees of the irreducible characters. This yields (33)

g ~ (h/2 -

l)(2h

+

2)2

+ 1+

(2h

+

1)2

+

(h

+

+

(h

+

Ö4)2

+

(k - h/2 - 3)4h2.

1)2

On account of (31), g can be written in the form g

=

4h2(t - 2)

+ 2h + 2h2.

Substituting this in ( 33), we have, after simplification, 4h2t ~ 4kh 2

(34)

+

2ö4 h - 2h.

If ö4 = 1, this yields t ~ k. Since k ~ t, we have k = t, and we must have the equality sign everywhere in ( 32). If ö4 = -1, we still can conclude k = t, and we see that the left side in ( 33) exceeds the 1ight side by 4h. Thus, we must have an inequality in (32) for some J°. If we write the inequality in (32) in the formf; > fj, then (28) shows that

li ~

1: + 2h.

Since f; ~ J'l2 + 4hfj + 4h2, we can add 4hfj + 4h2 on the right-hand side of (33) and (34). This leads to a contradiction. Thus, (II.J)

If r

= 1, we

have g

f =

+

Ja=

2h

=

4h2k - 6h2

2,

f4 = h

+

92

l,

J; =

+ 2h, ö2

= öa = Ö4 = 1,

!2 = 2h

+

2h

for j

FINITE GROUPS

~

1, 5.

733

ONE-DIMENSIONAL UNIMODULAR PROJECTIVE GROUPS

The Gase r = -1. Here 02 = -1 by ( 29) and, after interchanging and X4 if necessary, we may assume that o3 = -1. Then (29) reads

!2 -

xa

= f, W e separate the cases 04 = 1 and o4 = -1. Subcase 04 = +1. lt follows from (28) that f (29*)

1

!2 ~ 2h - 1. Also f; ~ 2h

Moreover f4 ~ h for j ~ 5. Instead of ( 33), we find here

(33*)

g ~ (h/2 -

+

1) (2h - 2)2

~ 2h - 2, and hence 1, and (29*) shows that f 3 ~ 3h - 1.

+ 1 + (2h +

(h

1)2

+

+ (3h +

1) 2

- 1)2

(k - h/2 - 3)4h2 •

Then ( 34) can be replaced by (34*) lt follows that we must have k = t, and that we must have equalities in all estimates f

= 2h - 2, f1 = 1, f2 = 2h - 1, fa = 3h - 1, Subcase 04

= -1. Here,

f ~ 2h - 2,

f2 ~ 2h - 1,

f3

~

h =

h

+

h - I, f4

~

h - 1, f;

I, f; = 2h for j

~

5.

~

~

5.

2h for j

Then g ~ (h/2 - 1)(2h - 2) 2

+ 1 + (2h -

1)2

+

+

(h -

(h -

1)2

1)2

+

(k - h/2 - 3)4h2•

This leads to (34**)

If one of the degrees f; has a value larger than the estimate ff used here, by (28), h ~ f7 + 2h, and we can add a term 4hf7 + 4h2 on the right-hand side of (34**). If j ~ 5, this is an additional term I2h2, which is impossible as t ~ k. If 1 ~ j ~ 4, it follows from (29) that we must have inequality for two values of j, and again, this gives a contradiction. Finally, if f > f*, we have an additional term (h/2 - 1)(4h(2h - 2) + 4h2 ). Again, we have a contradiction. lt follows that the degrees have the values used in the estimates and that the equality sign holds in (34**), that is, that t = k - 2. Thus (II.J*)

If r

= -1, then we can assume

f = 2h - 2,

!2 =

93

02

2h - 1,

REPRINT OF [70]

03

= -1,

J; =

2h

f or j

~

5.

734

R. BRAUER, M. SUZUKI, AND G. E. WALL

ff

/l4

= 1, then

Ja = 3h - 1,

f4 = h

Case (a)

Case (b)

fa

ff /l4 =

+

k = t,

1,

and

g = 4kh2

-

6h 2

+ 2h.

-1, then

= f4 = h -

1,

t = k - 2,

g

9. The order

(J

= 4kh2

-

14h2

+

2h.

lt follows easily from the basic properties of the group characters that the coefficients c; in ( 30) are given by the formula g

(35)

c;

k

= c(T)2 ~

xei,'(T)\ 1 is either conjugate to an element of H or to an element R. If we had P- 1QP = Q", Q' = Q±1 ; cf. (11.A), (11.N). But since

+

+ r,

c(P) = 2h

we cannot have P- 1QP = Q. If P- 1QP = Q- 1, P would have even order, which is equally impossible. Hence we have a contradiction. Thus, (11.P)

The p-Sylow group $ of order 2h

+r

= pn of ® is abelian.

Two elements of $ are conjugate in ® if and only if they are conjugate in in($). Since c(P) = 2h r for every P ,6- 1 in $ (P being of type ( S) ), we see that the number of conjugates of P belonging to $ is equal to (in($):$). But since we have two classes of elements S, it follows that

+

2(in($) :$) = pn - 1. Hence in($) has the order 1

2P

n(

n

P -

)

1 =

+

{(2h l)h ( 2h - 1 ) ( h - 1 )

for for

T

r

= 1, = - 1.

12. Proof of the mein theorem Since ® has a subgroup in($) of order (2h + l)h for r = 1 and of order ( 2h - 1) ( h - 1) for r = -1, it follows that ® has a representation 3 as a transitive group of permutations in 2(h 1) or 2h letters respectively. The case r = 1. After removing the unit character from the character of 3, we have a character of degree 2h 1 which no longer contains the unit character. lt follows from (11.J) that this character must be irreducible and equal to x2 . Hence 3 has the character x1 x2 . Since two distinct

+

+

+

98

FINITE GROUPS

739

ONE-DIMENSIONAL UNIMODULAR PROJECTIVE GROUPS

irreducible constituents appear, 8 is doubly transitive. x1(G)

+

The number

x2(G)

gives the number of symbols left fixed by ß(G). lt follows from (27) that this is 2 for G = Ir ;= l. For G = S, it is 1 by (37), and since x2(R)

=

-r

as remarked in connection with (38), it is O for R. Hence no ß(G) with Because of the double transitivity, the subgroup leaving two letters fixed has order g/(2h + 2)(2h + 1) = h, and the subgroup leaving one letter fixed has order h(2h + 1). The elements of order 2 leave two letters fixed. The case r = -1. Here, the character of 8 has the form x1 + x where x is a character of degree 2h - 1. lt follows from (II.J*) that x = x2 • Again, 8 is doubly transitive. lt follows here from (27) that the elements 8(H') for H' ;= 1 do not leave any letter fixed. The elements 3(R) leave two letters fixed, and the elements ß(S) leave one letter fixed. No element ß(G), G ;= l, then leaves three letters fixed. The subgroup for which ß(G) leaves two letters fixed has order G ;= l leaves three letters fixed.

(2h - l)(h - 1),

and the subgroup for which ß(G) leaves two letters fixed has order h - l. In both cases, 8 is faithful. Indeed, the degree is at least 3 and only 8(1) leaves three letters fixed. We now apply Zassenhaus' method; cf. [2]. We have a group of permutations of N + l letters, doubly transitive, such that only the identity leaves three letters fixed. The order of the group is ½(N

+ l)N(N

- 1),

N

= 2h

+



In the case r = -1, Zassenhaus' assumptions are not quite satisfied, since the subgroup leaving two letters fixed does not contain elements of order 2. However, the method still works. This yields the result: @,...., LF(2, 2h

III.

+

THE CASE

(r = ±1).

r)

B

1. Assumptions We assume here (I) @ is a finite group of type (S). (II) The 2-Sylow subgroup ~ of@ is abelian of type (2, 2, · · · , 2), of order 2a > 2. 6 (III) @ does not have a proper normal subgroup which includes ~. and @ ;= ~6

Fora= 2 assume also that ([(T) =

99

~

for TE~, T

~

REPRINT OF [70]

I; cf. (1).

740

R. BRAUER, M. SUZUKI, AND G. E. WALL

As shown in I, it follows from (1) and (II) that for a ;;;: 3, we have CE(T) =

~

for T E ~, T ;,!; l. The case a = 2, CE( T) ;,!; ~ for some T E ~, T ;,!; 1 has been treated in II. Hence we assume that if a = 2, we still have CE(T) =

(1)

~

for

T

E ~,

T ;,!; l.

2. The classes of involutions

(111.A)

All elements of order 2 of @ belang to the same class of conjugate

elements. Proof. Suppose that X and Y are two involutions which belong to different classes. Then by Lemma (3A) of [l], there exists an involution Z such that Z ECE(X), Z E _z_;_ = (2a/z;) + 1 = 2a + 1'

and the sum is at least equal to 1 ~2 l + 2a + 1 ~ Zj

cf. (10).

1 2a - 1

=

l

+

2a-s-l 2a + 1 -

8

1 2a - 1 ;

Thus, (12) yields

(2a - 2)22a :2: _g

( 13)

8

-

22

-

1 + 22a - s2a - 2a - 2a + (2a+l)(2a-1)

8

+ 1 - s2a -

(2a - 2)2 2a(2a + 1 )(2a - 1) ~ g(ia+l - 2a+l8 - 2a+l),

Combining this with (2), we find

On the other hand, by ( 11) and ( 10) (14)

103

REPRINT OF [70]

8

'

744

R. BRAUER, M. SUZUKI, AND G. E. WALL

whence 2s ~ 2a. lt follows that 2s = 2°; moreover, in (13) and (14) the equality sign must hold. This implies that g = (2° + 1)2"(2a - 1), that z1 = z2 = · · · = z, = 1, and, finally, that r = s. This yields the results (III.D)

@ has the order g

= (2" + 1 )2"(2" - 1).

(III.E) @ has exactly 2•- 1 - 1 degrees 2" + 1, and 2"-1 degrees 2" - 1, and one degree l . All other degrees are 2". lt remains to find the number of degrees 2". Combining ( 4) with the value of g, and the equation t = k, we have (k - 1)2" - 1 = 1(2 - 1)(2" 1), whence k - 1 = 2". Since we have 2" degrees 1, 2" + 1, 2" - 1, we have exactly one degree 2".

+

(III.E*)

There is exactly one degree 2"; k = 2"

+

1.

6. The main result Since@ has a subgroup 91 of order (2" - 1)2", that is, of index 2" + 1, it follows that @ has a transitive representation 3 by permutations of 2" + 1 objects. If the character of 3 is x1 + x, then x is a character of@ of degree 2° which no longer contains x 1 • Comparison 'with (III.E), (III.E*) shows that x =. Xk • Since Xk is irreducible, 3 is doubly transitive. If R is an element of@whose order is divisible by a prime factor p of 2" + 1, then all characters Xi of degree 2" + 1 vanish for R. Likewise, if S is an element of @ whose order is divisible by a prime factor p' of 2" - 1, then xi(S) = 0 for all xi of degree 2" - 1. Thus x;(R) x;(S) = 0 for 1 < j < k. N ow the orthogonality relations for group characters yield Xk(R)xk( S) + 1 = 0. Since Xk is the only irreducible character of its degree, its values are rational integers. lt follows that Xk(R) = ±1, Xk(S) = =Fl. Thus Xk(X) for X~ 1 is 0, +1, or -1, and the character of 3 for X ~ 1 has only the values 1, 2, 0. In particular, the representation 3 is faithful. Moreover, no ß(X) with X ~ 1 leaves three objects fixed. lt follows that 3 is triply transitive: The subgroup leaving one letter fixed has order 2" ( 2" - 1) ; the subgroup leaving two letters fixed has order 2° - 1; the subgroup leaving three letters fixed has order 1. Now, Zassenhaus' results apply. lt follows that@ = LF(2, 2").

7. Groups

@

which satisfy the assumptions (1), (II), but not the assumption (III)

If @ satisfies the assumptions (I) and (II), but not the assumption (III), let @0 be a seminormal subgroup of @ of minimal order which includes the 2-Sylow subgroup '.r. Then @0 ~ @. Again @0 satisfies the assumptions (I) and (II). If ®o ~ '.r, then @0 will satisfy the assumptions (I), (II), (III). Hence ®o = LF(2, 2"). Let @1 be a group which precedes @0 in a composition series from @ to @0 • Then ®o is normal in ®1. lf T E '.r, T ~ l, and if XE ®1, then X- 1TX is an

104

FINITE GROUPS

c

ONE-DIMENSIONAL UNIMODULAR PROJECTIVE GROUPS

745

involution of @o and hence conjugate to T in @0 • Thus x-1 TX y-1TY 1 with Y e @o . lt follows that XY- e f5.( T). Since c( T) = 2\ f5.( T) = st, and we find X EstY C ®o . Hence ®1 = @0 , a contradiction. Thus, ®o = st. Suppose @ :)

~1 ::) ' ' • : ) ~r

=X

is a composition series from @ to st. Suppose we know already that st is normal in ~z for some l. Then st is characteristic in ~z and hence normal in ~z-1. This shows that st is normal in@, @

= 91('.r).

Since 91(st)/f5.(st) is isomorphic with a subgroup 9)1 of LH(a, 2) in the usual manner, we have here ®/st "-' W. In our case, no element M ~ l of 9Jc has a fixed point. Also, 9Jc has odd order. lt follows thatall Sylow subgroups of 9)1 are cyclic, and this implies that 9)1 is soluble. Hence @ is soluble too. Thus, we have

(III.F) Let @ satisjy the assumptions: (I) @ is of type (S). (II) The 2-Sylow subgroup X of@ is abelian of type (2, 2, · · · , 2), order 2a ~ 4. For a = 2 assume also that C5.( T) = st for T e st, T ~ 1. If@ does not satisjy the assumption (III), then Xis normal in®, and@ is soluble. REFERENCES

K. A. FowLER, On groups of even order, Ann. of Math. (2), vol. 62 (1955), pp. 565-583. ZASSENHAUS, Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen, Abh. Math. Sem. Univ. Hamburg, vol. 11 (1936), pp. 17-40.

1.

R. BRAUER AND

2.

H.

HARVARD UNIVERSITY ÜAMBRIDGE, MASSACHUSETTS UNIVERSITY OF lLLINOIS URBANA, lLLINOIS UNIVERSITY OF SYDNEY SYDNEY, AUSTRALIA

105

REPRINT OF [70]

ANNALS

OF

MIATHEMATICS

Vol. 68, No. 3, November,1958 Printed in Japan

ON A PROBLEM OF E. ARTIN BY RICHARD BRAUER AND W. F. REYNOLDS *

(Received March 14, 1958)

To E. Artin on His SixtiethBirthday 1. Introduction E. Artin,in two importantpapers [1], has studiedthe orders of the knowntypesof simplefinitegroups. His resultsindicatea new approach to the investigationof the simplefinitegroupsG. In this connection,it is of interestto studythe largest primepowerwhichdividesthe orderg of such a group. The simplestquestionof this type is the followingone whichArtinasked us in a letter: what are the simplefinitegroupsG of 3? He conjecturedthat an order g whichis divisibleby a prime p > g11 all such groupsG belongto the knowntypes. We shall show that thisis actually true. Following Artin,we shalldenoteby L,(m) the group of unimodularlinearcollineationsof a Desarguean projectivelinewhichbelongsto a finitefieldwithm elements. We will prove groupwhoseorderg is THEOREM1. Let G be a finitesimplenon-cyclic divisiblebya primep > g'13. Then G is isomorphiceitherto L.(p) where p > 3 is a primeor to L2(p-1) wherep>3 is a Fermat prime,p=2T2+ 1, withintegraln. It is seen easily that the assumptionsof Theorem 1 implythat g is divisibleby p with the exact exponent1. We shall investigategroupsG of an orderof this type. Sylow's theoremasserts that G has subgroups P of orderp and that the numberof such subgroupsis of the form1 +rp where r > 0 is an integer. It seems of interestto studythis numberr furtherand to obtain refinementsof Sylow's theoremforour class of groups. We provethe followingresultfromwhichTheorem1 can be obtainedas a corollary. THEOREM2. Let G be a finitegroupwhoseordercontainsa primenumber p with the exactexponent1. AssumethatG is its own commutator subgroup,G = G'. If thenumberof subgroupsof G of orderp is 1 + rp thenr is of theform r = (hup + u2 + u + h)/(u + 1) ( 1) withpositiveintegersu and h, exceptwhen G has a homomorphic image * This researchwas supportedby theNationalScience Foundationunder GrantNSF-G1123. 713

RICHARD BRAUER AND W. F. REYNOLDS

714

G* of one of thefollowingtypes: ( I ) G* L,(p), p > 3 a prime. (II) G* L2(p - 1), p > 3 a Fermat prime. In the case (I), we have r = 1 and in the case (II), r = (p - 3)/2. Our proofof Theorem2 is based on methodswhichwere developed in two previouspapers [2], [3] in whichsimilarresultswere obtained for a more restrictedclass of groups. In a later paper, we will improveour resultsfurther. It is perhapsof interestto mentionthat Theorem1 is of importanceif one wants to determineall simplegroupswhose orderg lies belowa given numericalbound,since we have here a boundforthe primefactorsp of g forwhichg t 0 (modp2). If p2 i g, elementarymethodsshowthat p4 3, thenpa+3 < g. This imposes strongconditions on the ordersg. 2.

Group-theoretical preliminaries

Let G be a groupof finiteorderg and assumethatthereexistsa prime numberp such that g = 0 (mod p), g E 0 (modp2). Then a p-SylowsubgroupP of G has orderp. It followseasily (forinstancefromBurnside's theorem[4, p. 133]) that the centralizerC(P) of P is a directproduct C(P) = P x W of P witha group W of an orderw primeto p. If z:is a generatingelementof P and if a-is an elementof the normalizerN(P) of P, we have an equation -1ira-= 7t' where the integer j=j(o-) depends on a-.

It is seen at once that the mapping a- -- j(o-) is a homomorphismof

N(P) intothe multiplicativegroupof residueclasses of integersprimeto Since the kernel is C(P), it follows that N(P)/C(P) is cyclic

p (mod p).

of an order q whichdividesp - 1. Then thereexist exactly q distinct powersof 7r whichare conjugate to Tr in G. By Sylow's theorem,the numberof distinctsubgroupsof orderp of G has the form1 + rp withintegralr > 0 and 1 + rp = (G: N(P)). Combiningthese facts,we have g

(2)

=

pqw(1 + rp).

We call (q, w, r) the systemof p-invariantsof G. If r = 0 and w = 1, then G has orderpq and G can be definedby the relations

7tP

=

1, s7 = 1, Ir'17K

=

7tJ

where j is an integer which belongs

to the exponentq (modp). If j is changed, we obtain an isomorphic groupof orderpq. group. We call this groupG the p-metacylic PROPOSITION1. If G is a groupwhoseorderg containstheprimenumberp withtheexactexponent1 and if G has thep-invariants(q, w, r) with

ON A PROBLEM OF E. ARTIN

715

r = 0, thenthereexistsa normalsubgroupH of G suchthatG/His isomorgroupof orderpq. In particular, G # G' in phic with the p-metacyclic thiscase. Indeed, if r = 0, thenP is normalin G, that is, G = N(P). Since W is a characteristicsubgroupof C(P), W is normalin G. It is seen easily that G/W is isomorphicwiththe p-metacyclicgroup of order pq. Since GIW is soluble,we have G # G'. PROPOSITION 2. Let G be a group whoseorder g containsthe prime numberp with the exact exponent1 and let (q, w, r) be the systemof pinvariantsof G. If S is a normalsubgroupof G of an orders t 0 (modp) and if G1= G/S has thep-invariants(q,, w1,rJ),thenq = q,. Moreover, thereexistsan integerz > 0 suchthatr = z + r, + zrlp. of G ontoG1= GIS. The PROOF. Let 0 be the naturalhomomorphism image P1 = 0(P) of P is a p-Sylowsubgroupof G1. It is clear that the image 0(N(P)) of N(P) is includedin the normalizerN1(P1)of P1 in G1. On the otherhand, if 0(T) e N1(P1)forsome r e G, then r-1Pz C PS. It followsfromSylow's theoremapplied to PS, thatthereexistsan element p e PS withT Pr = p1Pp. Then rp1 e N(P), r e N(P)PS = N(P)S. It is now clear that 0(N(P)) = N1(P1). Let 7r again denotea generatorof P. Then 7t1= O(r) is a generatorof P1. If an elementr, of G1transforms zr,intoa powerz7r,then r, e N1(P1) and, as was just shown,we can set r, = 8(T) withr e N(P). It follows = 7rT with integrali. Applicationof 0 that we have an equationr-17TT into74'. Hence 7r' = 7ra. Since S has an order shows that rl transforms7w1 powers s t 0 (modp), two powersof 7t1are equal onlyif the corresponding of Trare equal. It is now clear that we may take i = j and that two powersof 7w1are conjugate in G1if and onlyif the correspondingpowers of 7r are conjugate in G. This impliesq = ql. Moreover,if r, belongsto the centralizerC1(Pl) of P1 in G1,we may take j = 1 and our argument shows that we can set ri = 0(r) withr e C(P). It followsthat 0(C(P))= C1(P1). If d is the orderof c(P) n 5, thensince C(P) has orderpw, the orderpw1of C1(P1)is equal to pw/d. This yieldsw = dw1. +rlp). Substitutingq=q1 and w=dw1 As in (2), we have g/s= pq1wl(1 and using (2), we find d(1 + rp) =s(1 + r1p). It followsthat d = s (modp). Since d dividess, we may set s = d (1 + zp) withintegralz > 0 and then our equation yields r =z + r1+ zr1p. This completes the proof.

PROPOSITION3. Let G be a groupwhoseorderg containstheprimenumberp withtheexactexponent1 and let (q, w, r) be thesystemof p-invari-

RICHARD BRAUER AND W. F. REYNOLDS

716

ants of G. Assume that no quotientgroupof G is isomorphicwith the group of order pq. Let Go be the unique minimal normal p-metacyclic subgroupof G for whichGIGois soluble. Thentheordergoof Gois divisible byp and if Gohas thep-invariants(qo,wo,ro), we have r = ro. PROOF. Suppose that p does not dividego. Using a principalseries fromG to Go, we can findnormalsubgroupsS and T of G such that Go C S c T : G and that (T: S) = p. Then G1 = G/S has a normalsubgroup T/S of orderp. Proposition2 shows that the firstp-invariantof G1is q and it now followsfromProposition1 thatG1has a quotientgroup isomorphicwith the p-metacyclicgroupof orderpq. However, this is impossiblesince G did not have such a quotientgroup. Consequently,p dividesgo. Since every p-Sylow subgroup of G is included in Go, we have r

=

ro.

It is easy to showthatin the case of Proposition3, we have qoI q, woI w. Proof of Theorem 2

3.

We now proveTheorem2 in a slightlyrefinedform. As a convenient abbreviation,we introducethe rationalfunction (3)

F(p, u, h)-(hup

+ u2

+

u + h)/(u+ 1) .

THEOREM2*. Let G be a groupwhoseorderg containstheprimep with

theexactexponent1. SupposethatG has thep-invariants(q, w, r). Let Gobe theminimalnormalsubgroupof Gfor whichGIGOis solubleand letS be themaximalnormalsubgroupof Goof an orderprimeto p. Theneither in theformr=F(p, u, h) withpositiveintegersu and r can be represented h or we have one of thecases: ( I ) G0/S L.2(p)withp > 3 a prime; r = 1. (II) GO/S-L2(p -1) withp > 3 a Fermatprime; r = (p - 3)/2. group of on the p-metacyclic (III) G can be mapped homomorphically orderpq. PROOF. Suppose that we do not have case (III). It is clear that S is characteristicin Goand hence normalin G. Proposition3 shows that p divides the order goof Goand, if Gohas the p-invariants(qo,wo,ro0) we and, if we have r rO. Obviously,Gois its own commutator-subgroup does not G* G*. Moreover, for holds set G* GO/S,the analogous fact have a non-trivialnormalsubgroupof an orderprimeto p. If G* has the p-invariants(q*, w*, r*), it followsfromProposition2 that there exists a non-negativeintegerz such that (4)

r

=

r

=

z + r* + zr*p.

ON A PROBLEM OF E. ARTIN

717

Since G* is its own commutatorgroup,we have r* >0, cf. Proposition1. Set now h = (z + 1)r*,u = z. Then, by (3) and (4), F(p, u, h) = r. If z > 0, r can be representedin the formr = F(p, u, h) with positive integersu, h. Assume that z = 0. It followshere that r = r*. If we write G for G*, it is clear that it will sufficeto proveTheorem2* forgroupsG which are equal to theircommutatorsubgroupsand whichdo not possess nontrivial normalsubgroupsof order primeto p. Under these additional assumptions,the theoremstates that if r cannotbe representedin the formF(p, u, h) with positiveintegersu, h, then either G- L2(p) with p > 3 or G L,(p -1) withp > 3 a Fermat prime. If w = 1, this follows at once fromTheorem10 of [3]. Hence we may assumethat w> 1. Considerthe p-blockB of irreduciblecharactersof G to whichthe unit characterXi of G belongs. As shownin [2], B consistsof q non-exceptionalcharactersX1,X2Y *- - X. and t = (p - 1)/qexceptionalcharacters X(x`,A= 1,2, *... t. For each xJe B, we have a sign a = ? 1 such that W). (foro- e Px W,o (5) Xf() = j Similarly,thereis a sign a, = ? 1 such that

f) EA=1~~~x.

( 6)

=a

(fora- e P x W. f W).

Moreover,all X(X"have the same degree f0and if X, has degreef , we have

( 7)

E

0 8f

is a sum of fj ptl Take o- as a generatingelementXrof P. Then XJ(o-) in of the fieldof the ideal divisor fixed p prime rootsof unity. If p is a pth

everypth rootofunityis congruent rootsofunityovertherationals,

to 1 (mod p). Hence (5) impliesfj _ , (modp) whence fj =

(8) forj = 1,2, ...

(9)

,

(modp)

q. Similarly,(6) yields

tfo a0

(modp).

Let c(o-)denotethe orderof the centralizerC(o-) of an elemento- of G. As is well known,the numbergxj(o-)/(Jc(o)fj) is an algebraicinteger. For G- = 7r, we have c(o-)=pw and it followsfrom(5) and (2) thatfj Iq(1 +rp) whence (j =1, 2, * ,q). t-fjI (p -1) (1 + rp) (10) Similarly,g1X()(af)/1c(o)fOis an algebraic integer for A 1,2, -, t. Addingover Aand using (6), we obtainthe analogousrelation

718

RICHARD BRAUER AND W. F. REYNOLDS

tf. I(p-1)(1 + rp) . (11) Assume that r cannotbe writtenin the formF(p, u, h) with positive integers u and h. An elementarydivisibilityconsideration[3, Lemmas 1 and 2] appliedto (8) and (10) showsthatwe musthave one of the cases (12) (13)

fj = 1, fj = p - 1

This applies forj = 1, 2, *

,

fj = 1 + rp fj = p(pr -r + 1) -1

(foraj= 1); (fora,

-1).

q.

Similarly,(9) and (11) implythattfoalso can have onlyone ofthe values in (12), (13), (14)

tfo= 1,

tfo= 1 + rp

(for 0= 1);

(fora0 = -1). (15) tfo= p(pr -r + 1)-1 tfo= p-1, In (7), we have q + 1 termsaifj with i = O. 1 ... , q. For at mostq of them,ai can have the value + 1. For each of these terms,fi < 1 + rp by (12) and (14). For i = 1, Xiis the unit characterof degree 1 and here a, = 1, as shownby (5). Since we could assume G = G' and hence r > 0 (cf. Proposition1), j1f1= 1 < 1 + rp. Thus, the sum of the termsof (7) with a,=1 is less than q(l+rp)= {(p-1)(1+rp)}It= {p(pr-r + 1)-1}/t. It now followsfromthe equation (7) that the cases fj = p(pr- r+ 1)-1 in (13) and tfo = p(pr - r + 1) - 1 in (15) are impossible. Since terms with a, = - 1 must appear in (7), one of the degrees p - 1 in (13) or (p - 1)/tin (15) occurs. If we havenf = p - 1 for some j with 11, S is a non-trivialnormal subgroupof G of an orderprimeto p. As shownabove, we could assume and Theorem2* that no such subgroupexists. We have a contradiction is proved. Theorem2 is an immediateconsequenceof Theorem2* since the assumptionG = G' impliesthat Go= G and that case (III) of Theorem2* is excluded. 4.

Proof of Theorem 1

Assume now that G is a finitegroupand suppose that there exists a primefactorp of the orderg of G such that g < P3. (18) If g 0 (mod p2) and if 1 + rp is the numberof p-Sylowgroups of G, theng > p2(l + rp) and (18) impliesthat r = 0. Hence a p-Sylowgroup P is normalin G. Suppose thenthat g t 0 (modp2). Let (q, w, r) denotethe p-invariants of G. If we assume that G = G', Theorem2 can be applied. The minimumof F(p, u, h) in (3) forpositiveintegersu and h is (p + 3)/2. Burnside's theoremshows that q > 2. Hence if r can be representedin the formF(p, u, h), (2) yields (19)

g = pqw(l + rp) > 2p(l + p(p + 3)/2)= p(p + 1)(p + 2),

contraryto (18). If r does not have the formF(p, u, h) withintegralu > 0, h>0, G has a quotientgroupG/H = G* of one of the forms ( I ) G* L.,(p)withp > 3; (II) G* L,(p -1) withp > 3 a Fermatprime. It followsfrom(18) that in the case (I), H can have at mostorder2. The same is true in the case (II) when p = 5. If p > 5, we must have

RICHARD BRAUER AND W. F. REYNOLDS

720

G = G* in the case (II).

This yieldsthe result:

THEOREM1*. Let G be a finitegroupof orderg and assumethatg has a primefactorp > g113.Assumefurther (a)

that G

=

y

() thatthep-SylowgroupP of G is notnormalin G,

(r) thatG doesnothave a normalsubgroupof order2. Then G-L,(p) wherep > 3 is a prime or G L,(p -1) wherep > 3 is a Fermat prime. Theorem1 is a special case of Theorem1*. It is of interestto replace (18) by weaker inequalities. As suggested by (19), the next class of simple groups encounteredare the groups L.(p + 1) wherep is a Mersenneprime. HARVARD UNIVERSITY TUFTS

UNIVERSITY REFERENCES

1. E. ARTIN, The ordersof thelinear groups, Comm. Pure Appl. Math., 8 (1955), 355-365. . The ordersof theclassical simplegroups, Comm. Pure Appl. Math., 8 (1955), 455-472. 2. R. BRAUER, On groupswhose order contains a prime number to the first powerI, Amer. J. of Math., 64 (1942), 401-420. groupsof primedegreeand relatedclasses of groups, Ann. , On permutation 3. of Math., 44 (1943), 57-79. 4. H. ZASSENHAUS, Lehrbuch der Gruppentheorie, 1937.

9 Mellis, 0., Goteborgs Kungl. Vetenkaps. och Vitterhets-Samhdlles Handl., 6, Folgden, Ser. B, .5. No. 13, 45 (1948). 0 Zeigler, J. M., W. D. Athearn, and H. Small, Deep Sea Res., 4, 238-249 (1957). 1 Rex, Robert W., and Edward D. Goldberg, Tellus, 10, 153 (1958). John Davis, Univ. of New Mex. Publications in Meteoritics, No. 2, 1 (1950). 2Buddhue, 3 Banfield, A. F., C. H. Behre, Jr., and David St. Clair, Bull. Geol. Soc. of Am., 67, 215 (1956). 4 Lewis, G. E., H. J. Tchopp, and J. G. Marks, Geol. Soc. Am. Memn.,65, 251 (1956). 5 Cromwell, T., R. B. Montgomery, and E. D. Stroup, Science, 117, 648 (1954). 6 Fisher, R. L., I.G.Y. General Report Series, 2, 58 (1958). 7 Ericson, D. B., and G. Wollin, Deep Sea Res., 3, 104 (1956). 8 Smith, George I., and W. P. Pratt, Geol. Surv. Bull., 1045A, 1-62, 1957. 9 Urey, H. C., Nature, 179, 556-5o7 (1957).

THE NUMBER

OF IRREDUCIBLE CHARACTERS IN A GIVEN BLOCK BY RICHARD

BRAUER

HARVARD UNIVERSITY

AND WALTER

OF FINITE

GROUPS

FEIT

AND CORNELL UNIVERSITY

Communicated January 16, 1959

Let ? be a group of finite order g and let p be a fixed prime number. The blocks irreducible characters of 6 with regard to p and their significance for the arithitic in the group ring of ( have been discussed in several previous publications.l particular, it has been stated that the number m of ordinary irreducible charters in a p-block of defect d is at most equal to pd(d+)/2. In the present note, is result will be improved. We shall show: THEOREM 1. Let B be a p-block of defect d of a finite group ?. The number m of linary irreducible

characters

xi, X2, ...

m

Xm in B satisfies

the inequality

1/4p2 + 1.

(1)

Our new proof which we shall now sketch is far more elementary than the pre)us proof of the weaker result. Let B consist of the modular irreducible charters j, l 2, . . pn- For p-regular elements R of @, i.e., for elements R of ? whose ler is prime to p, we have formulas xi(R)

=

E

dj(pj(R),

(1 < i < m)

(2)

j-1

Lere the dij are nonnegative longing to B. Set

rational integers, the decomposition

aj = (pd/g) E xi(R)xj(R), R

(1 < i, j < m)

numbers of ?

(3)

LereR ranges over all p-regular elements of ?. Let A denote the m X m matrix D and let m denote the n If D' denotes the transpose of D, matrix X j) (dij). an C = D'D is the n X n matrix of Cartan invariants of ? belonging to B. It lows from (2), (3), and the orthogonality relations for the modular group charters that A = pdDC-1D'.

(4)

[t is known that C has integral coefficients and that its largest elementary divisor pd. Now (4) shows that A has integral coefficients. Clearly, A is a symmetric ,trix. Fhe nth determinantal divisor of the m X n matrix D is 1.2 Using the normal m of D, we deduce from (4) that there exists an m X m matrix U with integral

afficientsand determinant 1 such that o\

/c-1 UAU' = pd

\o

o/

follows that not all coefficients of UAU' are divisible by p and hence not all of A are divisible by p. Let R,, R2, . . ., Rh} be a system of representatives for the classes of p-regular If the class of Ra consists of g, elements, and if Xi = xi (1) ljugate elements of ?. he degree of xi, then it is well known that the number ifficients

wi((R,)

in algebraic integer.

= g,Xi(R,,)/xi

Introducing these quantities in (3), we obtain h

aij=

(pc/g)x, E

a=l

(5)

(R.)ij(R.).

[f Xr is another irreducible character in B and if wc has the analogous significance Coi,then modulo a suitable prime ideal divisor p of p, we have w,(Ri)

W,(Ra)

(mod p).

)w, (5) yields easily the congruence (g/pd)

(aij/xi) = (g/pd) (arj/xr)

(mod p).

Let v denote the p-adic exponential valuation, v(p) = 1. Lectd of a block B, we can set

v(xi) = v(g) - d + X,

By the definition of the

(6)

bh Xi > 0 for 1 < i < m. We shall term the integer Xi the height of the charter Xi in B. There exist characters of height 0 in B. aracter, Xr = 0.

We choose Xr as such a

If congruences are taken as congruences in the ring of local integers for p, our ;ult becomes aij -

(Xi/xr)arj

(mod pl+Xi).

sing the symmetry of ai, we find aij^

(XiX/xlr)arr

(mod pl+Xi).

(7)

follows that a,,rris not divisible by p, since otherwise all aij would be divisible by and we have already seen that this is not true. Taking j = r in (7), we obtain v(air) = Xi.

particular, air s 0 for 1 < i < m.

(8)

It follows from (4) together with D'D = C that A2 = pA.

(9)

Mnce m

i=l

air2 =

(10)

pdarr.

ice air 7 0, this yields 1) + arr2 < parr.

(m -

Le maximum of pdx - x2 is obtained for x = pd/2 and hence m iis completes the proof of Theorem 1.

1 < p2d/4.

The formulas developed here can be used to obtain some further results. It lows from (3) in conjunction with the orthogonality relations for group charters that 0 < ai < pd.

By (9), m

E aj,2 = pdaii

(11)

?i = 1, 2, ... , m. If aii = p, all aji with j 4 i vanish. Then (8) implies that r. Since v(arr) = 0, it follows that d = 0. Thus ai < pd for d > 0. This can o be seen easily directly. We defined the height Xi of a character Xi of B by the equation (6). If Xi > 0, m (7) shows that ai is divisible by pXi+l. Hence Xi + 1 < d. We thus have a w proof of the following result.3 THEOREM 2. 1f B is a block of defect d > 2, the height Xiof a character Xi in B is

most d - 2.

For d = 01, and 2, we have Xi ==Ofor all xi in B.

There exist examples of blocks of arbitrary defect d > 2 which contain characters

height d - 2.

Let mx denote the number of characters Xi of B of height X; m=

mx.

(12)

Lenas shown by (8), mx terms air in (10) contain p with the exact exponent pX. ence the method used above yields (mo -

1) + mlp2 + m2p4- .+

.

< 1/4 p2d.

(13)

particular mx < l/4p2d-2x

(14)

?X > O, as it is easy to see that mo > 1. On the other hand, if B contains characters Xi of positive height X, = X > 0, an mo terms aji in (11) contain p with the exact exponent X. Hence mo
pa+a.

In particular, (C) holds for simple groups in the case a = 3. The proof of Theorem 8 is elementary but complicated, and we shall not give it here. We shall deal with another special case: THEOREM 9. Assume that G is a grouP of finite even order g. SuPPose that there is a Prime P such that (i) the P-Sylow group P of G is nonAbelian of order p\ (ii) There are no elements of order 2P in G. Then g > p 2a.

174

FINITE GROUPS

60

RICHARD BRAUER

PROOF,. W e must have P > 2. Let 11: =/= 1 be an element of P and suppose that there exists an involution T/ EG such that TJ- 111:YJ = 11:- 1 • Let r =/= 1 be an element of the center Z(P) of P, and !et Po be a P-Sylow group of the centralizer C 0(11:) of 11: in G with Po 2 {11:, r}. Then there exists an element a E C0(11:) such that r;a tranforms 11:---+ 11:- 1 , Po---+ Po. Replacing YJtl by a suitable power, we obtain a 2-element -r in the normalizer N0(P0) of Po that carries 11:---+ 11:- 1 • Since r- 2 then commutes with 11:, it follows from the assumptions that r- 2 = 1. If we map ~---+ -r- 1~-r for ~ E Po, Po is mapped onto Po a~d only the unit element of Po remains fixed. Since Po has odd order, it follows that P 0 is Abelian and that -r- 1~-r = ~-, for all ~ E Po. In particular, -r-'r-r = r- 1 • We now apply the same argument, replacing 11: and T/ by r and -r. We see that a p-Sylow group P, of C0(r) is Abelian. Since r E Z(P), it follows that P, is a P-Sylow subgroup of G. This contradicts assumption (i). Hence no element 11: =/= 1 of P can be transformed into its inverse by an involution. Now, a theorem of K. A. Fowler and the author [7] shows that there exist irreducible characters Xµ. of Gof P-defect 0. But this implies g > xµ.(l)2 ~ p2•. Q.E.D. Remark (1). In a recent letter, J. A. Green communicated to me the theorem that the defect group of a P-block can always be represented as intersection of two P-Sylow subgroups. This shows that actually not only (C) but also (C*) holds under the assumptions of Theorem 9. Remark (2). J. G. Thompson raised the question whether there exist groups G that satisfy the assumptions made in conjectures (Bi) and (B ii). 4. In concluding, we remark that the problem concerning simple groups whose order g is divisible by exactly four primes leads to difficult numbertheoretical questions. For instance, if P were to be a prime of the form p = 3 · 2n - 1 with n ~ 2 for which q = 3 · 2n-i - 1 = (P - 1)/2 is also a prime, then G = LlP) would satisfy our requirements. lt seems to be unknown whether there exist infinitely many such primes p. The first case one may consider is that of an order g = pap, p2 p3 , where P, P,, P2, and P3 are primes (P, > P2 > P3). It seems advantageous to distinguish three cases: (I) There exists an irreducible representation of degree P2. (II) There does not exist an irreducible representation of degree P2, but there exists an irreducibl1:: representation of degree p, in the principal P2· block or in the principal P3·block. (III) We have neither Case I nor Case II. In each of these cases, results can be obtained. For instance, in Case (III) the primes P, , P2, and P3 must satisfy the equation u, P, + u2P2 + U3P3 = 1, where u; is the unique integer such that

(j

= 1, 2, 3).

In addition, u; has to divide Pi - 1. Suzuki's simple group of order 29120 forms an example of this type of groups. I don't know the complete answer to our question. Of course, the orders discussed here are of a special type. However, the

175

REPRINT OF [80]

CONJECTURES CONCERNING FINITE SIMPLE GROUPS

61

order g of L 2(m) for any odd prime power m can be written in the form g = 2ap 1p 2pa so that, although P1, P2, and Paare not necessarily primes, the group behaves in many ways as if they were primes. In the case in which m is a power of 2, we even have the simpler case g = 2aP1P2 of Theorem 2. Coming now to the final conjecture (D), I have to be rather vague. There is some evidence that there is a situation similar to that mentioned for L 2(m) for each dass of classical groups of a fixed degree n over a finite field. There should be finitely mariy types, dependirig on the number m of elements of the finite field, and for each type, we should have a factorization (5)

of the group order into powers of integers P1 , · · ·, Pr so. that the characters of the group behave as if these factors P1 , P2, · · · , P, were primes. If this question can be settled, we may think of extending conjecture (A) as follows: (A *) There exist only finitely many simple groups G whose order g is divisible by exactly r distinct Primes and which do not belong to a class of classical groups f or which the number r in (5) does not exceed n. Harvard University REFERENCES (1]

ARTIN, E., The Orders of the Classical Simple Groups, Comm. Pure Appl. Math., 8 (1955), 455-72.

BLICHFELDT, H. F., Finite Collineation Groups. Chicago: Univ. of Chicago Press, 1917, chap. IV. [3] BRAUER, R., Number Theoretical Investigations on Groups of Finite Order, Proc. Internat. Symp. on Algebr. Number Theory, 1955. Tokyo and Nikko: The Science Council of Japan, 1956, pp. 55-64. (On p, 59, line 18, the word "abelian" should be replaced by the word "non·abelian.") [4] BRAUER, R., Zur Darstellungstheorie der Gruppen endlicher Ordnung, II, Math. Z., 72 (1959), 25-46. [5] BRAUER, R., Investigation on Groups of Even Order, I, Proc. Nat. Acad. Sei., U.S.A., 47 (1961), 1891-93. [6] BRAUER, R., and W. FEIT, On the Number of Irreducible Characters of Finite Groups in a Given Block, Proc. Nat. Acad. Sei., U.S.A., 45 (1959), 361-65. [7] BRAUER., R., and K. A. FOWLER, On Groups of Even Order, Ann. Math., 62 (1955), 565-83, Th. (5E). [8] BRAUER, R., and H. F. TUAN, On Simple Groups of Finite Order, I, Bull. Amer. Math. Soc., 51 (1945), 756-66. [9] BRODKEY, J. M. (to be published in Proc. Amer. Math. Soc.). [10] FEIT, W., and J. G. THOMPSON (to be published in Pacific J. Math.). [11] REE, R., A Family of Simple Groups Associated with the Simple Lie Algebra of Type (F,), Amer. J. Math., 83 (1961), 401-20. [12] REE, R., A Family of Simple Groups Associated with the Simple Lie Algebra of Type (G2), Amer. J. Math., 83 (1961), 432-62. (13] SPEISER, A., Theorie der Gruppen von endlicher Ordnung, 3d ed. Berlin: Springer, 1937, Sec. 70. (14] SUZUKI, M., A New Type of Simple Groups of Finite Order, Proc. Nat. Acad. Sei., U.S.A., 46 (1960), 868-70. [2]

176

FINITE GROUPS

ON FINITE GROUPS AND THEIR CHARACTERS RICHARD BRAUER

The idea of a presidential address seems to require a lecture delivered in the most refined and dignified scientific atmosphere yet understandable to the layman, a lecture which treats a difficult field of mathematics in such a complete manner that the audience has the excitement, the aesthetic enjoyment of seeing a mystery resolved, perhaps only with the slightly bitter feeling, of asking afterwards: Why did I not think of that myself? Well, I don't know how my predecessors did it, but I know t h a t I can't do it. Since the founding fathers of the Society have placed the presidential address at the time in the life of the president when he disappears into anonymity among the ranks of the Society, I shall not even try it. The choice of the field about which I am going to speak was a natural one for me, not only because of my own work in the theory of groups of finite order, but because of the new life which has appeared in this field in recent years. However, in spite of all our efforts, we know very little about finite groups. The mystery has not been resolved, we cannot even say for sure whether order or chaos reigns. If any excitement can be derived from what I have to say, it should come from the feeling of being at a frontier across which we can see many landmarks, but which as a whole is unexplored, of planning ways to find out about the unknown, even if the pieces we can put together are few and far apart. My hope then is that some of you may go out with the idea: "Now let me think of something better myself." Let me first mention one difficulty of the theory. We have not learned yet how to describe properties of groups very well; we.lack an appropriate language. One of the things we can do is to speak about the characters of a group G. I cannot define characters here. Let me only mention t h a t we have a partitioning of the group G into disjoint sets K%t Ki, • • • , Kh, the classes of conjugate elements. The characters then are k complex-valued functions Xi» ' • ' > X*» e a c n constant on each class Ki, They have a number of properties which connect them with properties of the group. These characters can be used to prove general theorems on groups, but we seem to have little control about what can be done and what not. You will see this more clearly later when I discuss specific results. I can give two reasons Presidential address delivered before the Annual Meeting of the Society, January 28,1960; received by the editors November 19,1962.

125

126

RICHARD BRAUER

[March

for this particular behavior of the characters. The first is that I believe that we don't know all about characters that we should know. My reason for this statement is the following: If you take groups of a special type, groups whose order g contains some prime number p only to the first power g = Pgo,

(P, go) = 1,

some powerful results can be proved. The groups G have a subgroup P of order p. If N is the normalizer of P, i.e. the subgroup of all 1 is an integer and if exactly m degrees Xi are equal to r, then a prime number p divides n = at most with

!GI

the exponent

In particular, e ~ mrp/(p - 1) 2 • Later, other conditions will be given which apply only for values n of a special kind, but which are far more suitable for an actual discussion. Their existence shows that certainly the last word has not been said about Problem 1. §4. The simplest examples show that nonisomorphic groups may have isomorphic group algebras over C. For instance, this is always true for two nonisomorphic Abelian groups of the same order n, since for both all degrees Xi are 1. If we wish to obtain a criterion for the isomorphism of two groups, we will have to strengthen our assumptions. Since we know that the degrees Xi are determined by . the scalars of multiplication aaß-y of the class algebra Z, we may require that these numbers aaß-y are the same for both groups, if the classes are indexed suitably. This is the same as to require that both groups have the same character table [xl provided, of course, that characters and classes are indexed suitably. Still, this is not enough as the two non-Abelian groups of order p 3 for primes p have the same character table. We strengthen our assumption further. If K is a conjugate class of G and m an integer, there is a conjugate class K[mJ of G which consists of the mth powers of the elements of K. We now ask: Problem 4. Let G and G* be two finite groups and assume that there exists a one-to-one mapping Ki ---, Ki * of the set of conjugate classes of G onto the set of conjugate classes of G*, and a one-to-one mapping Xi - t Xi* of the set of irreducible characters of G onto the set of irreducible characters of G* such that

(a)

(b)

Xi(Ki) = Xi*(K/) (K/ml)*

= (K/)[mJ for all integers m.

Are G and G* isomorphic? No counterexamples seem tobe known, but nobody seems to have

* Here,

[t] denotes the largest integer which does not exceed the real number t.

188

FINITE GROUPS

Representations of Finite Groups

139

looked closely. Perhaps it would be worthwhile to do some checking; it may not even be difficult to find counterexamples. On the other hand, if the answer was affirmative, this would be wonderful. My next problem is somewhat vague. Problem 5. Investigate connections between the values of the characters and the mappings K - K lml .

There are a number of obvious remarks one can make here (Supplements, §3). The question is of practical importance, if characters are used to study properties of particular groups. We can now state all we can ever hope for in the theory of characters in form of a problem. Problem 6. Give necessary and sujficient conditions for a square matrix (with operation on the columns, correspondir,,g to the mapping K - Klml thrown in) tobe the character table of a finite group. ·

There are good reasons to believe that important properties of characters have not yet been discovered. This means of course that any attempt to solve Problem 6 now would be futile. One may even doubt whether a satisfactory concrete solution can ever be given. §5. I continue my discussion of group characters by stating a more recent result. A subgroup E of a group G will be called elementary, if it is a direct product of a cyclic group A and a p-group P (that is, a group of prime power order pm). Necessary and sufficient conditions can be given for a function 8 defined on G to be an irreducible character of G. They are as follows. 1. The function 8 is constant on each class K;. 2. For every elementary subgroup E of G, the restriction to E is a character of E. *

3. We have

l

olE of 8

lo(u)l 2 = IGI.

uEG

In order to apply this result to a construction of the irreducible characters of G, it will be sufficient to consider a set of elementary subgroups (4)

* lt is sufficient to

require here only that oJE is a generalized character of E.

189

REPRINT OF [82]

140

Lectures on Modern M athematics

of G such that each elementary subgroup of Gis conjugate to one of the groups (4). If we know the irreducible characters of the groups E 1 and, if for each conjugate class of the E 1, it is known to which of the k conjugate classes K 1, K 2 , • • • , Kk of G it belongs, then the irreducible characters Xi of G can be constructed. For the first part, it would be highly desirable to have more information on the following problem: Problem 7.

Study the irreducible characters of p-groups.

We say that two conjugate classes of a subgroup E of Gare fused in G, if both lie in the same conjugate class of G. For the second part in our construction of the Xi we have to know what the fusions of E-classes in G are. We would like to know more about the ways in which the classes of a given group E can be fused in !arger groups. For instance, if we are interested in all finite groups which contain a given p-group P as their p-Sylow group, we are led to the question: Problem 8. Given a p-group P, can we find constructively all possible ways in which the conjugate classes of P can be f used in finite groups G with P as their p-Sylow group?

There are a number of ways in which the above procedure can be modified. In particular the groups (4) can be replaced by a set of subgroups H 1, H 2, • • • of G such that every elementary subgroup of G is conjugate in G to a subgroup of some Hi. The following special case seems to deserve attention. Pick a representative ui in each of the k conjugate classes Ki, (i = 1, 2, . . . , k) of G and let Ci be its centralizer in G; Ci = C(ui). If, say, uk = 1, then Ck = G. lt is obvious that the remaining groups (5)

satisfy the requirements imposed above on H 1 , H 2 , ask:

Wenow

Problem 9. H ow much information about the group G can be obtained if all or some of the groups Ci in (5) are given (possibly combined with information of the type mentioned above in connection with the Ei)?

For some comments, see the Supplements, §4. We shall see later that in some special cases amazingly much can be said, if a single Ci is known.

190

FINITE GROUPS

Representations of Finite Groups

141

§6. W e shall conclude our survey of the classical theory of characters with a brief discussion of some of the questions about groups G which can be answered if the characters of G are known. First, we can find all normal subgroups N and determine the characters of G/N. In particular, we can recognize whether or not G is simple and whether or not G is soluble. Among the normal subgroups N of G, we can identify the members of the ascending and descending central series of G. Also, we can determine the Frattini subgroup of G. There is more difliculty about getting information concerning the structure of normal subgroups N of G. For instance, I do not know the answer to the following question. Problem 10. Given the character table of a group G and the set of conjugate classes of G which make up a normal subgroup N of G, can it be decided whether or not N is Abelian?

This would be of importance if we want to identify the members of the derived series of G. If G possesses an arbitrary subgroup H of index r, then G possesses a transitive permutation representation of degree r, and, correspondingly, a reducible character of degree r. If we were able to recognize which characters of G belong to transitive permutation representations, we could determine the orders of all subgroups of G. However, no criterion for permutation characters is available, only a number of necessary conditions. For this reason, we are usually only able to show that G cannot have subgroups of certain orders. In a more refined form, we can use Frobenius' idea of induced characters (see Appendix, §6). Suppose we have two groups G and H such that the order of H divides that of G and suppose we know the character tables [ x] of G and [ip] of H. If H is a subgroup of G, we have a mapping 0: Li----, Ki of the set of conjugate classes of H into the set of conjugate classes of G determined by Li C Ki. Moreover, we have formulas for Ki = O(Li)

(6)

" with non-negative integral coeflicients aP" which are independent of i. If His no longer assumed tobe a subgroup of G, we may still have a mapping O and formulas (6) such that all conditions known for

191

REPRINT OF [82]

142

Lectures on Modern M athematics

the case of a subgroup Hof Gare satisfied. W e should include here the conditions w hich are connected with the mappings K - K [ml • lt would be a generalization of Problem 4 if we ask whether H then has to be isomorphic with a subgroup of G such that 0 and the formulas (6) have the meaning given them in the case H s;;;; G. In a less ambitious mood, we formulate here the questions: Problem 11. Given the character table of a group G, how much information about existence of subgroups can be obtained? Problem 12. Given the character table of a group G and a prime p dividing n = \G\, how much information about the structure of the p-Sylow group P can be obtained? In particular, can it be decided whether or not Pis Abelian?

For a number of comments, see the Supplements, §6. Every automorphism of a group G produces a permutation of the irreducible characters of G. Because of this, we may ask: Problem 13. How much information about the automorphißm group A(G) of a group G can be obtained if the characters of Gare known?

Some remarks are given in the Supplements, §7. §7. So far we have taken the underlying field Q as the field C of complex numbers. lt would not make any difference if instead we use an algebraically closed field Q of characteristic 0. W e now drop the assumption that Q be algebraically closed, but, for the time being, we shall stick to fields of characteristic 0. Then the group algebra r of G with respect to Q is still semisimple. As in (2) we can write r as a direct sum (7)

of simple algebras. Each Ap corresponds to a class of equivalent representations of G, now with coefficients in Q and irreducible in ü. All irreducible representations L of G in Q are obtained in this fashion. In order to investigate such representations L of G, we may see how they behave in the algebraic closure ö of n. Of course, in ö we have the situation studied above. This approach was first carried out by I. Schur who found that each Ap is in one-to-one correspondence with a family T P of irreducible characters Xi, taken in Ö and algebraically conjugate with regard to ü. If LP is an irreducible representation of G in ü, which is associated with the

192

FINITE GROUPS

Representations of Finite Groups

143

term Ap in (7), the character Ap has the form (8)

where Xi ranges over the characters in TP and where the mp are natural integers. This formula (8) corresponds to the breaking up of the representation Lp in the algebraic closure n of n. At this stage, we know little about the integers mPI except that they divide the degree x; of Xi E T P· We call the mp the Schur indices. So far we have disregarded any connections with Wedderburn's theory of simple algebras. According to this theory, each Ap is isomorphic with a complete matrix algebra of a certain degree qp with coeflicients in a division algebra l:i.p over 0. Let O(xi) denote the field obtained from Q by adjoining all values x;(g) with g EG of the character x;. lt turns out that the center Zp of l:i.p is isomorphic to O(xi) for Xi E Tp, This implies that the degree Zp = [ZP: O] is equal to the number of characters Xi in Tp, Moreover, the dimension of l:i.p over ZP is exactly mp 2 , Finally, the Wedderburn degree qP is equal to xi/mp, If the irreducible characters x1, x2, • • . , Xk of G are known, we know of course the number r of terms in (7), the fields ZP, and their degrees. lt is more diflicult to determine the Schur indices mp, Still, a great deal of information is available. lt is possible to reduce the same question to a treatment for the case of solvable groups. If the field O contains the nth roots of unity for n = \GI, then all mp are equal to 1, and it follows that the irreducible representations of G in O remain irreducible in the algebraic closure n. They can then be used as our old X 1, X 2, • • • , X k· As has been shown by L. Solomon [55), all mp are already 1, if O contains a primitive pth root of 1 for every odd prime divisor p of n = IG\ and, in addition, a primitive fourth root of 1, if n is even. Since in the case of an algebraic number field n a complete characterization of all central division algebras by arithmetical invariants has been obtained by R. Rasse [37], the problem arises to develop methods by which the Rasse invariants of the Ap in (7) can be determined. Such methods have been given by E. Witt [71]. lt is clear that the preceding results can be used to discuss special cases of Problems 1 and 2, cf. Supplements, §9. I conclude §7 by formulating a rather old problem which is still open.

193

REPRINT OF [82]

144

Lectures on Modern M athematics

Problem 14. Characterize by means of group theoretical properties of G the number of irreducible representations Xi of G in C which are equivalent to representations with coejficients in the field R ~/ real numbers.

§8. As our next generalization, we consider fields n of arbitrary characteristic. If we go back to Problem 2 for a moment, we may have a better chance of success if we admit fields of a different type. At least, no counterexample is known for the following version. Problem 2*. If two groups G1 and G2 have isornorphic group algebras over every ground field n, are G1 and G2 isomorphic?

Little work seems to have been done on this question. Incidentally, the assumption is not weakened if only prime fields n are co:ilsidered. If it is assumed that the group rings of G1 and G2 over the ring of integers are isomorphic, this implies isomorphism of the group algebras, and we should have a better chance of success if we try to prove isomorphism ofG1 and G2 under this-assumption. The theory developed for fields n·of characteristic p enables us to deal at least with some special cases. Let n now be a field of prime characteristic p. The interesting case is that p divides the ordern of the group G. The first marked difference is that in this case the group algebra r is no langer semisimple, the radical N of r is not (0). lt seems natural to formulate the following problem. Problem 15. Characterize by group theoretical properties of G the dimensions of the radical N and of its powers N 2, N 3, • • • • In particular, determine the smallest exponent e for which Ne = 0.

Unfortunately, nothing is known about Problem 15 except in the case of a p-group where p is the characteristic of n. In this particular case, a satisfactory theory has been given by S. A. Jennings [41]. Returning for a moment to Problem 1, we observe that, in the case of fields of characteristic p, a new type of necessary condition for group algebras r can be formulated: r belongs to a special class of algebras; it is a symmetric algebra. lt would be interesting to find other necessary conditions for Problem 1 of a similar nature, that is, to find general classes of algebras which include the group algebras. A variation of this leads to the problem:

194

FINITE GROUPS

Representations of Finite Groups

145

Problem 16. Obtain classes of groups G by imposing group theoretical conditions which can also be characterized by algebra-theoretic conditions imposed on the group algebra r (Supplements, §10).

§9. While it is possible to start an investigation of the irreducible representations of G in the field n of characteristic p directly, we prefer a different approach. If p is a given prime, we denote by vp(a) the exact exponent with which p divides the integer a =l= 0. If we set vp(b/a) = vp(b) - vp(a) for two integers b, a =l= 0 and set vp(O) = oo, we have the p-adic (exponential) valuation vp of the field Q of rational numbers. lt follows from a basic theorem of algebra * that vp can be extended to a valuation of the field Q = A of all algebraic numbers. We shall denote a fixed such extension again by vp. The set of elements a of A, with vp(a) ~ 0 forms a subring o of A, the ring of local integers with regard to vp, and A is the quotient field of o. The elements a of p with vp(a) > 0 forma maximal ideal~ in o. The residue class ring o/p = n is a field algebraically closed of characteristic p. For each a E o, we have a corresponding element a* E n; the residue class map a --t a* be denoted by 0. We now start from a matrix representation X of Gin A. Actually, since only finitely many coefficients appear in the X(g), g EG, the representation X will lie in a subfield A 0 of finite degree over Q. lt can be seen easily that after replacing X by an equivalent representation we may assume that all coeffi.cients of all X(g) lie in o. N ow the residue class map 0 of o --t n changes X into a representation X 9 in n. lt is quite possible that X is irreducible in A and that X 9 is reducible in n. Let again X 1, X 2, • • • , X k denote the irreducible representations of G in the algebraically closed field A of characteristic 0 and let F1, F 2, • • • , F 1 denote the different (that is, nonequivalent) irreducible representations of G in n. The number l of distinct representations Fi turns out to be the number of conjugate classes Kx of G consisting of elements ,, whose order is prime to p, or, as we shall say, consisting of p-regular elements. We can then speak of the multiplicity di,j with which Fj appears in the representation X/ obtained from Xi, by means of the residue class map 0. Actually, 0 may transform equivalent representations Xi and X/ 1> with coeffi-

will

* See, for instance, Zariski-Samuel [72, Chap. 6) or van der Waerden [67, Chap. 10).

195

REPRINT OF [82]

146

Lectures on Modern M athematics

cients in o into inequivalent representations X/ and xf1> 9 in n. However, it can be shown that in spite of this the decomposition numbers d,,; are well defined. Each F; appears in some X ,,8. We shall refer to the representations of G in A as to the ordinary representations and to the representations of G in n as to the modular representations of G. We now define the character et, of a representation Y of G in n. We restrict ourselves to p-regular elements g of G. If g has order m prime' to p, the characteristic roots of the matrix Y(g) are mth roots 111, 112, • • • of unity in n. For each such m,, there exists a unique mth root of unity E>. in o mapped by 0 on 11>-· We set ct,(g) = E1 + E2 + · · · . Thus, et, is a complex-valued function defined on the set G0 of p-regular elements in G. Two representations of G in n have the same character if and only if they have the same irreducible constituents. Let ct, 3 denote the character of the irreducible representation F;. If restriction of a function defined on G to the set G0 is indicated by a superscript zero, the definition of the decomposition numbers di; yields immediately the formulas l

(9)

l

x„ 0 =

d,,;cJ,;



= 1, 2, . . . , k)

j=l

In the further development of the theory, the integers by the formulas

Ci;

defined

k

(10)

Cij =

l

d„i d„ 3

(i, j = 1, 2, . . . , l)

µ=l

are of importance. They are the Cartan invariants which play a role in the investigation of the algebra r. N ow orthogonality relations for the modular characters cJ,; can be given. In accordance with the general spirit of the lectures, I shall talk about two questions which I cannot answer: Problem 17. lf fi = 1 where we have all we can wish for (see, for instance, the Supplements, §15). The trouble is that usually the number of possible types is so tremendously large that the actual discussion is out of question. In fact, we do not even know the p-groups of order pd except when d is small. lt is certain that most of the types which we cannot exclude on the basis of our present knowledge may never occur. Thus, the following rather vague problem is important. Problem 29. To obtain information on p-blocks of defect dfor given pd which restrict the possibilities f or types.

§15. The results outlined here must appear complicated, but this has tobe expected. The characters of many classes of groups have been determined and few, if any, regularities are evident. lt was my wish to show that in spite of this, there is a strong interrelation between properties of the representations and properties of the abstract group. This frequently makes it possible to obtain information about the characters, if we know rather little about the groups. Based on this, we can often study groups about which we have made only rather weak assumptions.

202

FINITE GROUPS

Representations of Finite Groups

153

The more connections we have between properties of the representations and properties of the abstract group, the better we can expect this to work. Most of the questions I have asked above can be summarized essentially in one single problem: to find new connections. §16. In the next sections, I shall leave the general theory of representations and discuss some topics of a more concrete nature where the results discussed so far can be applied. Recently, W. Feit and J. G. Thompson [28] succeeded in proving the famous old conjecture which states that all finite groups of odd order n are solvable; the complete proof is to appear in the Pacific Journal of M athematics. If we are interested in nonsolvable groups, we can now concentrate on groups of even order. Assume that n = is even. Then the group G contains elements j of order 2 or, as we shall say, involutions j. If j belongs to the conjugate class Ka of G, the scalars of multiplication aa"'Y of the class algebra Z of G have some special properties. Expressing the aaa'Y by means of the characters and using the blocks to investigate the resulting relations, results on groups can be obtained which we shall now discuss. lt will be necessary to introduce some notation. Let p be a fixed prime dividing n and Jet 1r be an element of order pa > 1 of G. The generalized centralizer C*(1r) of 1r is the subgroup of G consisting of all u EG for which either u- 11ru = 1r or u- 11ru = 1r- 1. A rather crude application of the method yields results of the following kind. Choose an arbitrary real number A > 1. We then have one of two cases.

!GI

Gase 1. (16)

There exists an approximate f ormula f or n. A

A

.

+ 1 ~c(3)2 ;;; n ;;;

A

.

A - 1 ~c(3)2

where ~ depends only on C*(1r), and the manner in which the intersection Ka n C*(1r) of the conjugate class Ka of j with C*(1r) breaks up into conjugate classes of C*(1r). If ko such C*(1r) classes appear in Ka, then (17)

where the real number X depends on A, p, and vp(c(j)).

203

REPRINT OF [82]

154

Lectures on Modern M athematics

Gase 2. There exists anormal subgroup N =l= Gof G such that (G: N) lies below [ßc(j)]I with ß depending on A, p, and vP(n). *

If 1r and 1r- 1 are not conjugate in G, only Case 2 has to be considered. For p = 2, the result can be improved somewhat. In (17) the factor k 0 can be deleted. If H is any finite group, we define its 2-regular core K2(H) as the maximal normal subgroup of Hof odd order. In the Case 2, (G: N) lies below [ß(C(j): K2C(j))]I. Let us restrict our attention to the case p = 2. We can obtain much more precise results, if we know the type to which the principal 2-block B 0 of G belongs. We shall then assume that we know the defect group of Bo, that is, the 2-Sylow subgroup P of G. This leads to the following problem. Problem 30. Given a 2-group P, investigate the finite groups G with Pas their Sylow subgroup. In particular, investigate the groups Gin which in addition the centralizer C(j) of an involution is given.

If j belongs to the center of P, then Pis determined by C(j) and we have a special type of Problem 9 with only one Ci given. lt may seem surprising that it is possible to say anything about G. Still, this is the case for a number of classes of 2-groups (Supplements, §16) and, in principle, the method can be used for any P. As a more precise question in the direction of Problem 30, we may ask: Problem 31. For which 2-groups P can there exist only finitely many (nonisomorphic) groups G with 2-Sylow group P and with 2-regular core K 2 (G) = {1} (that is, without normal subgroup of odd order larger than 1)?

This can at least be answered for Abelian P; see R. Brauer [10]. In §4 of this paper a number of other possibilities are mentioned which can arise in connection with Problem 30. The answer to the following question seems to be unknown. Problem 32. If Gis a simple (noncyclic) group of order n, does there exist an involution j such that n ;;;,i c(j) 3 ?

If G has at least two conjugate classes of involutions, this can be * A similar result is proved by elementary methods in R. Brauer and K. A. Fowler [13].

204

FINITE GROUPS

Representations of Finite Groups

155

proved by quite elementary methods even without the assumption of simplicity of G (see R. Brauer and K. A. Fowler [13]). The formula (17) with 1r = j shows that the methods discussed here might be applicable. lt would be useful if we had the answer to the following question. Problem 33. For which 2-groups P can it happen that there existjinite groups G with P as their 2-Sylow group and with only one conjugate class consisting of involutions?

§17. One of the principal outstanding problems in the theory of finite groups is that of determining all simple finite groups. Families of simple groups and a f ew individual simple groups ,are known (Supplements, §17). However, we have no idea how many of the simple finite groups have not ,yet been discovered. While it is not unthinkable that some day a direct and uniform approach to the study of all finite simple groups might be found, we do not know when this day will come. In the meantime, there are at least a few things we can try if we are much more modest. In some of the more approachable cases, one can find group theoretical characterizations of particular simple groups by using the idea of Problem 30 and imposing further conditions on C(j). If we knew more about the 2-Sylow subgroups of the simple groups and more about Problem 30, we would have at least an opening. Of course, for the simple groups of Lie type, it is a different prime, the characteristic of the field, which plays a distinguished role. But at least in some ways, the prime p = 2 seems to be second best. I am aware that I am going far out on a limb if I now make some other suggestions about things one might try to do. The answers may not be what I hope them tobe, and the direction may not be the right one. But at least the first steps do not look hopeless, and, without trying, we do not know where they may lead. As my starting point, I use a well-known theorem of Burnside: there do not exist any finite simple groups whose order n is divisible by exactly two distinct primes. I ask: Problem 34. I s it true that there exist only jinitely many simple groups whose order is divisible by exactly three distinct primes?

If we wish to go on to the case of four primes, we have to change the question somewhat.

205

REPRINT OF [82]

156

Lectures on Modern M athematics

Problem 35. I s it true that there exist only finitely many simple groups whose order n is divisible by exactly f our distinct primes and which are not of the type L2 (q) ?

Before turning to group orders with larger number of prime factors, we note that the orders of the simple groups in a given family \j (Supplements, §17) with fixed dimension can be written in the form (18)

where fr(q) denotes the rth cyclotomic polynomial, where a and ar are positive integral exponents, and where r ranges over a certain finite set of positive integers. Let M(\j) - 1 denote the number of values r appearing here. We now can pose the question: Problem 36. Given an integer s ~ 3, is the number of simple groups G finite whose order n is divisible by exactly s distinct primes and which do not belang to families \j with M(\j) ~ s?

A number theoretical theorem of C. L. Siegel [54] shows that the known simple groups will not provide counterexamples. Also, for given s, there are only finitely many families with m(\j) ~ s. lt is very likely that before much progress is made on questions concerning simple groups, more attention will have tobe paid to the known families of simple groups. We discuss some points which fit in with the general program discussed in this report. First of all, we would like to know the characters of the groups in the various families. By operating rather recklessly, it is often possible to make some good guesses. lt seems not unreasonable to hope that the structure of the groups Gina family \j depends somewhat regularly on the parameter q. A first question in this direction is: Problem 37. Consider the groups G in a fixed family \j of simple groups. Do there exist polynomials X1(q), X2(q),

, Xt(q),

n1(q), n2(q),

,nt(q)

with the following property: for any fixed value of q, the group G possess exactly n;(q) irreducible characters X; of degree x;(q), (i = 1, 2, . . . , t)?

206

FINITE GROUPS

Representations of Finite Groups

167

The primes p dividing the values r occurring in (18) play a somewhat exceptional role. If they are removed from fr(q) (for some fixed q, or, more generally, for some class of values of q determined by congruence conditions), the retnaining part hr of fr(q) often seems to behave as if it was a prime factor of n. This is specially useful, if the exponent ar is 1, since we can then apply tentatively the strong results known for the case where the group order is divisible by a prime with the exact exponent 1. For all families ij, such factors f r(q) with ar = 1 occur. If we have guessed the values of some irreducible character of G, it may be possible to prove that we actually have a character by showing that the necessary and sufficient conditions for characters are satisfied (see §5 in this section). Of course, ideas of the same nature can be tried for other group theoretical properties of G. lt is not necessary for our discussion that the polynomial (18) actually belongs to a family of simple groups. We ask: Problem 38. Consider a polynomial n(q) and assume thatfor all q in a certain set of integers there exists a group Gq depending on q of order n(q). Give necessary and .sufficient conditions of a group theoretical nature whick guarantee that the keuristic principles described above are true statements.

§18. As my last topic, I wish to discuss briefly the finite linear groups of a fixed degree m. By a linear group G of degree m over a field n, we mean a group G which has a faithful representation of degree min n. Given n, the question then is: what are these linear groups of a given degree? If m = 3 and if Q is the field R of real numbers, this problem is essentially equivalent to a study of the Platonic solids. We only have to add some degenerate solids, the dihedra. The linear transformations mapping the solid onto itself form a finite group. Given a finite linear group G, we can find the vertices of an associated solid by looking for points for which the total number of distinct images under transformations of G is relatively small. Our general question about the linear groups is an extension of the old question of Greek mathematics. Again, it is natural to work in the field Q = C of complex numbers. The basic result is a theorem of Camille Jordan (1878) which states that, for a given degree m, there exist bounds ßm

207

REPRINT OF [82]

158

Lectures on Modern M athematics

depending on m only such that a linear group G of degree m has a normal Abelian subgroup A for which the index (G: A) lies below ßmIn particular, there are only finitely many simple linear groups of degree m. Explicit bounds for ßm have been given by Blichfeldt, Bieberbach, and Frobenius; see H. F. Blichfeldt [3] for references. The small dimensions m = 2, 3, 4 have been studied, and, with the newer results on characters, the case m = 5 and probably the next cases can be handled. On the whole, we know very little about this subject. lt would be of interest to obtain substantially better values for the bound ßmSince for large m, more and more linear groups of degree m occur, it would be futile to try to list them all. Rather, we should concentrate on the interesting cases. Before this can be done, it would be necessary to decide which groups deserve special attention. lt may also be worthwhile to look more closely at the case where the dimension m is a prime power. lt has been shown by R. Steinberg [59] that if G is a finite simple Lie group and if pa is the highest power of the characteristic dividing the group order n, then G has an irreducible representation of degree pa. These Steinberg linear groups seem to play an important role, and it would be of interest if they could be singled out from all the linear groups of prime power degree. With the whole field wide open, there may not be too much point in placing emphasis on any particular question. I am mentioning the next problem because it plays a role in Blichfeldt's work and because recent progress has been made on it. Problem 39. For which linear groups G of degree m does it happen that there exist primes p > m 1 dividing the group order n such that G does not have a normal subgroup of order pa )> 1 ?

+

Recently, W. Feit and J. G. Thompson [26] proved that if the prime p divides the order n of the linear group G of degree m, and if the p-Sylow subgroup of G is not normal in G, then p ;;;:; 2m + 1. Their proof is based on a result of the author dealing with the case n $ 0 (mod p 2 ) which depends on properties of group characters of the type discussed here. I have been informed by W, Feit that he can now show that if the linear group G of degree m has order n = n 1n 2, where all prime 1 while the prime factors of n 2 factors of n1 are larger than m are at most equal to m + 1, then G has anormal subgroup either of

+

208

FINITE GROUPS

Representations of Finite Groups

159

order n1 or of order ni/p with p a prime. This disposes of a question which I asked at the original lectures. For some other results, see the Supplements, §18. There is no reason to exclude fields of characteristic p. The celebrated Theorem B of P. Hall and G. Higman [35] deals with a property of p-soluble linear groups over fields of cliaracteristic p. This theorem, which has already proved its value in J. G. Thompson's work, opens a number of new questions. If n is a finite field, say with q elements, we are dealing with the subgroups G of a particular finite group, the general linear group GL(m, q). Still, we know very little about the linear groups of degree m. If the case m = 4 had been studied systematically, Suzuki's simple groups would have been discovered earlier. lt may then not be superfluous to ask: Problem 40. finite fields.

Determine all linear groups of small degrees m over

No result in direction of Jordan's theorem can exist for algebraically closed fields of characteristic p instead of C. For some remarks, see the Supplements, §19. §19. I could go on. There are many topics about which we would want to know more, for exam.ple, invariants of linear groups and linear groups with integral coefficients. But the questions formulated here will suffice to show that a great deal of work remains to be done in the theory of group representations, even if we restrict ourselves to finite groups. If more mathematicians become interested and help to answer the questions, I have accomplished what I set out to do. SUPPLEMENTS §1. Introductory Remarks. lt is our aim to demonstrate that characters form a powerful tool for the study of finite groups. lt is essential that it is often possible to say quite a bit about the characters, if only very scanty information about the group is available. On the other hand, a great deal of information about the group is hidden in the characters. Thus, the interrelations between properties of characters and of group theoretical properties is of great interest. I have already said that I believe that for further progress combinatorial group theoretical methods will have to be used

209

REPRINT OF [82]

160

Lectures on Modern M athematics

together with methods from the theory of representations. At first glance, it may sometimes appear as if we try to let characters do all kinds of tricks just for the love of it. What we have in mind are the first steps of the program outlined here. There are indications that for the eventual success of the method, it will not only be the general principles which will matter, but also the ability to adapt ourselves to a particular situation. §2. N umerical Examples. lt may be instructive to mention an example in connection with Problem 1. I have chosen the case n = 60. The conditions (a), (b), (c) in Survey* §3 for the degrees x 1, x2, . . . can be satisfied in 65 ways. Actually, there are only 9 group algebras as can be seen by the methods developed in the later sections of the part on Survey. On the other hand, the number of (nonisomorphic) groups of order 60 is 13. If we are interested only in groups G for which G is its own commutator group G', exactly one of the Xi is 1. With this condition added, there are 6 systems satisfying (a), (b), (c). Only one of these, the system {1, 3, 3, 4, 5} corresponds to ~ group algebra, that of the icosahedral group. In order to show the mysterious nature of the degrees x1, x 2, . . . , I give another example, that of the Mathieu group M 24 of order 244,823,040. The characters of M 24 were obtained by Frobenius in 1904. There are 26 irreducible characters. The degrees are 1, 23, 45, 45, 231, 231, 252, 253, 483, 770, 770, 990, 990, 1035, 1035, 1035, 1265, 1771, 2024, 2277, 3312, 3520, 5313, 5544, 5796, 10395. There are all kinds of hidden relations, for instance, of the type discussed in finite Groups, §13. §3. Relations between the Characters and the M appings K --t K [mJ. As a first remark, we observe that if we know the characters of G, we can determine the orders c(CT) of the centralizers of the elements CT E K; we also write c(K) for c(CT). lt is obvious that c(K) divides c(K[ml) and this gives a necessary condition for the class K[mJ for given K. Moreover, if m is relatively prime to n = !GI, the values Xi(K[ml) are algebraically conjugate to Xi(K) in a definite manner and then K[ml is uniquely determined in this case. More generally, we can use the fact that if CT has order r and if x is a character of

* "Survey" refers to section headed SURVEY OF THE THEORY OF REPRESENTATIONS OF FINITE GROUPS.

210

FINITE GROUPS

Representations of Finite Groups

161

degree x, then x(K) is a sum of x rth roots of unity Ei aii.d x(Klml) is the sum of the The number r itself is a divisor of c(K). A result of a related nature is a theorem of Blichfeldt which gives a relationship between the irrationalities appearing in the values of the characters and the orders of the elements of G. A very short proof is given in a paper of the author, forthcoming in the Proceedings of the American Mathematical Society. I mention a rather special question which I cannot answer.

er.

Problem 41. Let p, q, r be three distinct primes and suppose that G has an irreduci'ble character x f or which there exist elements a, ß, 'Y of G such .that x(a), x(ß), x('Y) are irrational and x(a) E Q(V ±qr), x(ß) Q('V±pr), x('Y) E Q(V ±pq). Does G contain elements of order pqr?

E

If the answer should be affirmative, various generalizations could be tried. If the characters of G are known and if p is a given prime, we may be interested in finding out which classes K contain p-elements (for instance, as a :first step in attempting to determine the structure of a p-Sylow group of G). If Xi(K) is rational, a necessary condition is given by x/K) = Xi(l) (mod p). If Xi(K) is irrational, this has to be replaced by a congruence modulo a suitable prime ideal.

§4. Remarks on Problems 3 and 4, If the notation is as in Finite Groups §5, and if we write Ci for c(ui), the k numbers Ci are connected by the equation (1)

1

1

-+-+ C1 C2

+-Ck1 = 1

For uk = 1, we have Ck = n. Landau's method mentioned in connection with Problem 3 is based on a simple discussion of this Diophantine equation (1). Of course, (1) represents a condition which the groups Ci in Survey, (5) have to satisfy. If n > 1, none of the Ci can be 1. The case where some Ci is 2 can be discussed easily. Recently, W. Feit and J. G. Thompson [27] determined all G for which some Ci is 3. For the case that some Ci is 4, see M. Suzuki [60]. The answer to the following question seems to be unknown.

211

REPRINTOF[82]

162

Lectures on Modem Mathematics

Problem 42, Let p be a fixed prime. Can there exist injinitely many simple groups for which some C(u) is cyclic of order p?

Groups G in which all groups Ci in Finite Groups (5) are nilpotent have been called CN groups. They have been studied rather exhaustively by M. Suzuki [63, 64]. An earlier paper of W. Feit, M. Hall, Jr., and J. G. Thompson [25] should be mentioned here. Though the results have now been superseded it has played an important role in recent developments. Suzuki in his work also determined all simple groups (of even order) for which the groups Ci in Survey (5) are either 2-groups or of odd order. lt was in this connection that he discovered his new class of simple groups. §5. Remarks on Induced Characters. In many applications of character theory to investigation of group theoretical properties, Frobenius' relations between characters of groups and subgroups (Appendix, §6) play a fundamental role. A generalization of Frobenius' reciprocity theorem has been given by G. W. Mackey [44]. In certain cases quite definite information on the characters of G can be obtained easily on the basis of Frobenius' idea. This led M. Suzuki and the author to a discussion of the so-called exceptional characters. For a very simple proof see R. Brauer and H. S. Leonard [14]. Refined versions appear in Suzuki's papers and in W. Feit [24]. A most ingenious use of ideas of this general nature is of importance in W. Feit's and J. G. Thompson's proof of the solvability of the groups of odd orders. Since the intermediate results are geared to the special situation encountered there, it is impossible to give a taste of them here. §6. Information about Subgroups Available If the Characters of G Are Known. If His a subgroup of Gof index (G: H) = r the principal representation h - 1 of H induces a representation X of G of degree r. Actually, X is the permutation representation of G belonging to the subgroup H. The character x is easily seen to have the following properties (see, for instance, D. E. Littlewood [43]), Chap. 9, §3. (a) The degree r of x divides n = IGI. (b) The values of x are non-negative rational integers. (c) x(Klml) ~ x(K) for all conjugate classes of G and all exponents m.

212

FINITE GROUPS

Representations of Finite Groups

163

(d) The irreducible character Xi of G appears in x at most with the multiplicity Xi = Xi(l). In particular, the principal character of G appears exactly once in x. (e) For each class K of G, the number nx(K)/(rc(K)) is an integer. (f) If u EG has an order m which does not divide n/r, then x(u) = 0. If the irreducible characters x1, x2, . . . , Xk of G are known, we can form the reducible characters x of G which satisfy these conditions. There can exist a subgroup H of G of index r only if r is the degree of such a character x. The permutation represei:J.tation X belonging to a subgroup H is doubly transitive, if and only if x - 1 is irreducible. lt has already been mentioned that if the characters of G are known, we can find the orders c(u) of the centralizers of elements u EG. lt is also not difficult to obtain the order of the normalizer of the cyclic group generated by u. In some special cases, methods are available which allow us to obtain new subgroups rrom given ones. For instance, it may happen that we know two subgroups A, B of G and D = A ('\ B. If x is an irreducible character of G, we can find the multiplicity a with which the principal character of A appears in A. Likewise, if ß and ll have analogous significance for B and D respectively, these integers can be determined. If a + ß > ll, then A and B generate a proper subgroup H of G. For instance, in his paper on simple groups of order 244, 823, 040, R. G. Stanton [58] succeeded in constructing the character tables of these groups. There exists exactly one irreducible character x of degree 23. There are normalizers A and B of cyclic groups of orders 253 and 55 respectively, and we may assume that D = A ('\ B has order 11. Here, a ;= 1, ß = ll = 3 and the method applies. lt can now be seen that H has index 24 in G. Then G has a permutation representation of degree 24, and it is not very difficult to show that G must be isomorphic to the Mathieu group M 24 • We have here a case where the character table of a group, actually a very small part of it, determines a group uniquely. There exist other methods to construct subgroups by means of characters, but they all are of a rather special nature, and we shall not discuss them. As another example of an application of characters in a numerical case, a paper of E. T; Parker [46] may be mentioned. lt seems to

x\

213

REPRINT OF [82]

164

Lectures qn Modern M athematws

have been conjectured that every (noncyclic) simple group can be represented as a multiply transitive permutation group. Parker showed that this is not true for the simple group of order 25920. If the conjecture stated in Problem 23 is true, this would allow us to decide whether the p-Sylow group P of G is Abelian. If we know the p-classes K, we have a necessary condition for this question: c(K) for these classes must be divisible by the full power of p dividing n. lt does not seem tobe known whether this condition is sufficient for P to be Abelian. §7. Remarks Concerning Automorphisms of Groups. The automorphism r of G which leave every irreducible character Xi of G fixed form a normal subgroup B(G) of the automorphism group A(G) of G. Thus, A(G)/B(G) appears as a permutation group on the set {x1, x2, . . . , Xk}. If p is a fixed prime, the p-blocks of G are permuted among themselves. In many cases, this can be used to study A(G)/B(G). The elements of B(G) can also be characterized as the automorphisms r of G which map every conjugate class of G onto itself. Of course, all inner automorphisms of G belong to B(G). §8. Some Comments on Survey, §7. The number r of (nonequivalent) irreducible representations of the group G in the field n of characteristic O can also be described in more direct group theoretical terms. Let e denote a primitive nth root of unity and let g denote the Galois group of ü(e) over n. Every u E g maps e on a p'ower e•Cu) where s(u) is a rational integer prime to n. For every u E g, we can form the permutation K - K[s(u)J of the set {K 1, • • • , Kk} and g is then represented as a permutation group II on this set. The number r is equal to the number of transivity classes of II. This result probably was known to I. Schur. lt is a corollary of a simple lemma of the author [6]. lt was obtained independently by S. D. Berman [2]. There are extensions to the case of fields of characteristic p. For references to work of P. Roquette, E. Witt, S. D. Berman, and the author on Schur indices and related questions, see the paper of L. Solomon [55] quoted in Finite Groups, §7. Solomon's method provides a good approach to much of this work. Another result of interest was given by A. Speiser [56]; see also I. Schur [53]. Speiser proved (with n C C) that if the values of

214

FINITE GROUPS

Representations of Finite Groups

165

x; are real, and if the degree xi is odd, the corresponding Schur index is 1. For real x; of even degree xi, mp is 1 or, 2, see R. Brauer [4] in conjunction with later results of H. Hasse [36] on the exponents of central simple algebras. There are connections with questions concerning invariants. As to the behavior of the irreducible representation X; of Gin C in the field R, it was shown by G. Frobenius and I. Schur [31] that the number

can have only one the values 1, -1, 0. We have ei = 1, if X/"can be written in R; we have e; = -1, if this is, not possible, but if Xi is real. Finally, e; = 0, if Xi is nonreal. If the notation is as in Survey, §7, we have e; = 1 if the division algebra 11p is R, and Ei = -1 if 11p is the quaternion algebra. Finally, Ei = 0, if 11p = C. For any u E G, the number

,.

=

N(u)

l

EiXi(u)

i=l

represents the number of solutions of the equation ~2

=

u

with ~EG

While it is easy to characterize the number of characters Xi with = 1 or Ei = -1, no direct characterization of the number of Xi with e; = 1 alone is known. This is the point of Problem 14. ei

§9. Some Remarks on Problems 2 and 2*. lt follows from the remark on the centers ZP of the algebras Ap in Survey, §7(7) that nonisomorphic Abelian groups have nonisomorphic group algebras over the field Q of rational numbers. The case of Abelian groups has been studied further by S. Perlis and G. Walker [47] and W. E. Deskins [20]. The two (nonisomorphic) non-Abelian groups of order p 3 have isomorphic group algebras over every field n of characteristic O if p is odd. lt follows from Jennings' results (Survey, §8) that for fields Q of· characteristic p, the algebras are not isomorphic. For p = 2, this is already true for the field n = Q.

215

REPRINT OF [82]

166

Lectures on Modern M athematics

Jennings' results allow us to define many numerical invariants of group algebras of p-groups over the prime fields of characteristic p by counting the number of elements of a power of the radical which satisfy given conditions. This suggests that it may be much easier to study Problem 2 for this particular case. §10. Groups with Cyclic p-Sylow Group. An interesting theorem of D. G. Higman [38] gives a class of groups of the kind required in Problem 16. A finite group G has a cyclic p-Sylow group if and only if the degrees of all indecomposable representations of the group algebra r over a field Q of characteristic p have degrees which lie below a fixed bound. §11. A Remark on the Dimension of the Radical of the Modular Group Algebra. If we know the degrees fj of the modular irreducible representations F j in Finite Groups, §9, we can easily find the

dimension h of the radical of r.

Indeed, h = n -

1J/.

If we

j

could characterize h directly as required in Problem 15, this would give an interesting connection. As an example, we mention the groups G = L 2 (p). Here, the h are known and we find h = ¼(p + 1)(2p 2 - 5p) for p > 2. §12. Blocks. For properties of blocks, see R. Brauer [7, 9]; a continuation is being prepared for publication. Modifications and simplifications are given in M. Osima [45], K. Iizuka [39], and A. Rosenberg [50]. In this connection, I wish to mention the interesting investigations of J. A. Green [32, 33]. §13. Blocks of Defect 0. Blocks of defect O have properties much simpler than those encountered in the general case. Each block B of defect O consists of one ordinary irreducible character Xi and one modular irreducible character which happens to be the restriction Xio of Xi to the set of p-regular elements of G. An ordinary irreducible character Xi belongs to a block of defect 0, if and only if its degree Xi is divisible by the full power of p dividing n. An equivalent condition is that Xi vanishes for all p-singular elements. If the representation Xi belonging to Xi is a constituent of a reducible representation, it is a direct summand. If the characteristic p of Q does not divide the group order n, all p-blocks have defect 0. lt follows from the results stated here

216

FINITE GROUPS

Representations of Finite Groups

167

that there is really no difference between the ordinary and the modular theory in this case. This has first been proved by A. Speiser. As to Problem 19, the situation is of a rather mysterious nature. For instance, if Gis the symmetric group ®m of all permutations of m objects and if p = 2, then G has blocks of defect 0, if and only if m = r(r + 1)/2 with r = 1, 2, 3 . . . . In this case we have exactly one such block. We remark that for the symmetric groups the blocks are known for all primes p. The result in question has been conjectured by T. Nakayama and proved by G. de B. Robinson and the author [49]. In the case of a solvable group G, sufficient conditions for the existence of blocks of defect O have been given by N. Ito [40]. For a large class of group orders n including all odd n, these conditions are necessary. For sufficient conditions of a different nature for arbitrary finite G of even order, see R. Brauer and K. A. Fowler [13]. §14. Comments on Problems 20-23. The answer to Problem 20 is affirmative for d ~ 2. For d ~ 3 the number of ordinary irreducible characters in a p-block of defect d is less than p 2d- 2 (R. Brauer and W. Feit [12]). For blocks B of defect 1, the number of Xi E B is of the form T + (p - 1)/T where T is a divisor of p - 1. In particular, this number is at least 2 V p - 1. If p but not p 2 divides n, the principal p-block (that is, the block Bo which contains the principal character xo = 1 of G) has defect 1. lt follows that if G has class number k, then p ~ 1 + (k 2 /4). This is a partial result in direction of Problem 3 for primes p with vp(n) = 1. An answer to Problem 21 might lead to an answer to Problem 3 in a similar fashion. Problem 22 can also be answered in the affirmative for blocks of defect 1. In fact, much more precise results on the values of the decomposition: numbers and the Cartan invariants are known in this case. If the answer to Problem 23 is affirmative, this would yield an answer to the last part of Problem 12. In the case of p-solvable groups, it has been shown by P. Fong [30] that if the center of the defect group D of a block B has index pc in D, then all Xi E B have height at most c. In particular, if D is Abelian, all heights are 0. Conversely, if the principal block of G contains only characters

217

REPRINT OF [82]

168

Lectures on Modern M athematics

Xi of height O, then the p-Sylow group of Gis Abelian. Fong has also solved Problem 22 for p-solvable G. For arbitrary finite G, it can be shown that blocks with cyclic defect group contain only Xi of height O and the answer to the question raised in Problem 20 is affirmative in this case. W e ask:

Problem 43. Can a theory be developed for blocks with cyclic defect group D which f or ID! = p yields the known results f or defect 1? §15. Remarks Concerning Types of Blocks. There are cases where rather complete information is available about the possible types of blocks. The discussion is somewhat easier for p = 2, since additional methods are available in this case. As an example, we mention the 2-groups P of order 2 2m+1 with m ~ 2 generated by three elements 0-1, 0-2, T such that

Each such P occurs as 2-Sylow group of infinitely many simple groups. Let G be a finite group with Pas its 2-Sylow subgroup such that G does not have a normal subgroup of index 2. Then the principal block B 0 of G can be shown to consist of ½(2 2m-l 3 · 2m+l 1) ordinary characters Xi· There exists an integer

+

f

+

= 1 + 2m(mod 2m+l)

but not necessarily positiv p > (3m - 1)/2 (H. F. Taun [66]). Very recently, S. Hayden aud the author have shown that the case (3m - 1)/2 ~ p > (6m - 1)/5 is impossible. The case m = 2 in Problem 40 has been treated by L. E. Dickson in his book on linear groups [21]. Using methods related to those discussed in Survey, §16, D. Bloom has recently handled the case m = 3. N othing seems to be known for higher dimensions. APPENDIX

§1. While a definition of the elements of the group algebra as formal sums "I;a0g as in Finite Groups (1) is hardly satisfactory from a logical point of view, the reader will have no difficulty in constructing an associative algebra r over the given field n with a basis whose elements form a group Go under multiplication isomorphic with the given group G. If we identify corresponding elements of the groups G0 and G, the elements of r appear in the form Finite Groups (1). §2. We speak of a matrix representation X of a group Gin the field n, if with every g EG, there is associated a nonsingular square matrix X(g) of a fixed degree x such that the mapping g - X(g) is a homomorphism, that is, that

X(g1)X(g2) =

X(g1g2)

for all g1, g2 EG. We should keep in mind that square matrices really are the analytic expressions for linear transformations of a vector space V into itself, if a fixed coordinate system (basis) has been chosen. Two representations X and X* of the same "degree" x are equivalent if there exists a fixed nonsingular matrix P of degree x such that p- 1X(g)P = X*(g) for all g EG. This means that X(g) and X*(g) describe the same linear transformation of V in two different coordinate systems. A representation X is irreducible if there does not exist a subspace W of V with W =l= (0), W =l= V such that W is mapped into itself by all X(g) with g EG. By a matrix representation of an algebra r with unit element 1, we mean an algebra homomorphism of r onto a set of square matrices such that 1 corresponds to the unit matrix I. Every

220

FINITE GROUPS

Representations of Finite Groups

171

representation X of G can be extended to a representation of the group algebra r by linearity, and every representation of r is obtained in this fashion. The radical N of an algebra r is the set of those elements 'Y Er represented by the zero matrix in every irreducible representation of r. Then N is a two-sided ideal of r. If it consists only of 0, we say that r is semisimple. §3. The center Z of an algebra r consists of those elements r of r which commute with all 'Y E r, that is, for which t"f = 'Yr for all 'Y Er. Of course, Z is a subalgebra of r. In the case of the group algebra r of G, we can easily construct a basis of Z. Let K 1 , K2, . . . , Kk denote the conjugate classes of G. For each Ki, let {Kd denote the sum of the elements of Ki taken of course as element of r. lt is seen immediately that {Ki}, {K 2 }, • • • , {Kd forma basis of Z. We have formulas k

(1)

{Ka} · {K1i\ =

l

aaß7

{K 7 },

(a,

ß = 1, 2, . . . , k)

-y=l

where the aaß-y are non-negative integers. (lt will be clear what is meant by this in the case that n has characteristic p =!= 0.) If g7 is a fixed element of K 7 , then aaß-y is simply the number of ordered pairs (t 7/) of elements ~ E Ka, 1/ E Kß such that ~7/ = g7 • Because of the connection with the conjugate classes of G, the algebra Z is called the class algebra of G over n. lts dimension is the class number k of G. The basis {Ki}, {K 2 }, • • • , {K,.} will be called the natural basis of the class algebra of G. §4. Assume that the underlying field n has characteristic 0. If X is a representation of G in n and if g EG, denote by x(g) the trace of the matrix X(g), that is, the sum of the coefficients in the main diagonal of X (g). The resulting function x defined on G with values in n is called the character of X. Since the trace of a matrix depends only on the linear transformation represented by the matrix, equivalent representations have the same character. Moreover, each character x is a class function, that is, the value of x is constant on each conjugate class K of G. Occasionally, this constant value will be denoted by x(K). The product of two characters is again a character. By a generalized character, we mean the difference of two characters. Then the generalized characters form a ring.

221

REPRINT OF [82]

171

Lectures on Modern Mathematics

W e speak of an irreducible character if the corresponding representation is irreducible. One of these irreducible characters is the constant 1. This particular character is called the principal character. The generalized characters are the linear combinations (2)

of the irreducible characters x1, x2, . . . with integral coefficients ai. We have a character x in (2) if all ai here are non-negative and not all are 0. Assume now that n is the field C of complex numbers. We have exactly k irreducible characters x1, x2, . . . , Xk where k is the class number of G. If O and '17 are two complex valued functions defined on the group G of finite order n, we define the inner product 1 '\"'

-

(0, 17) = ;;: L, O(g)'ll(g) D

Then, x1, x2,

, Xk

form an orthonormal system:

(3)

lt follows that the ai in (2) are given by ai = (x, Xi). Moreover, (x, x) = ~ai 2There is an equivalent way in which we can write (3). If u EG, we denote its centralizer in G by C(u), that is, C(u) consists of all r EG with ur = ru. The order \C(u)\ will be denoted by c(u). Then, for u, 0-1 E G, (4)

where the, upper case applies, if and only if u and 0-1 are conjugate inG. By the degree x of a character, we mean the degree x of the matrices of the corresponding representation X. Then x = x(l). The degrees x1, x2, . . . , xk of the k irreducible characters all divide n. The values of each character x lie in the field A of all algebraic numbers, and nothing is changed if we replace the field C by A. On the other hand, if n is any algebraically closed field of charac-

222

FINITE GROUPS

Representations of Finite Groups

173

teristic 0, then A can be identified with a subfield of n. There is no change, if we replace A by n. §5. Assume further that n is an algebraically closed field of characteristic 0. The class algebra Z of G is a direct sum Z = ü1 EB · · • EB ük

(see Finite Groups, §3, (3)) where each ni is a field isomorphic with n. Consider a fixed i, 1 ~ i ~ k. U sing a fixed isomorphism, identify Üi with n. Let Wi denote the mapping which maps the arbitrary element E Z on the corresponding element of Üi, Then, Wi is clearly an algebra homomorphism of Z onto n. Moreover, w 1, w2, • • . , wk are distinct, and every homomorphism of Z onto n is obtained in this manner. Each Wi corresponds to an irreducible character Xi of G. Explicitly, this correspondence is given by

s

(5)

wi({K;..}) = nxi(crx)/(xi(l)c(crx))

where er;,, is an element of K;,,. §6. If His a subgroup of G, it is obvious that the restriction of a character x of G to His a character of H. Thus, if x1, x2, ... , Xk are the irreducible characters of G and 1/; 1, 1/; 2, • • • , fm are the irreducible characters of H, we have formulas (as in Finite Groups §6 (6)) m

(6)

Xi\H

=

l

aiiY1i

j=l

with non-negative rational integers aii· If 1/; is a character of H, Frobenius constructed a character 1/;* of G by means of 1/;. This is the character 1/;* of G induced by 1/;. lt is given by 1/;*(g) = ___!__ \'' i/;(xgx- 1) JHJ L.,

with the sum extending over those x EG for which xgx- 1 EH. For 1/; = Y1i, k

(7)

1/;/ =

l

Xiaij

i=l

where the aii are the same as in (6). Actually, if Y is the representation of H with the character 1/;,

223

REPRINT OF [82]

174

Leetures on Modern M athematics

Frobenius gave an explicit construction of the representation Y* of G with the character ,j;*. This method plays an important role in all applications of the theory. REFERENCES 1. E. Artin, Comm. Pure Appl. Math. 8 (1955) 355-365; 455-472. 2. S. D. Berman, Dokl. Akad. Nauk SSSR (N. S.) 86 (1952) 885--888 UniVf rsity of Chicago Press. 3. H. F. Blichfeldt, Finite eollineation groups, Chicago (1917). 4. R. Brauer, Sitzber. Preuss. Akad. Wiss (1926) 410-416. 5. R. Brauer, Amer. J. Math. 64 (1941) 401-420. 6. R. Brauer, Ann. of Math. 42 (1941) 926-935. 7. R. Brauer, Math. Zeit. 63 (1956) 406-444. 8. R. Brauer, Seminaire P. Dubreil, M. L. Dubreil et C. Pisot (1958-1959) 6.1-6.16. 9. R. Brauer, Math. Zeit. 72 (1959) 25-46. 10. R. Brauer, Areh. Math. 13 (1960) 55-60. 11. R. Brauer, Proe. Amer. Math. Soe., in press. 12. R. Brauer and W. Feit, Proe. Nat. Aead. Sei. U.S.A. 46 (1959) 361-365. 13. R. Brauer and K. A. Fowler, Ann. of Math. 62 (1955) 565-583. 14. R. Brauer and H. S. Leonard, Can. J. Math. 14 (1962) 436-450. 15. R. Brauer and W. F. Reynolds, Ann. of Math. 68 (1958) 713-720. 16. R. Brauer and M. Suzuki, Proe. Nat. Aead. Sei. U.S.A. 46 (1959) 1757-1759. 17. W. Burnside, The theory of groups of finite order, Cambridge (1911) Cambridge University Press, 2nd edition. 18. C. Chevalley, Tohoku Math. J. (2) 7 (1955) 14-66. 19. C. W. Curtis and I. Reiner, Representation theory of finite groups and assoeiative algebras, New York (1962) Wiley. 20. W. E. Deskins, Duke Math. J. 23 (1956) 35-40. 21. L. E. Dickson, Linear groups, Leipzig (1901) Teubner. 22. J. Dieudonne, Hermann (1948) Paris. 23. J. Dieudonne, Ergebnisse der Mathematik und ihrer Grenzgebiete (N. F.), Heft 5. Springer-Verlag. Berlin-Göttingen-Heidelberg, 1955. 24. W. Feit, Proe. Symp. in Pure Math. 6 (1962) 69-70. 25. W. Feit, M. Hall, Jr., and J. G. Thompson, Math. Zeit. 74 (1960) 1-17. 26. W. Feit and J. G. Thompson, Paeifie J. Math. 11 (1961) 1257-1262. 27. W. Feit and J. G. Thompson, Nagoya Math. J. 21 (1962) 185-197. 28. W. Feit and J. G. Thompson, Proe. Nat. Aead. Sei. U.S.A. 48 (1962) 968-970. 29. W. Feit and J. G. Thompson, Paeifie J. Math., in press. 30. P. Fong, Trans. Amer. Math. Soe. 98 (1961) 263-284. 31. G. Frobenius and I. Schur, Sitzber. Preuss. Akad. Wiss (1906) 186-208. 32. J. A. Green, Math. Zeit. 70 (1959) 430-445. 33. J. A. Green, Math. Zeit. 79 (1962) 100-115.

224

FINITE GROUPS

Representations of Finite Groups 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72.

175

M. Hall, Jr., The theory of groups, New York (1959) MacMillan. P. Hall and G. Higman, Proc. London Math. Soc. (3)7 (1956) 1-42. H. Hasse, J. Reine Angew. Math. 167 (1931) 399-404. H. Hasse, Math. Ann. 107 (1933) 731-760. D. G. Higman, Duke Math. J. 21 (1954) 377-381. K. lizuka, Math. Zeit. 76 (1960-1961) 299-304. N. Ito, Nagoya J. Math. 3 (1951) 31-48. S. A. Jennings, Trans. Amer. Math. Soc. 60 (1941) 175-185. E. Landau, Math. Ann. 66 (1903) 671-676. D. E. Littlewood, The theory of group characters, New York (1940) Oxford Univ. Press, Chap. 9, §3. G. W. Mackey, Amer. J. Math. 73 (1951) 576-592. M. Osima, Math. J. Okayama Univ. 4 (1955) 175-188. E. T. Parker, Proc. Amer. Math. Soc. 6 (1954) 606-611. S. Perlis and G. Walker, Trans. Amer. Math. Soc., 68 (1950) 420-426. R. Ree, Amer. J. Math. 83 (1961) 401-420; 432-462. G. de~- Robinson and R. Brauer, Trans.Roy. Soc. Can. Sect. !_II 41 (1947) Springer, 3rd 709-716. A. Rosenberg, Math. Zeit. 76 (1961) 209-216. I. Schur, Sitzber. Preuss. Akad. Wiss. (1905) 406-432. L. Schur, Sitzber. Preuss. Akad. Wiss. (1905) 77-91. I. Schur, Math. Zeit. 6 (1919) 6-10. C. L. Siegel, Math. Zeit. 10 (1921) 173-213, Theor. 7. L. Solomon, Jap. J. Math. 13 (1961) 144-164. A. Speiser, Math. Zeit. 6 (1919) 1-6. A. Speiser, Die Theorie der Gruppen endlicher Ordnung, Berlin (1937) Springer, 3rd edition. R. G. Stanton, Can. J. Math. 3 (1951) 164-174. R. Steinberg, Can. J. Math. 8 (1956) 580-591; 9 (1957) 347-351. M. Suzuki, Ill. J. Math. 3 (1959) 255-271. M. Suzuki, Proc. Symp. in Pure Math. 1 (1959) 88-100. M. Suzuki, Proc. Nat. Acad. Sei. U.S.A. 46 (1960) 868-870. M. Suzuki, Trans. Amer. Math. Soc. 99 (1961) 425-470. M. Suzuki, Ann. of Math. 76 (1962) 105-145. T. Tsuzuku, N. Ito, A Mizutani, and T. Nakayama, mineographed notes tobe published by Nagoya University. H. F.Tuan, Ann. of Math. 46 (1944) 110-140. B. L. van der Waerden, Modern algebra, New York (1949) Ungar, Vol. 2. H. N. Ward, Bull. Amer. Math. Soc. 69 (1962) 113-114. H. Weyl, Symmetry, Princeton, N.J. (1952) Princeton University Press. E. Witt, Abh. Math. Sem. Univ. Hamburg 12 (1937-1938) 256-264. E. Witt, J. Reine Angew, Math. 190 (1952) 231~245. 0. Zariski and P. Samuel, Commutative algebra, New York (1960) Van Nostrand, Vol. II.

225

REPRINT OF [82]

On quotient groups of finite groups By RICHARD BRAUER

§ 1. Introduction Let G be a group and H a subgroup. Every normal subgroup G0 of G gives rise to a normal subgroup

(1) of H. If, in addition, (2) we have isomorphic quotient groups G/G 0 ~H/H0 • Conversely, given a subgroup H of G and a normal subgroup H 0 of H, we can ask: Does there exist a normal subgroup G0 of G such that (1) and (2) hold? Theorems guaranteeing the existence of such normal subgroups G0 under suitable assumptions can be used to obtain criteria for simplicity of groups. The problem has been studied by a number of authors. I mention here only an interesting paper of WIELANDT [9]. lt is my aim to give some new contributions to this question. As in the preceding work, the group G is always supposed to be finite. The main result (Theorem 1, § 3) is somewhat complicated. lt is proved in § 5. A partial converse is given in §7. In § 8, it is shown that Wielandt's theorems [9] can be deduced from Theorem 1. § 9 contains a set of necessary and sufficient conditions for the existence of a normal n-complement in a finite group. In § 10 a further corollary of Theorem 1 is obtained, and in § 11 a generalization of the Theorem of P. HALL and GRÜN, (cf. M. HALL [4], Theorem 14.4.6) is deduced from it. In order to have a convenient tetrninology, we give a definition. Definition. Let G be a group, H a subgroup, H 0 a normal subgroup of H, H 0 0, ß'I' is an irreducible character of G. Now, (11) shows that every irreducible character V' of H/H0 can be extended to a character ß'I' of G. The same then is true for every character 1P of H/H0 . Choose 1P as the character of the regular representation of H/H0 • By (10),

ß'l'J G0 =ß,p(1),

ß'/i(G;) =O

for

i> 0.

Hence G0 is the kernel of the representation of G with the character ß 'I'. This proves that G0 4. Then (2.3)

c (J) = I -~ [ = (q - 1)z q (q + 1)/1 = 2" u 2 l v q.

We collect here some results concerning the group ~ which will be used repeatedly. The proofs are easy and well known and we shall omit most of the details. Denote by p a homomorphism of GL(2, q) onto ~ with the kernel 2. Let ~ denote a primitive root in the Galois field GL(q 2) with q2 elements. Then 7o = 7 q+l is a primitive root in GF(q). 1. The center 3 ( ~ ) is a cyclic group of order (q-1)/l=2u. If we set J =

p ( - I ) , U=p(v~ I), 11--(U) we have (2.4)

3 ( ~ ) = ( J ) x 11= ( J U )

with 111[=u. 2. We introduce the involutions

Then ~ ) = ( J , J r ) is elementary abelian of o r d e r 4 and contains the three involutions J, J1, and J2 =JYa. Since T (under conjugation) maps (2.6)

J--* J,

Jx~ Jz ~ J1,

126

RICHARDBRAUER:

the group 131 =(13, T ) is dihedral of order 8 with the central involution J. The centralizer ~(13) of 13 (in 9 or in (5) consists~of~the p-images of the diagonal matrices. We may then set (2.7)

~(13) = I3 • lI x

where ~ is cyclic of order u l. For instance,

can be used as generator of ~ . Then (2.8)

uT= U,

w T = U W-1.

For any ~ff(~3) which does not belong to 3 ( 9 ) , we have (2.9)

ff~(~)=ff(13).

We also note (2.10)

~ (131) = ~ ((13, T)) = 3 (9).

3. There exist elements MeGL(2, q) with the characteristic roots ? and ?q and then M has order q + 1 over 3(GL(2, q)). It follows that R = p ( M ) has order q + 1 over 3(9)- The order of R itself is (q2_ 1)/l and we may set

R=S VU k where S, V, U k are powers of R of orders 2 " - 1 v, u respectively. After replacing R by a conjugate, we may assume that S lies in an S2-group ~ 1 3 t of 9Here, $ has order 2" and ( S ) is a cyclic subgroup of order 2"-1. Then ( S ) c~ 131 has necessarily order 4. This implies that 131 contains the element F = S 2"-3. Without restriction, we may assume that

F=S2"-3=TJ1

(2.11)

and this implies that S 2"-= =F 2 =J. The element J1 of ~ must transform S into a power S' 4=S of S. Since the two characteristic roots of a reciprocal image p-1(S) of S are conjugate over GF(q), we can only have S' = S q. It follows that ~ = ( S , J1) is quasi-dihedral,

$2"-1=1,

J2_-_l,

J['~ S J I = J S -1

with J = S 2"-2. We also note that (2.12)

~Rm((S)) = 9t~((S)) = (r

J1)-

4. The reciprocal images in GL(2, q) of the elements of G(F) are the matrices of the form

On finite Desarguesian planes. II

127

Because of q--- - l(mod 4), they are the non-zero elements of a field isomorphic to GL(q2). This implies c(F)=(qZ_ 1)/l. It follows easily that (2.13) 5. For (2.14)

E(S")=E(F)=(S)x(V)xlI a~GF(q), set X~=p(la~), X2=p

(for

S"4=I,J).

(10 1).

The elements X1 with arbitrary a form a subgroup ~1 of order q and the elements X2 form a subgroup 1;2 =3; r. Clearly, J1 inverts X1 and X 2. It is seen without difficulty that if ~ is a po-element of ~ which is inverted by J1, then ~ i or r We have % ( ~ i ) = ( ~ 1 , 3(-~)). The group g(~3) normalizes 3~1 acting transitively on 3~1 -{1}. As is well known, (2.15) ~ = (r x1, ah). w 3. Summary of the results of [31, IH, w 8 Assume that (5 is a group of finite order g with quasi-dihedral S2-group ~, with the notation chosen as in w2. We shall assume that F and S J1 are conjugate in (5. We summarize the results of [3], III, w8; the integer denoted there by m will be denoted by - q . We do not know yet that this is the same q as above. The principal 2-block Bo =Bo((5; 2) of (5 consists of 4 + 2 "-2 irreducible characters. Four of these, denoted by Zo---1, )~1, Z2, Z3 have odd degrees xi=zi(1). There exist signs 65, 62, fi3 such that (3.1)

l+61xl+82Xz+83xa=0.

Two of 81, 82, 83 are - 1 and one is +1. The remaining characters of B o can be denoted by Z4, Z(") with 1 and hence ~ l _ ~ ( J ) , a contradiction. As remarked in w3, the elements F and S J1 are conjugate in ~ and hence in 15. The following result is now evident. (4B) The results of w 3 apply to 15 and we have Case L The irreducible character q~ of ~ ( J ) = ~ in (3.3) has degree q - l , and e = - l . Here q is the same prime power q =Pro which appears in GL(2, q). As a corollary, we note (4C) I f p is an odd prime dividing q - 1 but not u, then the irreducible character q, of ~(J) has defect 0 for p. (4D) There exists an element Ye15 which maps (4.1)

Y: J ~ Jt-* Je--* J.

Proof. Since J1 is conjugate to J in 15, a Sylow group argument shows that there exist ~e15 with J f = J such that ~ 3 r Since any elementary abelian subgroup of order 4 of ~ is conjugate to ~ in ~, we may assume that ~r Hence J and J~ are conjugate in gt(~3). Since T in (2.6) also belongs to 9I(~3), the group 9l(~)/~(~) is isomorphic with the group of all permutations of the three symbols J, J~, J2. This shows the existence of an element g which satisfies (4.1). (4E) I f two elements Zi and Z 2 of 3 ( ~ ( J ) ) are conjugate in t5, they are equal. Indeed, since ~ belongs to ~(ZI) and ~(Z2), a Sylow group argument shows that Z 1 and Z 2 are conjugate in 91(~). As in the proof of (4A), we see that Z i and Z 2 are conjugate in ~(J), and since they lie in the center, they are equal. We now come to the application of the results of w3. On substituting (2.3) in (3.5) and using (2.7), we have (4F) The order g of 15 has the form (4.2)

g = ( q _ 1)2 q3(q + 1)3 #/l

where the rational number # is given by (4.3)

#=(1-1)

xl(x 1+61)/(x~+6~ q ) 2 = ( l + l ) x z ( x 2 + f i z ) / ( x z - f z

q)Z.

130

RICHARDBRAUER:

The remainder of this section will be occupied by the proof of the following proposition. (4G) In the proof of (1A), it suffices to assume that 3 ( t 5 ) = 1 . If this assumption is satisfied, we have (4.4)

for z~3(r

ff(Z)=ff(J)

z , 1.

The proof of the first statement is not very difficult. Suppose that 15 satisfies the assumptions (I), (II) of (1A). Since 3 (15) -- ~ (J), we have 3(15)_~3(~(J)). By (4D), J~3(15) and it follows from (2.4) that 3(15)_clI. Set 15=15/3(15) The centralizer of the involution J = J 3 (15) of @ can be seen to be ~ ( J ) / 3 (15). It is now clear that 15 satisfies again the assumptions of (1A), with l replaced by l [3(15) F=7. Suppose that (1A) has been proved for groups of order smaller than g. If 3(15)+1, then (1A) holds for Ig. Since 7+1, we must have ~ PSL(3, q) and q - l ( m o d 3). Then 2=3 and hence I=1, 13(15)1 =3. It follows from the results of STEINBERG[8], w that 15" SL (3, q) or that 15" P SL (3, q) x 3 with ]3 ] = 3. In the latter case, the assumption (I) is only satisfied when q ~ 1 (mod 9). It is now clear that 15 has a form given in (1A). It will therefore be sufficient to assume that 3 (15)= 1. Before coming to the proof of the second part of (4G), we prove a lemma. (4H) In (4F), we have (4.5)

5 (5q+l)(q_l) 5q. (l-1)@~(5q+1)/q)2

µ.,v

1

x,(ytll)x1'Y~2>)x;(7rp)/x;

i j

with x1 = 5(;(1). This holds for allp-regular p E G:{Tr). Use I, (5.3) to express Xi( TTP) by means of the members cf,,,_1r of basic sets cf,b for the various p-blocks b of G:{Tr) and the corresponding decomposition numbers df,,_ of G. Likewise, express X;(7rp) by the cf,,/ and the decomposition numbers a;"' of ü. Since the resulting formula holds for all p-regular p E G:a{Tr) = G:o(Tr) and since

330

FINITE GROUPS

228

BRAUER

the cf,"''1T are linearly independent, cf,,,:' appears with the same coefficient on both sides. ,-This yields

g

!,a:. xi(y1)xiY2)fxi

= cG(yl)cG(y2)i ia;;.h}1>h}2>/x;

(2.5)

i

i

where we introduced the abbreviations

h}1> =

k x;(y~ >)/c0 (y~u), hJ2> = k x;(Y;2)1ec:,(y,) 1

(2.6)

µ

with µ, and v ranging over the same values as in (2.4). This holds for each ex. If cf,,.'1T belongs to the block b of li:(71'), then df,. = 0, except when Xi belongs to B = bG, cf. I, (2.6). Likewise, a'[.,. = 0, except when Xi belongs to the block 11 = bö of G. Hence it suffices to let Xi range over the irreducible characters in Band X; over the irreducible characters in 11. W e have now shown: THEOREM (2A). Let G be a group of even order g. Let 1T be a p-element of G for some prime p dividing g and let G be a subgroup of G with G :2 li:*(71'). If y 1 aud y 2 are two involutions of G, then

g 2'df„xiY1)x.(Y2)/xi i

= c(y1)c(y2)i

k J;;,hJllhJ2>/xi ;

f or each member cf,,.'1T of a basic set cf,b of a p-block b of Ir(71') and the corresponding decomposition numbers d'[.,., J;,. of G and G respectively. On the left, i ranges over the values for which Xi E B = bG and on the right, j over the values for which Xi E 11 = bö. The rational numbers h?>, M2> are deftned in (2.6).

Let B be a fixed p-block of G. Let p be a p-regular element of li:(71'). Multiply our formula with cf,,,."(p) and add over all cf,,," Ecpb for all p-blocks b of li:(71') with bG = B. Then for 11 = bö, we have ßG = bG = B. Conversely, if 11 is a p-block of G with ßG = B, and if not each Xi E 11 vanishes for all elements 1TP with p-regular p E li:(71'), then by I (2.6), there exist blocks b of li:(71') with bö = fj and then bG = B. We obtain: THEOREM (2B). Let G, G, 71', y 1 , y 2 have the same signiftcance as in (2A). If Bis aftxed p-block of G, then,for all p-regular elements p of li:(71'),

g kXh1)Xi(y2)Xi(1Tp)fx.

= c(y1)c(y2)i2'h?>h}2>x;(1Tp)/x;, i

i

where on the left, Xi ranges over all irreducible characters in B and, on the right, ranges over all irreducible characters of G which belong ·to p-blocks 11 of 0 with 11° = B.

Xi

331

REPRINT OF [92]

229

BLOCKS OF CHARACTERS III

III.

APPLICATION OF THE BASIC FORMULAS

The following results are obtained by a somewhat crude application of the preceding results. THEOREM (3A). Let G be a group of finite even order g, let y 1 and y 2 be two involutions of G, let 7T be a p-element of G for some prime p, and let O ~ l, at least one of the following two possibilities occurs: 1

Case I.

1

There exists an approximate formula for g:

(3.1) Here, w depends only on O and the intersections

Gn

ccl0 (Yi), 0

n

cclo(y2).

Case II. There exists anormal subgroup N of G with Sl"'G k N C G such that

(3.2) Here, pn is the exact power of p dividing g and pm is the exact power of p dividing 1 ,.(G, 7rp) = ">,.(G::c(7T), p). i

In giving estimates for w, use 15(;(7rp) 1 ,:(X;. We obtain THEOREM (3E). Let G, 7T, y 1 , y 2 , G and A have the same significance as in (3A). If p is a p-regular element of~ we have at least one of the following two possibilities:

The approximate formula (3.1) holds with w depending on G, w < W2l~ 12 where 11 , l2 have the same significance as in (3B). For p = 2, we have w < 1. Case I.

G n cclc(Y1 ), G n cclc(Y 2 ) and p. Moreover, Case II.

There exists anormal subgroup N of G with ftv

~

N C G such that

Note that by II , Section III: (i=l,2) and, for p = 2,

CoROLLARY (3F). If y is an involution of the finite group G and if A > 1, then (Case I) g ,:( A/(A - l)c(y) 3 or (Case II), there exists anormal subgroup N of G with .RP ~ N C G such that

1G: N 1 ,::;; [22n+1A">,.(G::(y))3/2]! where 2n is the power of 2 dividing I G 1This is immediate from (3E), if we take p = 2, 7T = y 1 = y 2 = y, = 1, G = G::(7r). Our method shows that in the Case II, N can be obtained as the kernel of an irreducible character Xi 1 in the principal 2-block B 0 of G.

p

*

(3G). Let G be a group of finite even order g Yr} be a set of involutions of G. Set

CoROLLARY

Let {y1

, .•• ,

C

335

= M!lxC(Ji). 1

REPRINT OF [92]

= 2ng0 , g0 odd.

BLOCKS OF CHARACTERS III

233

There exists a function ß(n, .\) depending only on n and A such that at least one of the following two statements is true: Case 1.

g

~

ß(n, .\)c3.

(3.8)

Case II. There exists a normal subgroup Hd. .R2G of G which does not contain any y, , 1 ~ i ~ r, and for which 1

Proof.

Apply (3F) with y

G:H

1

~

ß(n, .\).

= Yi taking say A = 2.

(3.9) If we have Case I,

g ~ 2c3• If we have Case II, we can find an irreducible character Xi =,t: 1 in the principal 2-block B 0 whose kernel N satisfies 1

G: N

1

~

[22n+2,\a/2]!.

If N still contains some y 1 , we can again apply (3F) to N. If we have here Case I, then

If we have Case II, then as in the proof of II, Theorem 4, we can find two characters Xi , x2 in B 0 such that if N 2 is the intersection of their kernels, 1 G : N 2 1 lies below a bound depending only on n and .\. If N 2 still contains some y 1 we can continue, and so on. Since B 0 contains at most 22" characters, this process must terminate after r ~ 22" steps. If we have Case I for Nr , then I Nr 1 ~ 2c3 and (3.8) holds for suitable choice of ß(n, .\). The other possibility is that Nr does not contain any element Yi , ... , Yr . If we set H =Nr, then (3.9) holds for suitable choice of ß(n, .\). We have H-:1. K 2G, cf. I, Section III, Theorem 1. CoROLLARY (3H). ff in (3G}, Yi , y 2 , ••• , Yr are chosen as the involutions in the center 3(P) of the Sylow 2-group P of G, then 1

G : .R2 G

1

~

ß(n, .\}c3•

Indeed, if we apply (3G), the Sylow 2-subgroup Q = H n P of H meets 3(P} in {l}. Since Q = c(J)-i 2c(P)-i, hi•> = c(])-i - 2c(P)-i, h~·> = hi•> = -c(]). Using (5.1) for g = J we obtain easily:

+

THEOREM (5.A). Let G be a group of order g = 4g0 with odd g0 • Let P be an S2 -group of G and let ] be 4n involution of P. ff G does not have a normal 2-complement, then

c(J)3

g

with

= 8a: c(P)2

(5.5)

H ere, 1, Xi , x2 , x3 are the degrees of the characters of the principal 2-block of G and x. = E t = ±1 (mod 4).

At least one ofthe Ez in (5.2) must be +I and one must be -1, say Ei = 1, -1. Set E2 = S = ± 1. In the case S = 1, we can apply (4B), (4C). Since J is not a square in G, we find (5.6)

E3 =

With Xa , the conjugate character 5(3 belongs to B0 , cf. I, Section III, Lemma 2. lt follows from (5.4) that 5(3 = xa and, as x3 is odd, E(x3) = 1. Now (5.6) shows that Xi cannot be real and that we must have Xi = x2 • In particular, Xi = x2 • If we set x3 = q, Xi = (q - 1)/2 by (5.3). Then (5.5) reads

g

+ 1)- c(J)3c(P)-2. algebraic integer for i = 1, 2, 3,

= 8q(q -

l)(q

2

Since gx,(J)/(c(J)x.) is an it follows from (5.1) and (5.5) that (q 1)/4 is an integer dividing c(])/c(P). We now have

+

340

FINITE GROUPS

238

BRAUER

THEOREM (5B). Suppose that in (5A), E1 E 2E3 = -1. ff the notation is chosen such that E3 = -1 and if we set q = x 3 -1 (mod 4), we have c(]) = mc(P)(q 1)/4 with integral m. Then

=

+

g = (m3c(P)/4)(q3

-

q)/2.

If in the case 8 = -1 we have x 2 = x3 and if we set q = x1 , we can rewrite (5.5) in a similar manner. lt follows as a corollary of deep results of Gorenstein and Walter (cf. the Introduction) that the degrees x 2 and x 3 are always equal. However, no direct proof seems tobe available. Remarks

l. 1 ex

< =

lt is seen easily that in (5.5) we have 3/8 ~ ex < 1 for 8 = l and = 3/8 occurs only if G/St. 2 G ~ A 4 and the case 15/8 only if G/St. 2 G ~ A 5 • In all other cases, 55/72 ~ ex ~ 91/72.

ex ~ 15/8. The case ex

2. The three involutions ], ] 1 , ] 2 = J] 1 are conjugate in G and hence in 91(P). lt follows that there exist elements t E G such that 1-1

Jt

= !1'

Here, t can be chosen as 3-element. 3. If we set G0 = CS:,(]), t) then G 0 satisfies the same assumptions as G. If we apply (5A) to G 0 and if ex0 here has the same significance as ex for G, then IG: G0 1 = ex/ex0 • Using Remark 1, we can now see that G = (CS:,(]), t) except when G/St.2 G ~ A 5 • 4. If c(P) > 4, there exist nonprincipal blocks B of defect 2. Each such B consists of four irreducible characters. Application of (2A) shows that ex can be expressed by the degrees of these characters and of suitable characters of CS:,(]) and CS:,(P). 5. If P is elementary abelian of order 4, but if G has a normal 2-complement K, we can still apply (2B). We obtain 1

K lcK(P) 2 = cK(J)cK(]1)cK(J2).

This is the special case of Wielandt's fixed point formula proved directly in II, Section IX.

VI.

AUXILIARY REMARKS

If the SP-group P is known explicitly, the decomposition numbers can be discussed by the methods of I, Section V. If g is even, for each type (I, Section V, Theorem 8) much more precise results than (3A) can be

341

REPRINT OF [92]

,

BLOCKS OF CHARACTERS III

239

obtained. In the following sections this will be clone for certain groups P. In order to avoid repetitions, we collect here a number of remarks. The p-block B will always be taken as the principal p-block B 0 •

1. As in I, Section V, we speak of a column a for B if with each Xi E B, there is associated a complex number, the "entry" for Xi. We shall denote this entry by (a)i (or if the character is denoted by xu>, by (a)(i>). For two columns a and b, the inner product (a, b) is defined as

with the sum extending over all Xi E B. A column a is integral, if all (a)i are integers. In particular, if a EG, we have the column x(a) with the entries Xi(a). If 7T E P, and if a fixed basic set form a basic set for B 0 and the corresponding Cartan matrix is

( 4 -2

348

-2

)

zn- + 1 2

.

FINITE GROUPS

(7.11)

246

BRAUER

THEOREM

(7K).

In the Case II,

_ 2 x1(x1 (x1

g -

-

+ S1)

c(J) 3 c(], a1')2

S1) 2

Remarks

l. Methods similar to those discussed in the Remarks in Section V can be used here. W e shall not go into details. 2. The character Xi is rational. Application of (4B), (4C) shows that either S1 , S2 , S3 is 1 or 2 = x3 , x2 = x3 ,

+

x2 , Xa are real and exactly one of S1 = -1, S2 = S3 = 1. 1

x

3. Suppose that we have Case I and that c1 = c(J, T) and c2 = c(J, aT) are not equal, say that c1 < c2 • Set y = (c1 + c2) 2/(c1 - c2)2. If the notation is chosen such that x 3 < x 2 , it can be shown that x3 < y(l + 2-n)(l + 2-n+l) when S1 = 1 and that x2 < 1 + 6y or x1 < 3y when S1 = -1. This can be used to prove that c2 :(; 3c1 where in the case 5l2 G = 1 the equality sign holds only for G ':'.'::' A 7 •

VIII.

GROUPS WITH QuAsI-DIHEDRAL

S 2-GROUP

The quasi-dihedral 2-group P

=

=/= 0 for some ex, we find {b/)1 = 81 , (b/) = 81 • This is impossible, cf. Section VI, 2. Thus (8/)< 00 >= 0 for all ex. lt follows in the same manner that (b/)i = -81 • If we interchange x 2 and Xa if necessary we can assume that (b/) 2 = 82 , (b/) 3 = -83 • The entry ±2 in d/ occurs for a new character x4 • Since br + b/ is orthogonal to x(l), (b/), = -282.

351

REPRINT OF [92]

249

BLOCKS OF CHARACTERS III

Thus,

+

Each column cß with ß -=I=- 1 has two nonzero entries and these are 1 and -1. In each column orthogonal to bß the corresponding entries are equal. Our preceding results imply that the nonzero entries of cß cannot belong to any character previously discussed. W e also see that there is a character for which all cß have the same entry E = ±1. This character will be denoted by x = -E. We now have identified 2n- 2 4 distinct characters of B 0 :

+

xo =

1,

(8.8)

X1 , X2 , X3 , X4 , x < $. We may finally suppose X - EFl, -EN EFa N, X EF.

(e) Since N -.NF2 - EN, there exist G1,G2e Y( such that G1:N- EN,

(pN) -+ (i!(EN) and G2: NF2-* EN, (i!(NF2) _+ (i!p(EN). In particular, G1 and G2 both map X into ccl(EFf) n (E$(EN). Conjugating if necessary by an element

in ($p(EN), we may assume both G1 and G2 map X onto EFP. Set G = G1G-; then G:J - J,X -X, N-*NF2. If W = ,=5 =((), then

is an S2-subgroup of .j. In particular, the S2-subgroups of .5/9 are elementary abelian of order 8. Modulo . the following elements represent the elements of

an S2-subgroup of .5/9: 1 F2 EF N EF3 NF2 ENF- 1 ENF. Moreover, N NF2 in 5j/% by the above. By Burnside's theorem we see that either all the nonidentity

representatives above are fused in .5/f, or 6 of them are fused in sets of 3 each. But this is impossible, since 5 of them have squares in and 2 of them have squares in ccl(X). This completes the proof.

REMARK. We note that if = $3(p,a,z) with p = O, a =zT = 1, and if = , then as a consequence of (e), the group C(E() has a normal 2-complement. PROPOSITION (2H). Let p =0, a = 1, r = 1. If (5 has no subgroups of index 2, then after a suitable relabeling of the generators E,N,F, there are exactly three possibilities for thefusion of involutions of .

I. J-X N EF E3 -NF-EN, II. JXNEF, EF3NFEN,

III. J -X-EF3, N-EFNF EN. In all three cases, E2 , ENX - ENF3, E2 - NXF ENF. Proof. The following are consequences of (2D) and (2F), F (2.1): E NXF, F2 ENX, since either fusion implies E2 -F2. Secondly, if

X - EFE for some oc= 1 or 3, then E2 -ENX, F2' NXF, NF -EN EFfl, where fl=1 or 3, fl# o. By (2G) and (2E), J- X, N- EF, E2 .ENF3,

F2- ENF. Suppose N - X. Then EF - X, and we have either cases I or II. Suppose that neither case I nor case II holds. If X - J is fused to no other class, then the only possible fusions in (f are between

X -J E2 ENF3 3 F2 ENF N -EF, EF3 | NF, EN by (2F), (2.1). But then < 24. Since the remaining Xie 1B have height _ 1, this would imply that B has ? 10 characters, which is impossible. Thus we may

assume Xi(x) = xi+1(x) = ti for i = 3,5,7, ri = + 1. The XieB for i ? 9 cannot have height > 2; otherwise the contribution from e2, f 2, x would be too large.

Hence these Xi have height 1, and necessarily xi(x) = 2i, =i ? 1. In particular, B has 14 characters and 5 functions in the basic set from the 1-section. The quadratic

forms QE and QF2 associated to cf(E2) and W(F2) are 3U2 - 2uv + 3V2 = 2u2 + 2v2 + (u - v)2, which has a minimum of 3 for nonzero rational integral vectors. If a Xie 14 has height 0, then the sum of the contributions from E2, F2, J, 1 must be 64- 34 = 30. We express this as

2QEL + 2QF2 + QJ+ Q' = 30. Now QJ, Q1 > 1. Since dE -dE and d F -diF are necessarily odd, QE2 and QF2 are < 11. If (u, v) are the decomposition numbers of Xi under E2 or F2, and if we allow for a possible change of sign (u, v)( - u, - v) and a possible interchange (u, v)-(v,u),

then the only possible values for (u, v) are (1, 0) and (2, 1). If Xi E B has height 1,

then u + v 2 (mod 4). The method of contributions implies that Q E2 and QF2< 30. With the same conventions as before, the only possible values for (u, v) are (1, 1) and (2, 0). Consider the entries under E2 for all Xi E h. If there are a,b,c,d rows

of type (1,0), (2,1), (1, 1), (2,0) respectively, then from the equality lu2 + Xv2 = 24, we have a+b=8, c+d=6, a+5b+2c+4d=24. Thus b=0, d=2. The entries can then be determined by the orthogonality relations. Similarly so for the entries under F2. The generalized decomposition numbers for 1 for the 2-singular

sections of 5 I have now been determined, and are as below. There are then 6 characters of B faithful on . Two of them, say Z15, Z16, are nonzero on ENF, ENF3. The columns under J and 1 can then be found by the methods of ?3. The

matrix for B in case I is (4.8), where ri = + 1. ZJ is then as in (4.2). 5. In addition to the assumptions made on (b on ?4, we also assume (5 has

more than one class of involutions. This excludes case I.

LEMMA (5A). Let X be a character of (5. Then (i) x(E) X(F) (mod 2),

(ii) X(E 2)-Z(F 2) =X(EN) -=%(J) (mod 4),

(iii) X(J) 2X(E) - Z(F2) 2X(F) - Z(E2) (mod 8), (iv) - 2X(E) - 2X(F) + X(F2) + X(E2) + 2X(EN) 0 (mod 8).

1966] A CHARACTERIZATION OF THE GROUP 9i112 37

O000000 ON NF NpO O Nh NO I ~

_~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I o o oi oo I FOi o _I .

-

o~~~~~~~~~~~~~~~~~~~~~~~~~~~~ o I I ooooI o >oo I >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I Oooi oo I ioi OI I

000

00

O-

N

w~0

N

N

0

0

NO

o

O

0

0

NO

No

NhFO

NO

N

NO

N

b

hOF

oopiO

O

;p

N

-OFh'-

~

IsH0

o o o o o o s o s o s o o~~~~~~~~~~I II

O O p Fh' O O Fh ~ ~ ~ ~ ~ ~ ~ ~ ~ ~i Ni IF IOOOOOO

N 000 N NO NO Nioo p -O N I OO '0w- O - 1oooo i b I o ~~~~~~~~~~~~~~~~~~~~~~~~i si I ooIo I

t I I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I I I

o o

38 RICHARD BRAUER AND PAUL FONG [March

Proof. (i) is obvious, since X(E) and X(F) are rational integers congruent to X(1) (mod2). The others follow by restricting X to 3 and computing (x,O)s, where 4 is a suitable generalized character of 3. For (ii) take 4 to be respectively

15 - (16s ( -10 ( 1 3 - (4 in the notation of (2.2). For (iii) take 0 to be

C = ( - (5, which yields 2X(E) + X(F2) + 2X(E2) + 2X(EN) + X(J)- 0 (mod 8). By (ii) 2X(F2) _ 2X(E2), 2X(EN)- 2X(J) (mod 8). Substitution of these gives the first part of (iii). The second part is proved similarly. For (iv) take 0 = 2, -Cg, and use the congruence 2X(EN) _ 2X(J) (mod 8). Let B be the 1-block of &. If Y is a representative of a 2-singular section of (5,

we assume a basic set 4,B = {Op} has been chosen for the 1-block of 4Q(Y) such

that the corresponding Cartan invariants are as in (4.1), and for Y= J, as in

(4.2). Denote by fpY the degree of 4, ; 4y is always the 1-character of (.(Y,),

so that f [ = 1. In particular, f EN -3 (mod 8) in case II, and f J--3 (mod 32). The corresponding columns of generalized decomposition numbers are then

denoted by bpY, except that in the case where (E(Y) has a normal 2-complement and there is but one such column, we will drop the index 1. For Xi e B,

xi(Y) = P(b Y)ifpy, where (b Y)i is the entry dY. Define the following columns of rational integers, indexed by the Xi E B.

bF - bE __+b_2 _ b_.II 24

bF + bE bEN _ bFin bj - _3b3 _ +2 b -_2b E b + bE2 =in + 2(bEN +II1, bEN) - 2bF - 2bE bb F2b4 8

8

in

II,

bF2 bE2 bF2 + bE2 + 2bEN - 2bF - 2bE in III. 4

L8

That these columns are rational integral follows from (5A) and the congruences

fEN -3 (mod 8), f -J 3 (mod 32). The inner products of these columns and the ones indicated are bj bEN bEN bEN

a b

4 0

0 0 1 0 4 0 -1 0

0 0 0 0 0 -2 0 0 0 0

(5.1) 0 0 3 -1 -2 0 0 4 2 2 1

91 u

0 0

-1

-1

0

-2

-2

0

4

1 1

2

4 0

1

0 3

0

0 0

0

0

0

0

0

0

4

2

2

where bEN occurs in III; bEN, bEN in II. By (5A), (iii) and the congruence

1966] A CHARACTERIZATION OF THE GROUP M12 39

Z(E) -(E2) (mod 2), it follows that VI -a (mod 2). We note that a and b are orthogonal to the column Z(1) of degrees. We now determine the entries of these columns. w and u have norm 3, and

thus have 3 entries + 1. a, b, Z, % have norm 4, and since a(3 = 1, b' =-1 each of these must have 4 entries + 1. We can choose five characters X such that

by replacing x by - x if necessary, we have

w %ta a 1

0

0

b1

1 -1 1 b2

1 -1 -1 b3 . 0 1 1 b4 0 1 -1 b5

By orthogonality b3 = b4, b2 = b5, bI = -b2-b3 . Suppose bi = 0 for some 2 < i ? 5. Then one of %,(E), xi(F) is + 2, and the other is 0; say that xj(E) = + 2, xi(F) = 0. If xi has height 1, then by (5A), (ii) xj(F2) xj(E2) -x(EN) _ 2 (mod 4). The contribution to xi from E, F2, E2, EN is then 64, which is too large. If xi

has height > 2, then Xj(F2) -X(E2)= 0 (mod 4). But (5A), (iii) implies xj(J) + xj(F2) _ 4 (mod 8) and the same congruence with the roles of E and F interchanged implies that xi(J) + xj(E2) 0 (mod 8). Thus one of xj(E2) Xj(F2) is nonzero, and the method of contribution again leads to a contradiction. The

argument for the case xZ(E) = 0, xZ(F) = + 2 is similar. Thus bi = 0 for 2 < i _ 5, and b1 = 0 as well.

We now rearrange the characters in B so that Xi is the 1-character and the others are as indicated!below:

b a E W u 1

0

0

0

0

1

0

0

0

-1

1

0

0

0

-1

1

0

0

0

0

0 1 -1 1 0

(5.2)

0

-1

-1

0

1

0

-1

0

0

0

0

1 1

1

0

0 0

0

0

0

1

0

0

0

1

The entries of ui are determined easily from (5.1). Since 9 is known, the method

of contributions implies that (bE2,bF2)i = + (1, 1) for 1 < i ? 4, and (bE2 bF2 )i = + (1, -3) or + (-3, 1) for 5 ? i < 8. Furthermore, knowledgeTof

40 RICHARD BRAUER AND PAUL FONG [March

u and w imply that (bE2 ,bF)i = (-2, -2) for i = 9, and (2,2) for i = 10. The signs of (bE2,bF2)i for i = 2,3,4 can be determined by the equation ubE =4. Since bE2, bF2 have norm 32, each has exactly two + 3's for 5 < i ? 8. The

products WbE2 = 0, bF2 = -8 determine the signs completely. If the characters are arranged as in (5.2), then

F E E2 F2

1 1 1 '' 1 1 1 -1 -1 1

1 1

-1

1

-1

1 -1

1

-1

1

1

1

1

-3

-3

1 -1 -3 1 -1 0 0

1 0

-3 -2

0

1 -2

2

2

In particular, Xg Z %1Q have height 1, and by (5A) (iii) Xi has height >3 for i > 11;

these Xi then actually have height 3 or 4 by [4]. The columns bEN, bN + bN in cases II, III respectively are now completely determined from w and bF . In II the entries of b ENb EN can be determined as

follows: (b EN + bEN)i = + 1 for 1 ? i < 8, 2 for i = 9, 10. Since bEN has norm 6, the nonzero entries are + 1. The product ubEN= 2 shows that one of (b1EN)2, (bN)3 is -1 and the other is 0, since (bEfN)1O = (bE2N)1O = 1. Since X2, X3 may be

interchanged at this point, we may assume (bEN)2 =-1. From the equality

bbEN = 0, we have (bfE)3 = (bElN)4 = 0. This completely determines the entries under EN except for the arrangement (bEN, bEN). for 5 ? i ? 8. Here (1,0) can occur for i = 5 and 8, or for i = 6 and 7; the remaining entries are (0, 1). This point will be settled below.

Let si = ()i. Then (B= -1, (u=1 imply that s2 + S3 + S4 -1 -S2-S3 + S1O=1, so that S4 =-slo. Also (ta=1, (SW = 1 imply that s5 = s8 and S7 - S61 . In particular, one of the numbers S6, S7 iS ? 1, and the other is 0. The quadratic form QJ is lOu2 + 2v2 + (u - v)2. The method of con-

tribution then implies the following: If Xi has height 0, then (bJ, bJ)i, up to a sign change, can only be (1,0), (O1), (1,2), (-1,2), (0,3), and for 5 < i ? 8, only the first three are possible. If Xi has height 1, then only (1,1), (0, 2) are possible; if Xi has height 3, only (1, 3) is possible. If Xi has height 4, the method of contri-

bution yields a contradiction. Since si = 0 or + 1, the definition of S forces the conclusion (bJ, bj)= (1,0) for i =5 and 8; thus s5 = = 0. Similarly,

(bJ, bJ)i= (0,1) or (-1,-2) for i =6,7, and the latter can occur only in III by the method of contribution. Thus, if S6 -1, S7 = 0, then (b, bJ)i = (0, 1); if

19661 A CHARACTERIZATION OF THE GROUP 9312 41

S= 0, s7 = 1, then (J, b)i =(-1, -2). We call these two possibilities Illa, IlIb respectively. Suppose S4 = 0. Then one of S2 ,S3 = + 1, and the other is 0.

There must then be characters X11,X12 in B such that s11 # 0, S12 # 0. For

i =11, 12, (bqJ,b-)i = + (1,3), and since bi has norm 6, (bj)i = 0 for i = 9 or 10. But this contradicts the fact that b- has norm 22. Thus S4 0 0, and we easily

see that S4 =-1, 9 = 1, S2 = S3= 0. Sg can be found from (3w =-1. If sg = 0, B contains one more character Xl1 for which slI = 1. The matrix of decomposition numbers can now be completed, the uncertainty remaining in II under EN

being settled from b-bfEN= 0. We omit the details. The complete matrices are in cases II, Illa, IlIb respectively F

E

F

2

1

1

1

1

1

1

-1

E2

1

-1

EN

0

-1

1

0

J

0

0

1

1

-1

0

0

2

0 1

1 1 -1 -1 0 -1 0 -1 -1 -1 -1

(5.3)

1

1

1

-1

-1 1

-1

1

1

0

1 1

1

1

-3

-1 1

1

-3

-1

0

1

2

1

0

-1

0

-1

0

0

0

1

1

0

0

1

0

0

1

-3

1

0

0

1

0

0

1

-1

0

-3

0

1

1

0

0

0 0 -2 -2 1 1 -1 -1 0 1 -1 0

0

2

2

1

1

0

-2

-1

1

0

0 0 0 0 0 0 -1 -3 1 -2 1 The matrix of Cartan invariants of B is 4 -2 2

-2 14 01

L 2 0 6 If f1'f2,f3 are the degrees of the corresponding basic set, then f1 = 1, f2- -9 (mod 32), f3 - 11 (mod 32). F

E

F2

1

1

1

1

1

E2

1

-1

1

-1

EN

1

-1

J

0

0

1

1

-1

0

0

0

1

2

0 1

1 1 -1 -1 -1 0 -1 -1 -1 -1 -1

(5.4)

1

1

1

-1

-1 1

-1

1

1

-1

1

1

-3

-3 1

1

1 1

-1 1

2

1

0

0

0

-1

0

0

-1

0

0

1

1

0

1

0

0

0

1

-3

1

0

1

0

0

0

1

-1

0

-3

1

1

0

0

0

0 0 -2 -2 2 -1 -1 0 0 1 -1 O

0

2

2

2

0

-2

-1

0

1

0

0 0 0 0 0 -I -3 1 0 -2 1

42 RICHARD BRAUER AND PAUL FONG [March

The matrix of Cartan invariants is

r 4 1 -2 2 ) 1

2

3

2

-2 3 14 0

l

2

2

0

6J

f= 1, f2-32(mod 64), f3=-7 (mod 64), f4=-11 (mod 32). F

1

E

F2

1

1

E2

1

EN

1

1

J

0

1

1

0

0

1 1 -1 -1 -1 0 -1 0 1 -1 1 1 -1 -1 -1 0 -1 0 -1 -1

(5.5)

1

1

1

-1

-1 1 -1

1

1

-3

-1 1

1

-3

1 1

0

0

-2

0

0

2

1 1

1

-1 1

1

1

-1

-3

1

-1

-3

1

1

-2 2

2

2

0

0

2 0

-1

0

-1

0

2

0

1

-2

0

-2 0

2

-2

0

-1

2

0

1

0

2

1

0

-3

0

0

-2

The Cartan matrix is

r 5 0 -9 1

0 2 ol.

L -9 0 29

fl = 1, f2-0 (mod 32), f3-29 (mod 64). 6. We now use the methods of [2] to complete the investigations of cases II and III. We recall the following facts: Let (f be a finite group of even order g. Let p be a prime, and P a p-element of (5. Let (5 be a subgroup of order g such

that ( > d*(P) = . If Ja and J are two involutions of (5, and {0P'} is a basic set for a p-block of d;(P), then (6.1) g Y Zx(J)xZ(Jp)dP Ix, = gc(JQ)c(Jc) I hA,,,hA#e7MPpd/x,. Here on the left xZ runs over the irreducible characters of the block B of (5 corresponding to the block b of (i(P), xA = Zx(i), and d.Pp are the corresponding general-

ized decomposition numbers. On the right, ha = XX.(J)/C(J), runs

over characters of the block B of (5 corresponding to b, x = %(1), d Pp are the corresponding generalized decomposition numbers, and {Ja} ranges over

a set of representatives of the conjugate classes of (5 contained in the conjugate

class of Ja in (5. For notational convenience, we will let L(Ja, J, bp) , R(Ja, J#, bp) denote the left-hand side, the right-hand side of (6.1) respectively.

1966] A CHARACTERIZATION OF THE GROUP 9M12 43

In the applications to follow, we will always take p =2 and b the 1-block of (i(P). It then follows that B and B are the 1-blocks of (5 and (i respectively. As a second remark, we note that if ( has a normal 2-complement,

then the characters j,, in the 1-block of C can be identified with the irreducible

characters 0,M of an S2-subgroup D of 63. Moreover, if Ja, Jp, P are in ?, then c(J,) lc(Jp) lR(Ja, Jp, , )P is equal to

E oJa2 0y(/ dl' u .l cJa A)v j(j-#,v) 01(()

(6.2) =f , 01(Ja,A) 0M(Jp,v) OM(P) c(JaA) CQG(J_,v) i. V itl 0'E(1) CQ;(jaJaAC;(j#'V) C(ja,A)j(j,#V)

where a,(ja, Ji, , P) is the multiplicity of the class sum of P in D in the product of the class sums of Ja,. and Jp,v . We assume in this that the elements J.,; and

JP v ,. Finally, we note that (6.1) can be added over linear combinations of the dP for varying 6, P, and p. If 1) is such a combination of the bp we understand by the equality L(J., Jp, ij) = R(Ja, Jp, ij) the result obtained by first applying (6.1) to the various b , and then summing appropriately. We now turn to the situation at the end of ?5.

Case III. Set /=fJ, /-3 (mod32). If we set 6=X*(F) and (i(E), and compute L(J, J, b) = R(J, J, b), and L(J, EN, b) = R(J, EN, b), we find that

1 2 + /2 + (2p - 1)2 0 X2 X3 X4

(6.3) 1/ 2+ I-1

I + - _ +=0.

X2 X3 X4

Here the right-hand sides are computed by observing that E*(E), V*(F) have normal 2-complements, and that E,F do not occur in III in the appropriate product of involutions. Subtracting the second from the first, and cancelling ,B - 1, which is permissible since /3 -3 (mod 32), we have

(6.4) ! + ! + 223 - 1 = O. X2 X3 X4

Substituting (6.4) into (6.3) then yields 1 - (23 - I)/x4 = 0, or X4 = 2B - 1. Thus J is in the kernel of X4. By (6.3) / /x2 + //x3 + 2 = 0 . Since Z2(J) = Z3(

= - f, and I|,? < Ix2j,Ix31, we see that x2=x3 =-/, and hence J belongs

to the kernel of Xi for 1 ? i ? 4.

In IIIb, x2 =f2-f3, x3 =-f2-f3; thusf2 =0, f3=l. But X(JJ) -1-2fl,

and x6 = , * Since I1 + 2B | < I ,B , we must have / =-I, which contradicts

44 RICHARD BRAUER AND PAUL FONG [March

the congruence ,B -3 (mod 32). Thus IIlb is impossible. We henceforth assume

we have IlIa. In this case, x4(J) = X4 implies that f3 = 1 - 2f.

If (5 = C*(EN), Ei*(F2) respectively, and we compute L(J, J,w) = R(J, J,W), then (6.5)

+,

+

X5

=

X6

=

Xg

0.

Here the right-hand side is computed by noting that i*(EN), i*(F2) have normal

2-complements, and by using (6.2). If we substitute x5 = -f3, x6 = f4, x9 =f3 -f4 into (6.5) and simplify, we obtain the equality f4 + /Bf3- 0, so thatf4 = 2132 - fl.

The equality 2(J) = X2 implies f2 = -2(f3- 1)2. Summarizing, we have f1 = 1, f2= -2(fl - 1)2, f3= 1 -2,B, f4= fl(2fl - 1).

Let R =fnkernel of Xi for i = 1, 2, 3, 4. Since J e , R must contain the sub-

group ? = of $ generated by the elements of 3 conjugate in (5 to J. W is abelian of type (2,2,2). Since F2, N O SR, 9 is an S2-subgroup of R. Assume now that (i has no normal 2'-subgroups # 1. Let % be a minimal normal

subgroup of (5 contained in SR. Since | I is even, W _ S.

Suppose % # R. Since 91 is characteristically simple and W is an S2-subgroup

of 9, it follows that 9 is simple. Hence 211 = 9(I)/(i() has order 7.3x with

x = 0 or 1. Clearly 3 stabilizes XP, so we may view 9/E as a group of operators

of W3. If P e $ and P91 centralizes 2D3, then J leaves W1 n (i(P) invariant. Since 9 is an irreducible 2-group, it follows that % ? gi(P), so Pei 13; n T()= W.

Thus $3/9 is faithfully represented on W5. This is impossible, since j13 j = 8

and | 1 divides 3 7. Thus W = %.

Since W is a self-centralizing normal subgroup of , it follows that

()= W x ZX, where Z has odd order. Hence Z char T(0- (5, so t = 1. Thus (5/9 is isomorphic to a subgroup of GL(3,2) of order 0_ (mod 8) which has no subgroup of index 2. Hence (5/9 - GL(3,2). The group (5 exists and is unique by [11].

Case II. Set a= fEN, fl=f J; a--3 (mod8), fl -3 (mod32). If we take (i =- *(E), T*(F) and compute L(J, J, b) =R(J, J, b), and L(EN, EN, b) = R(EN, EN, b), we find that

1 p2 + #2+ (2l- 1)2 -O X2

X3

X4

I oc2 oc2

1 + -+ -+ -= 0. X2 X3 X4

The right-hand sides are 0 by observing that T*(F), V*(E) have normal 2-complements, and that in II, E, F do not occur in the appropriate products of involu-

tions. Expressing the degrees xi in terms of If1 ,f2 ,f3 and simplifying, we get

1966] A CHARACTERIZATION OF THE GROUP 9312 45

(6.6) oc 2 _ f2(M 2 + f f3) = ? (6.6) ~~~~~~2f2 + f3 = (6.7) fl2(f2-2+4(1 + f2+f3))-4p(1 +f2 +f3) - (1 +f2 +f3)(f2 - 1)=O, (6.6) and (6.7) can be used to eliminate f3; a little manipulation then gives the equation

(6.8) f2(p2 - 22) + 22f2( - 1)2 + O2(O2p2 - (23 -1)2) = 0.

jf ~2 + p2, then the above equation factors into (p2 - 22) (f2-i) (f2-j) = 0, where

(6.9) i= -2 3p _ C2- o + 2Xp - -2f If 22 = p2, then a = / from the congruences a--3 (mod 8), p -3 (mod 32). In this case, f2 =- (2 + 2a - 1)/2.

If we take (5 = V*(F2), ci*(EN), and compute L(J, J,w) = R(J, J,w), we have _____ +____ 2 1 _2 -c(F2)

C(J)2 f2 -f3 f2 f3 = c(F2 N)2 or

~c(F 2) (flf2 f2 f3 (f2-_f3) (6.10)(6.10) g = C(j)2 c(F2,N)2 + f3)2 Apply now the methods at the beginning of this section to the situation (3 = W(J),

- = (*(F2), and L(X, X, F2 ) = R(X, X, bF2). Using (4.7) ,we see that L(X, X, bF2) = 2c(J) (f + 3)2 //3(P + 1). On the other hand, in Ci(J) X is fused only to EF, and (ccl (X) u ccl(EF))2 does not involve F2. Thus R(X, X, bF)= 0. and we must have /B = -3. This already shows that oc 0 /B; otherwise, f2 = -1, whereas

f2--9 (mod 32). Hence f2 = i or jt of (6.9). Iff2 = i, then a -24- _ 022 = (mod ,B + oc), and c + ,B divides ,B(1-_3)2. Iff2 =, then- a + 24cl _ a2 P0 (mod ,B-c), and oc - / divides /3(1 _ /3)2. In either case, ac + 3 divides 48. Only a small number of possibilities can occur for ac, and since f2 - 9, f3 - 11 (mod 32), a check of these shows that only the following solution exists:

a = 5, fp = -3, f2 = 55, f3 = - 11. The degrees of the characters in B are then 1, 55, 55, 55, 11, 11, 99, 45, 66, 54, 120.

In particular, (5 has a rational character X6 of degree 11 faithful on $3. Assume now that Qi has no normal 2'-subgroups : 1. Then by a theorem of Schur's, the order g of (5 divides 26.36.52.7.11.

The characters in the 1-block of (E(J)I have degrees 1, 1, 3, 3, 3, 3, 3, 3, 6, 2

by (4.7). Let 0 = Oj, for some j where 3 < j ? 8. If SI is the kernel of 0, then for a suitable choice of j, the only nontrivial elements of $3I in SR/ are the elements in the class of e2 . Since 0 is a rational character, (((J): A) is of

46 RICHARD BRAUER AND PAUL FONG [March

the form 2s3t, so that %i(J)/R is solvable. If we now choose 0 so that the only

nontrivial elements of j3 in the kernel Sl/ of 0 are the elements in the class of f2, then a similar argument shows that (2(J)/S1 is solvable of order

2s'3". But SR r) R 1n = . It now follows that if 9M is the maximal normal 2'-subgroup of C(J), then (E(J)/9 is solvable, and its order is of the form 2a3b, Since (iE(J)/9R has only one block for the prime 2 by [5], the order of fi(J)/9M is

then 192. Set m=|192R1 The values of a, fi, f2, f3, and (6.10) show that the order of (5 is

(6.11) g = 253 3 5 *lm c N -(F )

c(F2, N) c(F2,N)'

We note that c(F2, N) c(F2), and c(F2, N) j 64m, the latter being true, since no

3-element of (i(J)/R commutes with N. In particular, 34 , m since g 1 26 * 36 * 52 * 7 * 11.

Suppose p divides m, where p = 7 or 5. Since 72 j g, 53 4 g, (6.11) implies that p j c(F2, N). Thus there exists an element Y in T. Io ((F2) of order p. In

particular, by (5.3) Z6(JY) = X6(F2Y) = 3 and so Z6(Y)- 3 (mod 4). Since Z(Y) can assume only the value 4 for p = 7, and the values 6 or 1 for p = 5, this is

impossible. Thus m 127 and g | 26*36 5.11. Let S be an element in (5 of order 11, and let ( = . If %(e) = W((), then ( would have a normal 11-complement. In particular, S would normalize an

S2-subgroup 1 of (5 and hence centralize the central involution of $1, which is impossible. Thus %(a) = (E(S). The order of 9(Q)Ai(s) is necessarily a divisor # 1 of 10. The index (6: 91($)) is then of the form 32 5 *3 , 32 *37, or 64 - 3 where 0 < a < 6. By Sylow's theorem, this index is -1 (mod 11), and thus must be 64 33. If m # 1, then there would exist an element T of order 3

in W~(S). Since %6(ST) Z6(T) (mod 11) and %6(ST) = 0, we would have

Z6(T) =O0 (mod 11) and this is impossible. Thus m = 1. We have now shown that g = 26.33.5 11 = 95,040. Suppose I5 has proper

normal subgroups; among these choose 9r minimal. If (fi/9 has even order, then there would exist an irreducible character - I in the 1-block of (5/91 which would then belong to the 1-block B of 6. But this is impossible, since the nontrivial

characters in B are faithful. If (5/9 has odd order, then 1 is an S2-subgroup of 91. If 91 has no subgroups of index 2, then 9 has necessarily two classes of

involutions, and the preceding work shows that 9 has order 95,040. Hence 9 is solvable, and 9 = 1. which is impossible. Thus X is simple. By a theorem of

Stanton [10], (5 must be the Mathieu group 92)12. Summarizing the results of this paper, we have

THEOREM (6A). Let (5 be afinite group of order 64g', where g' is odd. S ippose there is an element F of order 8 in (5 SliCh that F is self-centralizing in some

S2-subgroup , an d F is conjugate to its odd powers in 43. Then on e of theotllowing possibilities hold:

1966] A CHARACTERIZATION OF THE GROUP 9112 47

(a) (5 has a subgroup of index 2. (b) 03 has one class of involutions. (c) If 02'(03) is the maximal normal subgroup of (5 of odd order, then

T)/02'(T)) '01344 or 9112, where 61344 is a uniquely specified nonsimple, nonsolvable group of order 1344, and 9212 is the Mathieu group on 12 symbols. COROLLARY (6B). The only simple group satisfying the assumptions of (6A) and having more than one class of involutions is 9212. REFERENCES

1. R. Brauer, A characterization of the characters of groups of finite order, Ann. of Math. (2) 57 (1953), 357-377.

2. ---, Investigations on groups of even order. I, Proc. Nat. Acad. Sci. U. S. A. 47 (1961), 1891-1893.

3. ---, Some applications of the theory of blocks of characters of finite groups. I, J. Algebra 1 (1964), 152-167.

4. R. Brauer and W. Feit, On the number of irreducible characters of finite groups in a given block, Proc. Nat. Acad. Sci. U. S. A. 45 (1959), 361-365.

5. P. Fong, On the characters of p-solvable groutps, Trans. Amer. Math. Soc. 98 (1961). 263-284.

6. G. Frobenius, Uber die Charaktere der mehrfach transitiven Gruppen, S-B. Preuss. Akad. Wiss. (Berlin) (1904), 558-571.

7. M. Hall and J. Senior, Thle groups of order 2" (n ?6), Macmillan, New York, 1964, 8. P. Hall and G. Higman, On the p-length of p-soluble groups, Proc. London Math. Soc, 6 (1956), 1-42.

9. I. Schur, Uber eine Klasse von endlichen Gruppen linearer Substitutionen, S-B. Preuss. Akad Wiss. (Berlin) (1905), 77-91.

10. R. Stanton, The Mathieu groups, Canad. J. Math. 3 (1951), 164-174. 1 1. W. J. Wong, A characterization of the Mathieu group M12, Math. Z. 84 (1964), 378-388.

12. H. Zassenhaus, The theory of groups, Chelsea, New York, 1949. HARVARD UNIVERSITY,

CAMBRIDGE, MASSACHUSETTS UNIVERSITY OF CALIFORNIA,

BERKELEY, CALIFORNIA

An analogueof Jordan'stheoremin p characteristic By Richard BRAUER*and Walter FEIT** 1.

Introduction

Let C be the fieldof complex numbers. A well-known result of C. Jordan

functionJ(n) definedon the set [10] assertstheexistenceof an integer-valued of positiveintegerssuch that every finitesubgroupG of GL"(C) containsa normalabeliansubgroupA withIG: A I < J(n). Severalproofsofthistheorem of a finite are known,see thereferencesin [6, (36.13)]. Since a representation group in an algebraicallyclosed fieldK of characteristic0 is equivalentto a in an absolutelyalgebraicsubfieldK0 of K, Jordan'stheorem representation 0. remainstruewiththe same J, if C is replacedby any fieldof characteristic p > 0. On theotherhand,theanalogousresultis falseforfieldsofcharacteristic p and set For example,let K be an algebraicallyclosedfieldof characteristic Gm= SL2(pm).Each Gmis a subgroupof GL2(K) anda normalabeliansubgroup of Gmhas orderat most2 whilethe orderI GmI of Gmis arbitrarilylarge. In this paper the followinganalogue of Jordan'stheoremfor fieldsof characteristic p is proved. THEOREM. Let p be a prime. There exists an integer-valuedfunction f(m, n) = fp(m,n) such that,if K is a field of characteristicp and if G is a has orderat mostpm,then finitesubgroupof GLn(K) whoseSylow-p-subgroup G has a normal abelian subgroupA with I G: A I < f(m, n). This theoremhad been conjecturedby 0. H. Kegel in a letterto us. Of to provethetheoremundertheadditionalassumptionthat course,it willsuffice K is algebraicallyclosed. Notation

subsetofG. ThenCG(B), NG(B), Let G be a finitegroup,and B a non-empty ofB in G, and IB I denoterespectivelythe centralizerofB in G, thenormalizer and thecardinalityofB. If H is a subgroupof G, IG: H I is the indexof H in G. We writeH < G to indicatethat H is a normalsubgroupof G. * The work of the firstauthor was partiallysupportedby the U. S. Air Force Contract AF 49 (638)-1381monitoredby the AF Officeof ScientificResearch of the Air Research and DevelopmentCommand. ** The work of the second author was partiallysupportedby the U. S. Army Contract DA-31-124-ARO-D-336.

120

BRAUER AND FEIT

If p is a prime,a p-elementis an elementwhoseorderis a powerofp whilea p'-elementis an elementof an orderrelativelyprimeto p. A groupis a p-group or p'-grouprespectively,if all its elementsare p-elementsor p'-elements. Throughout thepaper,p willbe a fixedprime,and K an algebraicallyclosed fieldofcharacteristic p. All groupsare assumedto be finite.The groupalgebra of G over K is denotedby K[G] with K consideredas subfieldof K[G]. All modules are assumed to be finitelygeneratedright unital modules. The representation of G associatedwitha K[G]-moduleis usually denotedby the Germanletter. An invariant elementz ofa K[G]-moduleL is an corresponding elementz C L forwhichzx = z forall x C G. For anygroupG, Lo(G)denotesthe trivialK[G]-modulewithdimKLo(G) 1. If L1,L2, , L, are K[G]-modules, L (EDL2 e**

Lm. =ey

Li

is thedirectsumoftheLi. If all Li are equal, we writemL1forthis sum. L* is theK[G]-modulecontragredient to L1. We say thatL1 is a constituentofL2, ifL1 is a quotientmoduleofa submoduleof L2. The notationL1 IL2 meansthat to a directsummandof L2. The kernelof L1 is the kernelof L, is isomorphic the associatedrepresentation 2,. Two modulesare said to be distinct,if they are non-isomorphic. If H is a subgroupof G, and M a K[H]-module, MI denotesthe K[G]moduleinducedby M. If x C G, thenMx is theK[x-1Hx]-module consistingof the same elementsas M with the operationsdefinedin the obviousmanner. Finally,if L is a K[G]-module,LH is the K[H]-module obtainedfromL by restrictingthe operatorsto K[H]. 2. Preliminaries For laterreference,we nextlista numberof well-known results. The first six holdforfieldsof arbitrarycharacteristic. (2.1) Let A and B be K[G]-modules,and let a and 53respectivelydenote the corresponding representations.Then A* 0&B is isomorphic to the K[G]moduleW obtainedin the followingmanner: W is isomorphic to HomK(A, B) as a K-space and if w C W, x C G, then WX =

f(x-')w5(x)

1

For our purposes,we even couldhave definedA* 0 B as the moduleW. Clearly, (2.2) The invariantelementsof A* 0 B in (2.1) correspond to theelements of HomK[G](A, B) in W. 1 Since we work with right modules, we write ad forthe transformation obtainedby performingfirsta transformation,) and then a transformationC;,2 = Ccp.

JORDAN'S THEOREM IN CHARACTERISTIC p

121

The nexttwo resultsare due to G. W. Mackey[11], cf. [6, ? 44]. (2.3) Let H be a subgroupof G and let M be a K[H]-module. Then (MG)W P3

L

((M")Hnx-lHx)H

wherex rangesover a completesystemof (H, H)-double cosetrepresentatives in G. (2.4) Let H be a subgroupof G. Let L be a K[G]-moduleand Ma K[H]module. Then MG?L

L(M0L&

L.

(2.5) (Clifford'sTheorem,cf. [6, (49.2)]). If H < G, and if L is an irreducibleK[G]-module,then LH

EX MX

whereM is an irreducibleK[H]-module, and x ranges over a subset of a completesystemof cosetrepresentativesof H in G. We notea corollary. (2.6) If M Lo(H) in (2.5), thenH is includedin thekernelof L. p. If L is an Fromnow on, it will be essentialthat K has characteristic indecomposableK[G]-module,and if H is a minimalsubgroupof G such that L IS forsomeK[H]-moduleS, H is calleda vertexofL. J. A. Green[7], using resultsof D. G. Higman,has shown: (2.7) If H is a vertexof the K[G]-moduleL, thenH is a p-groupwhich is uniquelydeterminedup to conjugation. A subgroupH1 of G containsa conjugate of H, if and onlyif L I (LHl)G. 2 ofG as thefuncWe definethemodularcharacterX ofa K-representation tionon G whosevalue \(x) is thetraceof 2(x) forx e G.2 We say thata module affordsa characterX, if X is the characterof the associatedrepresentation. (2.8) ([5, ? 6]; [2, ? 3]; or [6, ? 82]). Let L1 and L2 be two irreducible K[G]if and onlyif theyaffordthe same modules. Then L1 and L2 are isomorphic, character. Let &2be a fixedalgebraicnumberfieldnormaloffinitedegreeoverthefield Gth rootsof unity, Q ofrationalnumbersand chosensuchthatt2containsthe IG of a subgroupof G is equivalentto an f2-repreand that each C-representation sentation.3We choosea fixedextensionv of the p-adic exponentialvaluation ofQ to &2. Throughoutthepaper,R willbe theringof local integersof t2 with In [2] and [5], the term modular character is used in a differentsense. As is well known, our conditionsare satisfiedif Q is the fieldof IG Ithrootsof unity over Q. This is immaterialfor our purposes. 2

3

122

BRAUER AND FEIT

regardto v, p will be the primeideal of R, and 0 will be the residueclass map of R ontoR/p= R0. Moreover,we identifyRI witha subfieldof K. If G is a p'-group,everyK-representation of G is completelyreducibleand of G is equivalentto a representation obtainedfroman everyK-representation of G by the residue class map 0. Thus, Jordan'stheorem R-representation implies:

(2.9) Thereexists an integer-valued functionJ(n) definedon the set of positiveintegerssuch that if G is a finitep'-subgroupof GL,(K), thereexists a normalabelian subgroupA of G with IG: A I < J(n). (2.10) [1, p. 101] If p is a modularirreduciblecharacterof G and if K1 is a subfieldof K suchthat p(x) C K1forall p'-elementsx of G, thenq' is afforded by someK1[G]-module. (2.11) [7,Th. 10]. Let L be an indecomposable K[G]-modulewhichaffords themodularcharacterX. If thevertexof L is properlyincludedin a Sylow-psubgroupP of G, thenX(x) 0 forall p'-elementsx of C,(P). In particular, X(1) = 0, i.e., dimK L is divisibleby p. For basic resultsin the theoryof blocks,the reader is referredto [2] or andmodularirreducible ap-blockBis a setofordinary [6,Ch.XII]. Bydefinition, characters. We shallalso speakofthe irreduciblemodulesand representations inB. A reduciblemoduleis saidto belongtoB, ifall its irreducible constituents belongto B. Everyindecomposable K[G]-modulebelongsto someblock,cf. [7]. The p-blockof G containingLo(G) will be denotedby BO(G). (2.12) [4]. Let P be a Sylow-p-subgroup of G. Each p-blockcontainsat mostIP I2modularirreduciblecharacters. (2.13) [5, p. 578]. Let Fj and Fj be irreducible K[G]-moduleswhichlie in the same block. If dimKFj E 0, dim, Fj E 0 (modp), then some irreducible constituentof R* 0&Fj is in BO(G). The remainerof this-sectionis concernedwith somesimplelemmaswhich are neededforthe proofof the maintheorem. (2.14) Let L be a K[G]-modulesuch that everyirreducible constituent of L is equivalentto L0(G). If H is the kernelof L, thenG/H is a p-group. to assumethatL belongsto a faithfulrepresentation PROOF. It suffices 2. By assumption,the characteristicrootsof 2(x) are 1 foreveryx C G. Hence 2(x) - I is nilpotentand so 2(XPa)

-

I =2;(Z)w-I

-

(2(x)

-

I)p=

0

forsufficiently largeintegersa. Since2 is faithful,each x is a p-element,and this is what we have to show.

123

JORDAN'S THEOREM IN CHARACTERISTIC p

(2.15) Let G be a finitegroup of theformP x D whereD is a p'-group. Let L be an indecomposableK[G]-module. There exists an indecomposable U ?& V.4 K[P]-module U and an irreducibleK[D]-module V such that L PROOF. Let 2 be the representation corresponding to L. The restriction 2 ] D of 2 to the p'-groupD is completely reducible. For y C P, 2(y) commutes withall 2(z), z C D. If 2 ID had twonon-equivalent irreducibleconstituents, it wouldfolloweasily fromSchur's lemmathat L was decomposable.Hence all irreducibleconstituentsof 2 ID are equivalent. Introducingsuitable coordinates and writingrepresentations in matrixformwe can set $(z) = I0 (9iY(Z)

(forz C D),

where1. is the unitmatrixof somedegreeu, and where 5i is an irreducible representation of D. Again by Schur's lemma,each 2(y) withy C P has the formU(y) 0&t(1) whereU(y) is a matrixof degreeu. Clearly,y U(y) is a 0 U of P. Then 2(yx) = 11() Q3(z). If U and V are the reprerepresentation sentationmodulesof U and S3 respectively,we have L U?g V. If U was decomposable,L wouldbe decomposable. -

(2.16) Let N be a group whichhas a normal subgroupP x D such that P is a Sylow-p-subgroup ofN. If L is an indecomposableK[N]-module,then LPXD is a direct sum of the form

a LLPXD

'

ei__U0TV Dfae=1 U"(g&Vi I

where the Vi are distinct irreducible K[D]-modules, the Ui are K[P]modules,and U10 V1,** , Ue0 Ve are conjugateunder theactionof N. If Lo(P) 0&W ILPXDfor someK[Dj-module W, L is irreducible. If L is irreducible, then P is included in the kernel of L, dim, L L 0 (modp), and P is a vertexof L. PROOF. Clearly,D is a p'-group. SinceP x D containsa vertexofL, there exists an indecomposabledirect summand S of LPXD such that L I SN, cf. (2.7).

U? V where U, V are as in (2.15). As By (2.15), S has the formS it follows from(2.3) that LPXDis a direct sum of modules LPXD I (SN)PXD, Ux 0 Vx whereU is an indecomposable K[P]-module, V an irreducibleK[D]module,and x rangesovera set of elementsofN. Let V1,***, Vedenotea full modulesoccuringamongthe Vx here. Let Ui denotethe set ofnon-equivalent direct sum of the modules Ux for which Vx LpXD Since LPXD 4

@e=Ui

u

Vi. Then

Vi

LPXD for x e N, it follows easily fromthe Krull-Schmidttheorem

Of course, U is identifiedwith a K[G]-module whose kernel includes D, and V is

whosekernelincludesP. witha K[G]-module identified

124

BRAUER AND FEIT

Ue0&Ve are conjugate under the action of N. If that U10&Vi, Lo(P) 0&W ILPXDforsomeindecomposable K[D]I-moduleW, it followsfromthe theoremthatLo(P) (0 W I Ui(0 Vi forsomei. HenceLo(P) I Ui, Krull-Schmidt and we see that Ui x uL,(P) forsomepositiveintegeru. Thismustthenhold forall i. Thus, P is includedin the kernelof L. Then L can be consideredas an indecomposable KI[N/P]-module.Since N/P is a p'-group,L is completely reducibleand consequently,L mustbe irreducible. If L is irreducible, thenas P < N, by [2, (9D)] P is includedin the kernel of L, and L may be consideredas an irreducibleK[N/P]-module. Hence dimK L t 0 (modp). By (2.11), P is a vertexof L. (2.17) Let L1 and L2 be twoindecomposableK[G]-moduleswhich afford

themodularcharacters AssumethatX1(1)# 0, X2(1) -X,X2,respectively.

# 0.

Let P be a Sylow-p-subgroup of G. Then L1 and L2 belongtothesame p-block of G, if and onlyif -XJ(X)/X1(1) =

_X2(X)/_2(1)

for all p'-elementsx of CQ(P). PROOF. Supposethat Li belongsto thep-blockBi, i = 1, 2. SinceXi(1)# 0, Bi has fulldefectand we maychoosean ordinaryirreducible characterXiC B withXi(1) t 0 (modp). ThereexistR-representations XiwiththecharacterXi. Let x C G, and let s be the sum of the conjugatesof x in the group algebra. Then Xi(s) = cji with

ai=I G: CG(x)lXi(x)lXi(l)C R.

If $ is a constituentof XV,then$i(s) = o41. If $ has the modularcharacter (pi we see on formingthe trace that G: CG(x)l0pi(x) = coapi(1)

Thisremainstrue,if(piis replacedbyanymodularcharacterinBi, in particular, if (pi is replacedby xi. Now, a knownlemmaon blocks([2, (6F)] or [6 (86.25)]) appliesand yieldsthe statement. (2.18) If q is a modularirreduciblecharacterin BO(G),everyalgebraically conjugatecharacterqp is in BO(G). PROOF. We may set qp = qI where 7 is an automorphismof R0. There exist

a of &i withRg = R, lp = p whichinduce7 in R0 = R/p. We automorphisms X of G forwhichXVhas can choosean absolutelyirreducibleR-representation a constituent withthemodularcharacterp. Then (XO)0has a constituentwith the modularcharacter i = cpi. WithX, the algebraicallyconjugaterepresentationXI lies inBO(G), cf. [3, Lem. 2]. We now see that (pibelongsto BO(G), as we wishedto show.

JORDAN'STHEOREMIN CHARACTERISTIC p (2.19) Let F bean irreducibleK[G]-modulesuchthatdim, F

125 0 (modp).

Then Lo(G) appears at least with multiplicity2 as irreducible constituent of F*OF. F* 0 F as themoduleW in (2.1), (2.2) withA PROOF. We can interpret B = F. Schur's Lemma shows that the submodule W1of invariant elements of W consists exactly of the scalar elements cI with c C K. Let W2 be the ofF intoF oftrace submoduleof W consistingofthe K-lineartransformations W2. Since dimKL _0(modp), it followsthat 0. Clearly, W1 Lo(G) WI W, - W2. Thus, Lo(G) occursas irreducibleconstituentof W of multiplicity at least 2. 3. Proof of the theorem

ofthe finitegroupG. The Sylow-p-subgroup Let P be a Sylow-p-subgroup of C,(P) is the centerZ(P) of P and Burnside'stheoremshows that C,(P) is the direct productof Z(P) and a p'-groupD. The p'-elementsof CG(P) are exactlythe elementsof D. Set N= N,(P). Then PC,(P)

= P x D < N = N,(P).

This notationwill remainfixedthroughoutthe rest of the paper. (3.1) Let F be an irreducible K[G]-module with dimKF t 0 (modp). Then we have a formula

(E,

FN

Mt,

whereeachM, is indecomposable, dimKM1E 0 (modp) and dimKM-- 0 (modp) for i > 1. If F and M1affordthemodularcharactersq and p, respectively, 9p(x)= p1(x)forall x e D. Also,F IMf1.Finally, (MlG)Nhas a unique indecomposabledirectsummandwith vertexP. PROOF. Set FN @,M, with indecomposableM,. By (2.11), Fhas vertex P. As P cz N, (2.7) showsthatF IMGforsomei, say F IMG. If M1has vertex H c--P, then M1 I ((Ml)Hf)Nwhence MIG I ((Ml)H)G. Since F I MG, a vertex of F is containedin H. Thus, H = P, i.e., M1has vertexP. of N = N,(P), it follows that P Since P is the onlySylow-p-subgroup x-1Nxforx e G, x X N. Then N n x-'Nx does not contain a Sylow-p-subgroup of G, and (2.3) shows that (MIG)N has a unique direct summandM1 with vertex P. The same is true for FN since FN I (MG)N and M1IFN. Now (2.11) can be applied to M, withi > 0. If M, affordsthe modularcharacterpi, then pu(x) 0 for xcDandi>O.

Thus

9(X)

In particular,jil(1) =

-

E,

A(X)=-

(X) .

(1) # 0, p(1) = 0, fori > 0. This completestheproof.

BRAUER AND FEIT

126

If F satisfiesthe assumptionsof (3.1) and if M1is definedas in (3.1), we shall say that M1correspondsto F. (3.2) If F and M1are as in (3.1), thenF is in BO(G),if and onlyif M, is in BO(N). We have F Lo(G), if and onlyif M1 Lo(N). PROOF. By (2.17), F is in BO(G),if and onlyif 9(x)/p(1) = 1 forall x E D. = 1. Again, this is so, if and onlyif M1is in BO(N). By (3.1), thenp1(x)/1p(1) If F Lo(G), clearlyM1= FN Lo(N). Conversely,assume that M1 Lo(N). By (3.1), FI Lo(N)G; and by (2.7), Lo(G) ILO(N)G. If F 6 Lo(G), this yieldsthat(Lo(G) F) I Wand so (Lo(N) eDLo(N)) I (MG)Ncontraryto thefact directsummandof (MG)N has vertex P. Thus, that onlyone indecomposable

e

F

Lo(G). The crux of the proofof the theoremis containedin the next lemma.

(3.3) SupposethatL is an irreducibleK[G]-modulewithn = dimK L > 1. Then at least one of thefollowingfacts holds. ( i ) G has a normal subgroupof index p. (ii) Thereexistsan irreducibleconstituentF ofL* 0 L 0 L* 0 L with F in BO(G),F o Lo(G). (iii) Thereexistsan irreducibleconstituentF ofL* (0 L withdim.KF t 0 (modp) and F 0 Lo(G), such that if E is the K[N]-module corresponding to F, and if H is the kernelof ED, then ID: H I < J(n) whereJ(n) is defined by(2.9). F8 PROOF. Assume that neither (i) nor (ii) holds. Let F0 = LO(G),Fe, of L* 0 L. denotethe distinctirreducibleconstituents Considera K[G]-moduleZ such that each irreducibleconstituentof Z is isomorphicto a constituentof L* 0 L 0 L* 0 L. Since (ii) is excluded,an to Lo(G) or it is not in BO(G). Since irreducibleconstituentof Z is isomorphic constituentof Z lyingin (i) is excluded,(2.14) showsthat an indecomposable of Lo(G) to Lo(G). This impliesthat if r is the multiplicity BO(G)is isomorphic the K-space of dimension ZOof -ofZ, then r is the as irreducibleconstituent invariantelementsof Z. This remarkwill be appliedseveraltimes. ( Fj with0 ? i, j < s. By (2.2), ZO Hom[,G](F., Fj). First,takeZ = 0i* For i = j, Schur's lemma shows that r 1, and it followsfrom(2.19) that 0 and now (2.13) dimKF. is not divisibleby p. Next, take i # j. Here, r ...,

implies that F0, F1, *, F. lie in s + 1 distinct blocks of G.

ofL* 0 L. Sinceall the irreduciconstituent Let Y be anyindecomposable of Y lie in the same block,theymustall be isomorphicto F. ble constituents seriesof Y. We for some i; 0 ? i ? s. Let b be the lengthof a composition maythenapplythe remarkabove to Z= F' 0 Y. From what has already

JORDAN'S THEOREM IN CHARACTERISTIC

p

127

b as been shown, we concludethat L4(G) appearswiththe exact multiplicity irreducibleconstituentof Z. Thus, herer = b and, by (2.2), b - dimK HomK[G] (Ft, Y) . Repeat now the same argumentwith Y replacedby the maximalcompletely reduciblesubmodule(the socle) Y. of Y. If b. is the lengthof a composition seriesof Y., we find bo= dimKHomKG[](Ft, YO).

Any e HomK[,G(F., Y) maps F. eitheronto a simple submoduleof Y or (0). withHome,[q (F., YO). Thus, )(F,) ' YOand HomK[,] (F., Y) can be identified Hence b = boand the submoduleYOof Y must coincidewith Y. Since YO is we see that Y is irreducible. completelyreducibleand Y is indecomposable, constituentof L* 0 L, it followsthat Sincethisappliesto anyindecomposable r

L*

0L

is completelyreducible, say

L*

(3.4)

0

L

(E=

aF

(ai > 0).

,

ofL* 0 L, all irreducible constituents F. SinceLo(G)occursas constituent of L* 0 L occur among the constituentsof L* 0 L 0 L* 0 L. We may thereforetake L* 0 L forthe moduleZ above. Here again r = 1 and, by (2.19), dimK L t 0 (modp). We also see that ao = 1 in (3.4). Since dimK L t 0 (mod p), (3.1) applies to L and we may set M eDXil eX2e ff ... whereM and the Xi are indecomposable K[N]-modules,the vertexof M is P whileeach Xi has a vertexproperlycontainedin P. Thus, LN

(L* 0 L)N

(3.5)

(M* 0 M) e

M' ,

whereM' is a directsum of termsof the formsM* 0 X,, Xi*0 M, Xi*0 Xj.

It is seen easily form(2.4), that each indecomposabledirect summandof M' has a vertex properlycontained in P. 0, 1, *.. , s. Thus, (3.1) also As we have seen, dimK F'Et 0 (mod p) for i applies to FoyF1, *.., F8. Let M, be the K[N]-module which correspondsto F.; MO= Lo(N). It follows from (3.4) and (3.5) that (3.6)

(M*

0 M) eDM'

fflEs=o aiMi eDM",

whereeach indecomposable directsummandof M" has a vertexproperlycontained in P. Clearly, MO= Lo(N) while M1, M2, *.., M1 do not lie in BO(N), cf. (3.2). By (2.16), we can set MPXD

e1

Ui O Vi,

BRAUER AND FEIT

128

Ve are distinct irreducible K[D]-modules while Ui, *, Ue are K[P]-modules which are mutually conjugate under the action of N. Thus,

where Vi, *,

(M* 0 M)PXD

= i (U,Uj0(V,Vj) )0 0 Uj) 0 Lo(D) e)S Ee=1(Up 30

where S is a K[P x D]-module which does not contain LO(P x D) as constituent. Since Lo(N) has vertex P, (3.6) shows that Lo(N) IM* (0 M. Thus LO(P x D) I (M* 0 M)PXD. It follows that Lo(P) I Uj* 0 Uj for some j. Since the Uj are conjugate under the action of N, this impliesthat Lo(P) J Uj* (g Uj for j = 1, *.. , e and eLo(P x D) I (M* 0 M)PXD. If e > 1, there exists an indecomposable direct summand T 0 MOof the right side of (3.6) such that LO(P x D) I TPXD. By (2.16) T has vertex P, and so T - M, for some i > 0. Also, T - M, is irreducible and by (2.6), P x D belongs to the kernelof Mi. Now, (2.17) shows that Mi lies in BO(N). This has already been seen to be false and hence we must have e 1. Thus,

=

(3.7)

MPXD

U0

V,

where V is an irreducible K[D]-module and U is a K[P]-module such that Lo(P) I U* $? U. Set dimK V= d. If d = 1, then V* 0 V Lo(D) and D is in the kernel of M*OM. By (3.6), Mi M*0Mfori = 0,1, s..,s. So,Disinthekernel of each Mi. By (2.17), Mi then is in BO(N). This is possible only for s =0. Then n -1 by (3.4) contraryto the assumption. Thus, d > 1. Let X be an irreducibleconstituentof M and let Hobe the kernel of X. By (2.16), P C Hoand so N/HOis a p'-group. Thus, by (2.9) there exists a subgroup AOsuch that Ho ' AO< N, I N: AoI < J(n), and that AO/HO is abelian. Set A1 = AOn D, H1 = Ho n D. Then A1 < N, ID: A, I < J(n), and A1/H1is abelian. By (3.7), XD uOV for some positive integer uO. It follows that H1 is the kernel of V. If A is an abelian group and V a K [A]-module, then V* 0 V contains LO(A) at least with multiplicitydimK V. Applying this with A = A1/H1,we see that (V* 0 V)A1contains LO(A1)at least with multiplicityd > 2. As V is irreducible, V* 0 V contains Lo(D) with multiplicity 1. It follows that there exists an irreducibleconstituent W 6 Lo(D) of V* 0 V such that LO(A1) WA1. By (2.6), A1 is in the kernel of W.

Since Lo(P) 0DWI (M* (0 M)PXD by (3.7), it follows from (3.6) that Lo(P) 0 WI TPXD for some direct summand T of the right hand side of (3.6). By (2.16), T is irreducibleand has vertex P. Then T Mi for some i with 0 ? i < s. Clearly, i # 0. Let H be the kernel of (Mi)D. Since A1 belongs to the kernelof W, and A1 < N, by (2.6) A1belongs to H. So, I D: H ?< ID: A1 I < J(n).

JORDAN'S THEOREM IN CHARACTERISTIC p

129

Set F = Fi, E = Mi, We have Case (iii), and (3,3) is proved; (3.8) Let L be an irreducible K[G]-module with dimK L

= n > 1. Sup-

pose that G contains no normal subgroup of index p, Then there exists an irreducible constituent F of L* ® L ® L* ® L such that, if G0 is the kernel of F, then 1

< 1 G: Go 1 < 1 GL1P1%,Jn4(p) 1 •

Let F be an irreducible K[G]-module with dimK F = d, let G0 be the kernel of F and let cp be the modular character aff orded by F. Suppose that cp has e algebraic conjugates in K. The subfield generated by all cp(x) as x ranges over the p'-elements of G has p• elements. By (2.10), G/G0 is isomorphic to a subgroup of GLip•) and so G/Go is isomorphic to a subgroup of GLd.(p). Hence I G:Go 1 ~ 1 GLd,(p) 1, By assumption, (3.3) (i) does not hold. If (3.3) (ii) holds, and if Fis chosen accordingly, G -=I=- G0 since Ff'/, Lo(G). By (2.12) and (2.18), cp has at most I P 12 algebraic conjugates in K. The result follows from the remarks above and the fact that dimK F < n4. Suppose then that (3,3) (iii) holds, and again choose F accordingly. As before, G0 -=I=- G. Let B 1 , B 2 , ···,Badenote all the distinct p-blocks which contain algebraic conjugates of the modular character cp afforded by F. Suppose that cp i E ß 3 where 0' 3 is an automorphism of R 8• Let F(il denote a K[G]-module which affords cp"i, let E(JJ be the K[N]-module corresponding to F(il and let µ 1 denote the modular character afforded by E(j). If µ is the modular character afforded by E, clearly, µ 1 = µ•i for j = 1, 2, · ·,, a. We have d = dimK F < n 2 < n4. By the remarks above, the proof will be complete when we can show that the number e of algebraic conjugates of cp is smaller than IP l2J(n). By (2.12), it suffices to show that a < J(n). If i -=I=- j, then cp"i and cp"i are in distinct blocks. Thus, by (2.17), cp"i(xi 3) -=l=cp03(xi1) for some xii ED; and by (3,1), µ"i(xi 1) -=I=- µ 01(xi 3). This shows that the restriction µ D has at least a distinct algebraic conjugates. On the other hand, the number of conjugates of µ 1D is at most equal to the index ID: HI of the kernel Hof En, By (3.3) (iii), a < J(n) and we are finished. Let G be a group and let L be a K[G]-module. The type t(G, L) of (G, L) is defined by t(G, L) = (m, n, a) where pm is the order of a Sylow-p-subgroup of G, n = dimK Land a is the multiplicity with which Lo(G) occurs as a constituent of L* ® L ® L* ® L. Define the ordering -< by (m1 , n 1 , a 1) -< (m, n, a) if one of the following is satisfied ( i) m1~ m, n, ~ n, Q. 1 ~ o..., PROOF.

0

1

399

REPRINT OF [94]

BRAUER AND FEIT

130

(ii) m1 m, n, < n, (iii) m1 m, n, = n, a, > a. If H is a subgroupof G, thenclearlyt(H, LH) -< t(G,L) or t(H, LH) = t(G,L). Observethatift(G,L) = (m,n, a), then a < n4. Thus thereare only manytypesless than a givent(G,L). finitely of G, and let IP = pt. Let X be a (3.9) Let P be a Sylow-p-subgroup where faithfulK[G]-modulewithdimK X = n. Let g(m,n) = IGLIpl2J(2,f)4(P)I exists there Then abelian. G is not that J(n) is definedby (2.11). Assume Go< G with I G: GoI < g(m,n) such that t(G,,XG) -< t(G,X). PROOF. It maybe assumedthat G has no normalsubgroupof index p; otherwisethe resultis trivial. Suppose firstthat everyirreducibleconstituentof X has K-dimension1. whichhas the same set of comreducibleKj[G]-module Let X1be thecompletely positionfactorsas X. By (2.14), P is thekernelof X1 and G/P is abelian. Thus G is solvableand by a Theoremof P. Hall [9], [8, p. 141] G containsan abelian P l. Let G0= nl x-'Ax. Then G. < G, G: Go < subgroupA with IG: A P I! _ g(m,n) and sinceG is notabelian,I P I 1 so that t(G.,X00) K t(G,X). L of X has K-dimensionat constituent Supposenextthatsomeirreducible least 2. Let F and G. be definedby (3.8). If t(G,X) = (m, n, a) and t(G.,X0) = (m1, n1,a1), then clearly m1 < m and n, = n. Since G. is the kernel of F and

F 0 Lo(G),a, > a. Thus t(G.,X,0) -< t(G,X). The resultfollowsfrom(3.8). (3.10) Thereexists a functionh(m,n, a) such that if X is a faithful K[G]-module,then I G: A I < h(t(G,X)) for somenormal abelian subgroupA of G. PROOF. Induction with respect to -).

We form the Gauss periods of

length mx

(2.4)

α>ίλ) = Σ t f V

where z> ranges over the integers, k, k 4- tx, k + 2 tx, . . . , Λ + (w λ - l)fχ. Clearly,

GROUPS WHOSE ORDER CONTAINS A PRIME TO THE FIRST POWER ( λ)

cn k remains unchanged, if k is changed (mod tλ).

If P is a fixed generator

ι

of φ and if F G % we have with d\V)

(2.5)

383

= θ{M~ VM)

T/\PV) = £ ( A 1 Σ V ( ^ ) ^ U

(2.6)

C ίλ)

|

V

λ )

with ε , e = ± 1 ; cf. [1] I, Theorem 4. We take Bo as the principal >block of ®. m0 = m.

Then 0o = l, r o = l , i.e. U = f, 0)

0)

We shall choose here the notation such that Cί, = 1 and that s = 1 0)

for i = 0, 1, . . . , a - 1 and s = - 1 for z = a, a + 1, . . . , m - 1. Set ω{k] = 7?/,.

The ^ are the Gauss periods of length m.

It is seen easily

that 7ι= τ Σω/ x Λ/.

(2.7)

v =0

As is well known, τ?ι, τ?2, . . . , ^/ form a Z-basis for the ring of algebraic integers of the field Q ί ^ c Q ^ ) (2.8)

*> we have ΣT?/-

-1.

t =l

With each ηt, t h e conjugate complex number rji a p p e a r s in {771,^2,... :ot). We shall use the notation (2.9)

Vi = Vi"

We also give some formulas for the multiplication of the 77/.

These can be

proved easily directly, but we apply a group theoretical method. Let Wp,t denote the metacyclic group of order p(p - I) ft defined as group with generators P, M with the relations Pp = l, Mm=l,

M

Then Ίίlpj satisfies our conditions for @ and t= {p — l)/m conjugate classes of elements of order p.

is the number of

Here, B) is the only >block.

The

s

non-excepetional characters can be identified with those of Dί/,,//, i.e. of a cyclic group of order m. 0)

k

Q = C , (0£k We use the notation Q for the field of rational numbers and the notation Z for the ring of integers.

384

RICHARD BRAUER

m, ε{0) = l. The product VTγf will contain : \ if and only if Zy0)C* = Z0). Taking the element P, we see that this is so, if and only if i- j . Hence we may set τn-1

ί

Li Ij

= 2 J CtfrAr

+ 0,7 2-1 C

r=l

with integral ajr^O and Kronecker it

follows that Σ (*μ + ε(0)2)Cμ0) -f Σ (y> - ε(012)Cl0)

μ =0

V =a

vanishes for ^-regular elements. Since the C/0) are still linearly independent on the set of i>-regular elements, we find #μ = - ε(0)2, y^ = ε(012. Now, the prinicipal character Cί0) appears with the multiplicity - 1 in Ξ; we have x o = - 1. Hence e(0ϊ = 2 = Σ2r, *μ= - 1 , ^ = 1.

(4.3)

r

Thus, (Λ

y(λ)y(λ) _ y(λ)y(λ) _

λλ

α -1 m-1 t ί0) ((P ( 0 ) "S^Λ _l. V Λ _l_ V ^