Theory of Finite Simple Groups [1 ed.] 0521866251, 9780521866255

This book provides the first representation theoretic and algorithmic approach to the theory of abstract finite simple g

508 25 38MB

English Pages 674 Year 2006

Report DMCA / Copyright

DOWNLOAD FILE

Polecaj historie

Theory of Finite Simple Groups [1 ed.]
 0521866251, 9780521866255

Table of contents :
Contents
Acknowledgements
List of symbols
Introduction
1. Prerequisites from group theory
1.1 Presentations of groups
1.2 Generalized quaternion groups
1.3 2-Groups without non-cyclic abelian characteristic subgroups
1.4 Transfer and fusion of elements
1.5 Coprime group actions
1.6 Simple groups with dihedral or semi-dihedral Sylow 2-subgroups
1.7 Simple groups with strongly embedded subgroups
2. Group representations and character theory
2.1 Algebras, modules and representations
2.2 Conjugacy classes of finite groups
2.3 Characters of finite groups
2.4 Characters and algebraic integers
2.5 Tensor products
2.6 Induction and restriction
2.7 Frobenius groups and exceptional characters
2.8 Brauer's characterization of characters
2.9 Projective representations and central extensions
3. Modular representation theory
3.1 Existence of splitting p-modular systems
3.2 Indecomposable modules
3.3 Semi-perfect rings
3.4 Group lattices and Heller modules
3.5 Relative-projective modules and homomorphisms
3.6 Blocks of finite groups
3.7 Defect groups
3.8 Brauer's first main theorem
3.9 Support and kernel of a block idempotent
3.10 Vertices and sources
3.11 Modular characters of finite groups
3.12 Blocks of defect zero
3.13 Green correspondence
3.14 Blocks and normal subgroups
3.15 Blocks with normal defect groups
3.16 Brauer's second and third main theorems
3.17 Blocks of defect one
4. Group order formulas and structure theorem
4.1 Suzuki's group order formula
4.2 Thompson's group order formula
4.3 Groups with a unique conjugacy class of involutions
4.4 Brauer's group order formula
4.5 A theorem of G. Frobenius
4.6 Brauer-Suzuki Theorem
4.7 Glauberman's Z*-Theorem
4.8 Structure theorem
5. Permutation representations
5.1 Permutation groups
5.2 Orbits, stabilizers and group order
5.3 Conjugacy classes of permutation groups
5.4 Endomorphism rings of permutation modules
5.5 Intersection algebras
5.6 Character formula of Michler and Weller
5.7 Algorithm for computing character values
5.8 Completion of concrete character table calculations
6. Concrete character tables of matrix groups
6.1 Norton's irreducibility criterion
6.2 From matrix groups to permutation groups
6.3 Conjugacy classes of matrix groups
6.4 Example of a concrete character table calculation
7. Methods for constructing finite simple groups
7.1 Free products with an amalgamated subgroup
7.2 Irreducible representations of free products with amalgamated subgroups
7.3 Kratzer's algorithm computing compatible characters
7.4 Michler's algorithm constructing simple groups from given centralizers
7.5 Uniqueness criterion
8. Finite simple groups with proper satellites
8.1 Mathieu groups M_{11} and M_{12}
8.2 Mathieu groups M_{22}, M_{23}, and M_{24}
8.3 The satellites of M_{24}
8.4 Generating matrices of the Held group He in GL_{51} (11)
8.4.1 Matrix \mathcal{A}= \kappa(b_i)
8.4.2 Matrix \mathcal{B} = \kappa(mljn)
8.4.3 Matrix \mathcal{C} = \kappa(ljn)
8.4.4 Matrix \mathcal{D} = \kappa(\gamma)
8.5 Janko's sporadic groups \mathsf{J}_2 and \mathsf{J}_3
8.6 Simple satellites of the alternating groups
9. Janko group \mathsf{J}_1
9.1 Structure of the given centralizer
9.2 Character table of groups of \mathsf{J}_1-type
9.3 Existence Proof
9.4 Uniqueness Proof
10. Higman-Sims group \mathsf{HS}
10.1 Structure of the given centralizer
10.2 Fusion
10.3 Existence proof of \mathsf{HS} inside GL_{22} (11)
10.4 Uniqueness of \mathsf{HS}
10.5 A Presentation for Aut(\mathsf{HS})
10.6 Representatives of conjugacy classes
10.6.1 Conjugacy classes of H = C_G(z) = \langle c, r, s \rangle
10.6.2 Conjugacy classes of D = N_H(A) = \langle r, s \rangle
10.6.3 Conjugacy classes of E = N_G(A) = \langle r, s, g \rangle
10.6.4 Conjugacy classes of U = C_G(v) = \langle v, y_1, y_3, y_4, e \rangle
10.6.5 Conjugacy classes of N_0 = N_G(3_A) = \langle a, e, y_3, k \rangle
10.6.6 Conjugacy classes of N_1 = N_G(5_A) = \langle a_1, b_1, x, y \rangle
10.6.7 Conjugacy classes of N_2 = N_G(5_B) = \langle c_2, y_3, w, v \rangle
10.7 Character tables
10.7.1 Character table of H = C_G(z)
10.7.2 Character table of D = N_H(A)
10.7.3 Character table of E = N_G(A)
10.7.4 Character table of C_G(2_B)
10.7.5 Character table of N_G(3_A)
10.7.6 Character table of N_G(5_A)
10.7.7 Character table of N_G(5_B)
10.7.8 Character table of N_G(7_A)
11. Harada group \mathsf{Ha}
11.1 The centralizer of a 2-central involution
11.2 The existence proof
11.3 Uniqueness
11.4 Representatives of conjugacy classes
11.4.1 Conjugacy classes of H = C_G(z) = \langle v_5, x_0, m, n, f \rangle
11.4.2 Conjugacy classes of D = N_H(B) = \langle v_5, x_0, m, n \rangle
11.4.3 Conjugacy classes of T = C_H(w) = \langle v_3, h, k \rangle
11.4.4 Conjugacy classes of E \cong N_G(B) = \langle v_5, x_0, m, n, s \rangle
11.4.5 Conjugacy classes of U \cong C_G(w) = \langle b, c \rangle
11.4.6 Conjugacy classes of N \cong N_G(3_A) = \langle v_2, q_2, a_3, y_1 \rangle
11.4.7 Conjugacy classes of M \cong N_G(3_B) = \langle n_1, m, z_7, n_2, n_3, n_4 \rangle
11.4.8 Conjugacy classes of N_G(5_A) = \langle f_4, v_7 \rangle
11.4.9 Conjugacy classes of N_G(5_B) = \langle f_3, f_5, v_8 \rangle
11.5 Character tables
11.5.1 Character table of H = C_G(z)
11.5.2 Character table of D = N_H(B)
11.5.3 Character table of E = N_G(B)
11.5.4 Character table of U = C_G(2_B) \cong 2.HS.2
11.5.5 Character table of N_G(3_A)
11.5.6 Character table of N_G(3_B)
11.5.7 Character table of N_G(5_A)
11.5.8 Character table of N_G(5_B)
11.6 Generating matrices \mathcal{A, B, C} \in GL_{133} (19) of \mathfrak{G}
12. Thompson group \mathsf{Th}
12.1 The centralizer of a 2-central involution
12.2 Conjugacy classes of elements of even order
12.3 Existence proof of \mathsf{Th} inside GL_{248} (11)
12.4 Determination of the 3-singular conjugacy classes
12.5 The 5-, 7- and 13-singular conjugacy classes
12.6 Group order
12.7 Uniqueness proof and concrete character table
12.8 Representatives of conjugacy classes
12.8.1 Conjugacy classes of H = \langle d, x, y \rangle
12.8.2 Conjugacy classes of E = \langle x, y, a \rangle
12.8.3 Conjugacy classes of D = \langle x, y \rangle
12.8.4 Conjugacy classes of H_0 = \langle k_1, k_2, k_3, k_4, k_5, d_1, d_2 \rangle
12.8.5 Conjugacy classes of E_0 = \langle k_1, k_2, k_3, k_4, k_5, d_5, t_0 \rangle
12.8.6 Conjugacy classes of N_0 = N_G(d_0) = \langle k_1, k_2, k_3, k_4, k_5, d_1, d_2, s_0, k_6, d_0 \rangle
12.8.7 Conjugacy classes of N_4 = N_G(d_4) = \langle d_0, d_8, k_9, k_{10}, k_8, d_4, b_1, b_2, b_3, b_4 \rangle
12.8.8 Conjugacy classes of N_1 = N_G(D_1) = \langle d_14, d_15, b_5, b_6, k_9, d_3 \rangle
12.8.9 Conjugacy classes of N_{10} = N_G(d_{10}) = \langle l_1, l_2, l_3, l_4, k_{12}, d_{16} \rangle
12.8.10 Conjugacy classes of N_7 = N_G(s_1) = \langle q_2, q_3, q_4, d_{12} \rangle
12.8.11 Conjugacy classes of N_5 = N_G(f) = \langle f, f_1, w_1, w, w, d_{13} \rangle
12.9 Character tables
12.9.1 Character table of H = C_G(z) \cong 2^{1+8}.A_9
12.9.2 Character table of E = N_G(A) \cong 2^5.GL_5(2)
12.9.3 Character table of D = N_H(A)
12.9.4 Character table of H_0
12.9.5 Character table of E_0
12.9.6 Character table of D_0
12.9.7 Character table of N_0 = N_G(d_0)
12.9.8 Character table of N_4 = N_G(d_4)
12.9.9 Character table of N_1 = N_G(D_1)
12.9.10 Character table of N_{10} = N_g(d_{10})
12.9.11 Character table of N_7 = N_G(s_1)
12.9.12 Character table of N_5 = N_G(f)
12.10 Partial character table of matrix group \mathfrak{G}
12.10.1 Intersection matrix M_2
12.10.2 Intersection matrix M_3
12.10.3 Intersection matrix M_5
12.10.4 Common Eigenvectors of M_i
12.10.5 Gollan-Ostermann Numbers of \mathfrak{G}
12.10.6 11 characters of \mathfrak{G}
References
Index

Citation preview

Theory of Finite Simple Groups GERHARD 0. MICHLER Institute of Experimental Mathematics University of Duisburg-Essen and Cornell University

CAMBRIDGE UNIVERSITY PRESS

Theory of Finite Simple Groups This book provides the first representation theoretic and algorithmic approach to the theory of abstract finite simple groups. Together with the cyclic groups of prime order the finite simple groups are the building blocks of all finite groups. The theory presented here is built on the intimate relations between general group theory, ordinary character theory, modular representation theory and algorithmic algebra. Each of these theories is developed in this book from scratch. The author then applies these theories to present proofs of classical and new group order formulas, and a new structure theorem for abstract finite simple groups. This, and the famous Brauer-Fowler theorem, provides the theoretical background for the author's algorithm which constructs all finite simple groups G having a 2-central involution z with a given centralizer Ca ( z) = H as matrix groups over finite fields. It also determines their conjugacy classes and character tables. The theory and algorithms have concrete applications and the author demonstrates this by constructing all the simple satellites of the known simple groups which are not uniquely determined by a given centralizer H. Uniform existence and uniqueness proofs are given for the modern sporadic simple groups discovered by Janko, Higman and Sims, Harada, and Thompson. This latter result due to Weller, Previtali and the author proves a longstanding open problem in the theory of finite simple groups. These applications show that the methods developed in this book can be used efficiently to calculate matrix representations, permutation representations and character tables of large groups. GERHARD MICHLER is an Emeritus Professor of the Institute of Experimental Mathematics at the University of Duisburg-Essen, and Adjunct Professor at Cornell University.

NEW MATHEMATICAL MONOGRAPHS

Editorial Board Bela Bollobas William Fulton Frances Kirwan Peter Sarnak Barry Simon Burt Totaro For information about Cambridge University Press mathematics publications visit http://www.cambridge.org/mathematics

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK www.cambridge.org Information on this title: www.cambridge.org/9780521866255 © Cambridge University Press 2006

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2006 Printed in the United Kingdom at the University Press, Cambridge

A catalog record for this book is available from the British Library ISBN-13 978-0-521-86625-5 hardback ISBN-10 0-521-86625-1 hardback

The publisher has used its best endeavors to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate. All material contained within the DVD-ROM is protected by copyright and other intellectual property laws. The customer acquires only the right to use the DVD-ROM and does not acquire any other rights, express or implied, unless these are stated explicitly in a separate licence. To the extent permitted by applicable law, Cambridge University Press is not liable for direct damages or loss of any kind resulting from the use of this product or from errors or faults contained in it, and in every case Cambridge University Press's liability shall be limited to the amount actually paid by the customer for the product.

To my wife

WALTRAUD

Contents

Acknowledgements List of symbols

page xi xii

Introduction 1

2

3

Prerequisites from group theory 1.1 Presentations of groups 1.2 Generalized quaternion groups 1.3 2-Groups without non-cyclic abelian characteristic subgroups 1.4 Transfer and fusion of elements 1.5 Coprime group actions 1.6 Simple groups with dihedral or semi-dihedral Sylow 2-subgroups 1.7 Simple groups with strongly embedded subgroups

1 8 9

12 15 25 34 38 40

Group representations and character theory 2.1 Algebras, modules and representations 2.2 Conjugacy classes of finite groups 2.3 Characters of finite groups 2.4 Characters and algebraic integers 2.5 Tensor products 2.6 Induction and restriction 2.7 Frobenius groups and exceptional characters 2.8 Brauer's characterization of characters 2.9 Projective representations and central extensions

42

Modular representation theory 3.1 Existence of splitting p-modular systems 3.2 Indecomposable modules 3.3 Semi-perfect rings 3.4 Group lattices and Heller modules 3.5 Relative-projective modules and homomorphisms 3.6 Blocks of finite groups 3.7 Defect groups 3.8 Brauer's first main theorem 3.9 Support and kernel of a block idempotent 3.10 Vertices and sources 3.11 Modular characters of finite groups 3.12 Blocks of defect zero 3.13 Green correspondence

110 112 123 128 137 144 153 160 163 170 174 181 187 189

43 52 55 64 68 74 82 90 97

viii

Contents

3.14 3.15 3.16 3.17 4

5

6

7

8

Blocks and normal subgroups Blocks with normal defect groups Brauer's second and third main theorems Blocks of defect one

200 206 212 222

Group order formulas and structure theorem Suzuki's group order formula Thompson's group order formula Groups with a unique conjugacy class of involutions Brauer's group order formula A theorem of G. Frobenius Brauer-Suzuki Theorem Glauberman's Z*-Theorem Structure theorem

242

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

243 246 248 256 259 260 266 269

Permutation representations 5.1 Permutation groups 5.2 Orbits, stabilizers and group order 5.3 Conjugacy classes of permutation groups 5.4 Endomorphism rings of permutation modules 5.5 Intersection algebras 5.6 Character formula of Michler and Weller 5.7 Algorithm for computing character values 5.8 Completion of concrete character table calculations

277

278 281 289 295 301 307 314 316

Concrete character tables of matrix groups Norton's irreducibility criterion From matrix groups to permutation groups Conjugacy classes of matrix groups Example of a concrete character table calculation

320

6.1 6.2 6.3 6.4

320 324 326 328

Methods for constructing finite simple groups 7.1 Free products with an amalgamated subgroup 7.2 Irreducible representations of free products with amalgamated subgroups 7.3 Kratzer's algorithm computing compatible characters 7.4 Michler's algorithm constructing simple groups from given centralizers 7.5 Uniqueness criterion

333

Finite simple groups with proper satellites Mathieu groups Mn and M12 Mathieu groups M 22 , M23 and M24 The satellites of M 2 4

376

8.1 8.2 8.3

334 341 348 354 364 377 388 397

Contents

8.4 8.5 8.6 9

Generating matrices of the Held group He in GL51(1l) Janko's sporadic groups J2 and J3 Simple satellites of the alternating groups

Janko group J1 9.1 Structure of the given centralizer 9.2 Character table of groups of Ji-type 9.3 Existence Proof 9.4 Uniqueness Proof

IX

406 409 418 433 433 435 447 452

10 Higman-Sims group HS 10.1 Structure of the given centralizer 10.2 Fusion 10.3 Existence proof of HS inside GL 22 (11) 10.4 Uniqueness of HS 10.5 A Presentation for Aut(HS) 10.6 Representatives of conjugacy classes 10. 7 Character tables

455 455 457 473 476 478 480 483

11 Harada group Ha 11.1 The centralizer of a 2-central involution 11.2 The existence proof 11.3 Uniqueness 11.4 Representatives of conjugacy classes 11.5 Character tables 11.6 Generating matrices A, B, CE GL 133 (19) of (5

487 487 491 497 525 530 546

12 Thompson group Th 12.1 The centralizer of a 2-central involution 12.2 Conjugacy classes of elements of even order 12.3 Existence proof of Th inside GL 248 (11) 12.4 Determination of the 3-singular conjugacy classes 12.5 The 5-, 7- and 13-singular conjugacy classes 12.6 Group order 12.7 Uniqueness proof and concrete character table 12.8 Representatives of conjugacy classes 12.9 Character tables 12.lOPartial character table of matrix group (5

554 555 557 561 571 598 607 611 616 622 641

References Index

645 656

Acknowledgements

Most of the computational work described in this book was done at the Institute of Experimental Mathematics of Essen University founded in 1989 by means of a substantial research grant of the Volkswagen Foundation. The author's research has also been supported from 1984 to 2003 by several grants of the Deutche Forschungsgemeinschaft. Therefore he wants to thank both in13titutions and Essen University (from which he retired on July 31, 2003) for their constant support for the research done by his former study group in representation theory of finite groups and computational algebra. The author has given lectures and seminars on the topics of the second and third chapters at Tulane University and the Universities of Tiibingen, Giessen and Essen between 1971 and 2003, and recently at Cornell University, where he also included the contents of Chapters 4 to 9 in his graduate courses. The participants of these courses made valuable suggestions to improve the presentation. He would like to express his thanks· to all of them. The experimental work done with his former assistants Gollan, Kratzer, Staszewski, Weller and colleague Professor W. Lempken at the Institute of Experimental Mathematics has inspired him and others to find several new theoretical results and algorithms which are treated in this book. Special thanks are also due to the author's colleagues at other Universities: H. Bender(Kiel), B. Fischer (Bielefeld), K. Harada (Columbus, Ohio), B. Huppert (Mainz), J.B. Olsson (Copenhagen), A. Previtali (Como), U. Stammbach (ETH Zuerich), J. G. Thompson (Gainesville) and K. Waki (Hirasaki). The author owes thanks to his former secretaries S. van Ackern, B. Hasel at Essen University and graduate student B. Chan at Cornell University for the enormous help they have given him with the typing of various versions of the manuscript. The difficult tables in the appendices of the last five chapters were produced by Mr. H. Schreven of Essen University using a special program developed by M. Weller. The diagrams in Chapter 5 were designed by M. Kratzer. The layout of the entire manuscript and the construction of the table of contents, list of symbols, references and index was done by B. Chan and Professor K. Dennis of Cornell University. The author is greatly indebted to all of them for their technical support. Finally, the author would like to thank the Department of Mathematics of Cornell University for its great hospitality during his stay as a visiting professor from 2003 till 2005.

List of symbols

Aut(A) cJ?(P) r(P) Z(P) Zi(P)

D1(P) M(2m)

u*v U* G'(p)

x~

Sz(q) Matn(R) EndR(M) rk(M) J(A) n ffiLfi i=l £di i

XO

Ca(x) gp 9p' GLn(F) Ker(ii) la tr( !i(g)) C-i = x 0 not dividing IDI. Furthermore, Algorithm 7.4.8 provides methods for calculating the concrete character tables of the simple target groups. In general, a finite simple group G of H-type is not uniquely determined by H. If G is a finite simple group of H-type, then any finite simple group X of H-type which is not isomorphic to G is called a proper H-satellite of G. The

Introduction

5

finite simple groups with proper simple satellites are dealt with in Chapter 8. M. Suzuki's survey article [139] lists the following examples:

Jz

H 2.84 21+o: L3(2) 21+4: A5

A6 As A12 A4k

Ds

Simple satellites of G P8L3(3) = L3(3) L5(2),He J3 L2(7) ~ L3(2)

2::s: 84 25 : 86 2~/o-l : 82k

Ag A13, Sp5(2) A4k+1, k = 4, 5, ...

G Mu M24

authors R. Brauer Held Janko M. Suzuki Held Yamaki Kondo

The existing literature does not contain a proof that this list shows all the known simple groups having a proper simple satellite. In the first three sections of Chapter 8 the (sporadic) simple Mathieu groups Mu, M 12 , M22, M23, M24 are dealt with. From J. A. Todd's abstract definitions [143] of these groups, presentations of the centralizers Hi = CM; (zi) of 2-central involutions Zi of the simple groups Mi, i E {11, 12, 22, 23, 24} are derived. The satellites £ 5(2) and Held's simple sporadic group He are constructed from H 24 by means of Algorithm 7.4.8. Similar proofs are given for Janko's sporadic groups J2 and J3 . Finally, Held's and Yamaki's Theorems are proved which classify the simple satellites of the alternating groups A4k for k = 2, 3. Kondo's Theorem [93] classifying the simple satellites of A 4 k for all k > 3 is stated without proof because of a lack of space. A finite simple group G is uniquely determined by the centralizer H of a 2-central involution z E G, if G does not have any non-isomorphic simple H-satellites. The author's uniqueness criterion is proved in the last section of Chapter 7. It is practical because it builds on the fact that one simple group .(tgi)- 1 Jfor all t ET, 1::; i::; n}. Proof. We have Utgi = U>.(tgi) for all t ET and 1::; i::; n by the definition of the map >.. Hence X is a subset of U. Since G is finite g;: 1 = gfi for some integer ai 2: 1 and all 9i E S. Let u E U. Then there is an integer m such that u = w 1 w 2 ... Wm for some w j E S. Let t 0 = 1, tj = >.(tj-l Wj) for 1 ::; j ::; m. Then

(*) Clearly, tj-l Wjtj E X ::; U for 1 ::; j ::; m - 1. As u E U equation (*) implies that tm-1Wm E u. Hence tm = >.(tm-lWm) = 1 = T nu. Now another application of (*) yields that u E (X), which completes the proof. Definition 1.1.9 Let U be a subgroup of the finite group G generated by S = {g1,92, ... ,gn}, Let T be a transversal of U in G such that 1 E T. The generators of U in X constructed in Theorem 1. 1.8 are called Schreier generators. Corollary 1.1.10 Each finite group G has a finite presentation G = (X I R), where X is a finite set of generators of G, and R, is a finite set of defining relations rk = l. Proof. Since G is finitely generated Theorem 1.1.6 states that there is a finitely generated free group F having a normal subgroup R such that F / R ~ G. Hence R has finite index in F. Thus it is finitely generated by Theorem 1.1.8. Now Theorem l.l.6(b) completes the proof.

Remark 1.1.11 Let G = (X JR) be a finitely presented group. In general, it is very difficult to determine the order IGI of G. In their seminal paper [144] J. A. Todd and H. S. M. Coxeter introduced a practical method for enumerating cosets Hg of a finite group G ,; (X I R) with respect to a given subgroup H -=J=. G. It is described in detail in M. Suzuki's book [135], its pp. 174-80. There are efficient implementations on the algebra software packages GAP [40] and MAGMA [12]. The recent versions can handle fairly large indices JG: HI on suitably large computers. For certain research projects it may be more effective to use G. Havas' standalone programs [64] without the tremendous overhead of the computer algebra systems. In general one does not know whether a given presentation G = (X I R) defines a finite group G. Let H be a subgroup of G generated by a finite

12

Prerequisites from group theory

number of words in the elements of the finite set X. Mendelsohn's Theorem asserts that the coset enumeration method of Todd and Coxeter terminates after a finite number of steps, if the index [C : H[ is finite. In this case it also yields a permutation representation of G on the cosets Hg of H. But there is no practical upper bound for [C : H[. Therefore applications of the implemented coset enumeration methods on computers may run out of enough memory space and one cannot calculate the index [C: H[ for the chosen subgroup.

1.2

Generalized quaternion groups

In this section we determine the structure of the finite 2-groups S which have a unique involution z -/- 1. This is done by showing that such a 2-group S is isomorphic to a generalized quaternion group defined below. In Chapter 4 this result will be used in the proof of the Brauer-Suzuki Theorem which asserts that a finite group G with such a Sylow 2-subgroup S cannot be simple. Definition 1.2.1 A finite group S is a generalized quaternion group of order [S[ = 2m 2: 8, if it has a presentation

s = (x, y I x2m-2 =

Y2, x2m-l

= 1,

xY

= x-l).

If m = 3, then S is called a quaternion group. Lemma 1.2.2 Let A = (a) be a cyclic p-group of order pm, where p is any prime number. Then the following statements hold:

(a) If n < 3 or pis odd, then the automorphism group Aut(A) of A is cyclic of order (p - l)pn-l _ (b) If n 2: 3 and p = 2, then Aut(A) is a direct product of two cyclic groups (a) and ((3) of order 2n- 2 and 2, where a°' = a 5 and af3 = a- 1 , respectively.

Furthermore, 1 and

Proof.

=

a 2 n - 3 , (3,

and 1 (3 are the only involutions of Aut(A),

See [156], its p. 146.

Lemma 1.2.3 Let s = (x, y I x 2 m- 2 = y 2, x 2m-l = 1, y- 1 xy = x- 1) be a generalized quaternion group of order ISi = 2m 2: 8. Then the following statements hold:

(a) N

=

(x) is a normal subgroup of S with

IS: NI =

2.

1.2 Generalized quaternion groups

(b) a 2

= z = x2

m -

2

13

for each a E S - N.

(c) Z(S) = (z) is the center of S. (d) Z(S) is the unique subgroup of order 2 in S. (e) Each subgroup U of S is either cyclic or a generalized quaternion subgroup of S.

(f) In particular, Q = (x 2 n- 3 , y) is a quaternion subgroup of order IQI = 8. Proof.

Clearly, z = x 2 m- 2 E Z(S). Since

the group S contains only one element of order 2. Furthermore, all elements a ES - N have order 4, and Z(S) = (z). The other statements are now clear. Lemma 1.2.4 Let A be a maximal abelian normal subgroup of the p-group P, where p is any prime number. Then Cp(A) = A.

Proof. Since A is normal in P, also C = Cp(A) is normal in P. Suppose that C # A. Then C = C / A # 1 is a proper normal subgroup of P = P / A. Let Z(P) be the center of P. Then C n Z(P) # 1. Therefore this intersection contains a cyclic subgroup U = U / A of order p. Hence U is an abelian normal subgroup of P such that U > A. This contradiction to the maximality of A completes the proof. Lemma 1.2.5 Let X = (x) be a cyclic normal subgroup of order 2k in the non-abelian 2-group P of order 2k+1. If k :::: 3, then there is a y E P - X such that P = (x, y), where y satisfies one of the following four sets of relations:

(a)

y2

= 1 and xY = x- 1 ,

(b) y 2

=

(c)

=1

y2

(d) y 4

1 and xY

and xY

=

x1+ 2 k-',

= x- 1+2 k-',

= 1 and xY = x- 1 .

Proof. Since Xis a maximal abelian normal subgroup of P Lemma 1.2.4 states that X = Cp(X). Hence P/X:::; Aut(X), and IP/XI= 2. Thus y 2 EX for

14

Prerequisites from group theory

each y E P - X. Now Lemma 1.2.2 implies that one of the following three cases occurs: (1) xY

=

x- 1 ,

(2) xY

=

xl+ 2 k-i,

or (3) xY

=

x-1+ 2 k-i.

Suppose y E P - X satisfies Equ. (2), but y 2 i- l. Then y 2 = x 2a EX for some 1 :S a :S 2k - 1. Since k ?: 3 there is an integer c such that

because 1 + 2k- 2 is a unit in 7L/2k- 1 z. Now let u = yxc. Then U2

X2a[xYrxc

=

X2a+2c+2k- 1 c = X2[a+c(1+2k- 2 )] =

l

(yxc)2

=

y2[xcjYxc

=

X2ax(l+2k- 1)cXc

' because 2[a + c(l

+ 2k- 2 )] = 0 mod (2k).

Clearly,

Hence (b) holds for u E P - X. Suppose that y satisfies Equ. (3), but y 2

i-

l. Then

Therefore P = (x,y) has center Z(P) = (x 2 k- 1 ). Since y2 EX commutes with 2 2k-l x and y it follows that y = x . Let u = yx. Then P = (x, u), and

Furthermore, xu = xyx = x- 1 xYx = x- 1 +2 k- 1 . Hence (c) holds for u E P-X. Suppose that y E P - X satisfies Equ. (1). If y 2 = 1, then (a) holds. If 2 y i- 1, then (x2k-1)Y

=

[xYJ2k-1

=

(x-1)2k-1

=

x-2k-1

=

x2k-1'

because z = x 2 k-i is an involution. Again Z(P) = (z) has order 2, and y 2 E Z(P). Thus y 2 = x 2 k - i , and y 4 = l. Thus (d) holds, completing the proof. Theorem 1.2.6 A non-cyclic 2-group P has only one involution z only if P is a generalized quaternion group.

i- 1 if and

Proof. If P is a generalized quaternion group, then it has a unique involution, and it is not cyclic by Lemma 1.2.3. Suppose that P is a counter example of minimal order to the converse statement. Then P is not abelian by the structure theorem of finite abelian

1.3 2-Groups without non-cyclic abelian characteristic subgroups

15

groups. Let X be a maximal abelian normal subgroup of P. Let IXI = 2k. Then X is cyclic by induction. Let X = (x). Now Lemma 1.2.4 states that X = Cp(X). Hence 1 I P/ X ::; Aut(X), because P is not abelian. Let l f T/X be any subgroup of order 2 in P/X. If k = 2, then ITI = 8. Since z = x 2 is the only involution of T there is no element t E T- X such that t 2 = 1. Hence T = (x, t) for some t ET- X with l f t 2 E X. Since T is non-abelian and t 2 E Z (T) it follows that t 2 = zx 2 • Thus t 4 = 1, and Tis a quaternion group of order 8. Since k = 2 an application of Lemma 1.2.2 yields that 1 I IT/ XI < IP/XI::; I Aut(X)I = 2. Thus P = T. Hence k 2". 3. Again let 1 I T / X be a subgroup of order 2 in P / X < Aut(X). Then T = (x, t) for some t ET - X. Since z = x 2 k - i is the unique involution of P we have t 2 I l for all t E T - X. Therefore Lemma 1.2.5 states that t 4 = 1 and xt = x- 1 . Hence T = (x, t) is a generalized quaternion subgroup of P. Suppose that PIT. Ask 2". 3, Aut(X) is a direct product of a cyclic group (a) of order 2k- 2 and a cyclic group (/3) of order 2 by Lemma 1.2.2. It also states that xf3 = x- 1 , and that Aut(X) has precisely two further involutions ry and 1 /3 which are defined by

Since P/ XI T/ X it follows that P has 2 subgroups U = (x, u) and V = (x, v) such that IU/XI =IV/XI= 2, xu = x1+ 2 k - i , and xv= x-1+ 2 k- 1 . Now Lemma 1.2.5 states that u E U and v E V may be chosen so that u 2 = l and v 2 = 1, respectively. Since P has a unique involution z E X this is impossible. Therefore P = T is a generalized quaternion group, and the proof is complete.

1.3

2-Groups without non-cyclic abelian characteristic subgroups

In this section we apply the main result of the previous section in order to prove Philip Hall's classical theorem characterizing all 2-groups S in which each abelian (commutative) characteristic subgroup is cyclic. Of course the generalized quaternion groups studied in Section 1.1 belong to this class of 2-groups. All other classes like the dihedral, semi-dihedral and extra-special 2-groups will be defined in this section. The statement and the proof of P. Hall's Theorem require some more notations, definitions and standard results on p-groups. Let p be any prime number. Then a finite group P is a p-group , if its order IPI = pm for some non-negative integer m. Since the elementary results on p-groups stated in this section can be found in many standard group theory books, they are not proved here.

16

Prerequisites from group theory

Definition 1.3.1 Let P be a p-group of order IPI =pm, where pis any prime number. The Frattini subgroup (P) of Pis the intersection of all the maximal subgroups of P. The P is an elementary abelian p-group, if H 1 > H 2 > · · · > Hk = l is a chief series of G, if each Hi is a normal subgroup of G such that for each i = 1, 2, ... , k the subgroup Hi is maximal among the normal subgroups of G which are properly contained in Hi-l ·

With these definitions we can now state the following elementary results on p-groups. Theorem 1.3.4 Let P be a p-group of order statements hold:

IPI =

pn. Then the following

(a) Z(P) =/= 1. (b) If N is a minimal normal subgroup of P, then N :S Z(P) and

(c) N

n Z(P)

!NI= p.

=/= 1 for each normal subgroup N =/= l of P.

(d) Np(U) =/= U for each subgroup U of P.

(e) P has a chief series of length n. (f) If K1 < K2 < · · · < Ks is a chain of normal subgroups of P, then there is a chief series of length n of P such that all given subgroups Ki, 1 :S i :S s, are members of this chief series.

1.3 2-Groups without non-cyclic abelian characteristic subgroups

17

Proof. See [135], its pp. 87-91. In the next lemma we collect several useful and well-known properties of the Frattini subgroup (P) of a p-group P. Lemma 1.3.5 Let P be a p-group with commutator subgroup P'. Then the

following statements hold:

(a) (P) is the smallest normal subgroup N of P such that P/N is elementary abelian. (b) (P) = P' PP, where PP= (xP Ix E P).

(c) (U) :S (P) for each subgroup U of P. (d) (P)N/N

= (P/N)

for each normal subgroup N of P.

11 :Si :S r(P)} is a basis of the F-vector space P = P/(P), where F = Z/pZ, then P = (p1,P2, ... ,Pr(P)/·

(e) If {pi(P) E P/(P)

(f) If p = 2, then (P) = P 2 = (x 2 Ix E P).

Proof. See [77], its pp. 268-73. Lemma 1.2.5 provides a classification of all 2-groups S having a maximal normal subgroup X of index IS: XI = 2. Besides the generalized quaternion groups dealt with in Section 1.1 there are three more families of such 2-groups. Because of their importance their formal definitions and properties are stated. Definition 1.3.6 A finite group Sis a dihedral 2-group of order ISi = 2m 2: 4 if it has a presentation

S = (x, y I x

2m - l

= y 2 = 1, xY = x- 1 )

If m = 2, then S is called a Klein four group.

= (x,y I y 2 = x 2 n = 1,xY = x- 1 ,n 2: 2) be a dihedral group of order at least 8. Set z = x 2 n - i , H = (x), To = (y, z), and T 1 =

Lemma 1.3.7 Let S

(xy, z). Then we have

(a) Z(S) = (z), S' = (S) = (x 2 ) and 0 1 (S') = Z(S).

(b) H contains every cyclic subgroup of S of order at least 4. (c) T0 and T1 are four groups and are not conjugate in S. Furthermore, any four-subgroup of S is conjugate to T 0 or T1 . (d) INs(Ti): Cs(Ti)I

= 2 and Cs(Ti) = Ti,

0 :Si :S 1.

18

Prerequisites from group theory

(e) S has three conjugacy classes of involutions, represented by the elements z, y and xy, respectively.

(f) Aut S is a 2-group. Proof.

See [47], its p. 261.

Definition 1.3.8 A finite group S is a semi-dihedral 2-group of order 2m ?: 16, if it has a presentation

ISi =

Lemma 1.3.9 If S is semi-dihedral of order 2n+l, then the following statements hold:

(a) S has two conjugacy classes of involutions and of elements of order 4. (b) S has one conjugacy class of Klein four subgroups and of quaternion subgroups. If T is a four subgroup or a quaternion subgroup of S, then Cs(T) = Z(T) and INs(T): Tl= 2.

(c) S possesses precisely three maximal subgroups: M generalized quaternion subgroup (x 2 , xy).

= (x), fh(S),

and the

(d) S' = (S) is cyclic of order 2n- 1 .

(e) Z(S) has order 2 and S/Z(S) is dihedral. (f) !1 1 (S) is dihedral of order 2n.

(g) A proper normal subgroup of S is either maximal or cyclic and contained in S'.

(h) If D is a dihedral subgroup of S of order at least 8, then the maximal cyclic subgroup of D is contained in the maximal cyclic subgroup of S. Proof.

See [2], its p. 9.

Definition 1.3.10 A finite group S of order be of type M(2m) if it has a presentation

ISi =

2m with m?: 4 is said to

Lemma 1.3.11 Let S be any dihedral, semi-dihedral, generalized quaternion or M(2m)-type group of order ISi = 2m. Then S has the following properties:

(a) S/(S) is a Klein four group.

1.3 2-Groups without non-cyclic abelian characteristic subgroups

(b) S has a cyclic normal subgroup M of index IS:

19

Ml= 2.

(c) Z(S), (S), IZ(S)I = 2, and S has 2m-l involutions.

+1

(f) If S is semi-dihedral, then S' = (S). Let U = (u). Then U is a normal subgroup of S such that (S)/U, ISi = IS/UI < ISi, and INI = N/U:::; (S) n M = (S)] (S). This final contradiction completes the proof.

20

Prerequisites from group theory

Lemma 1.3.13 Let S be a non-abelian 2-group. If the center Z = Z[ : G -----t U / K be the transfer from G to the abelian group U / K. Then cI>(u) = unK for each u EU, where n = IG: UI.

(e) If(IU/Kl,n) = 1, then cl>: G

-----t

U/K is surjective, and UnKer(cI>)

= K.

1.4 Transfer and fusion of elements

29

Proof. (a) By definition the commutator subgroup U' of U is the smallest normal subgroup of U with abelian factor group. For each u E U and g E G we have u- 1ug = [u, g]. Hence u- 1ug E U' for each g E U. Thus U':::; U* = (u- 1ug I u E U,g E Gwithug EU), and U* GLn(..,

where pis a non-negative integer and>.. is an integral

i=l

linear combination of the irreducible characters 1Pk E Irric(W). Thus

x(b) = -p + >..(1) for all 1-/= b EB. This completes the proof.

2.8

Brauer's characterization of characters

In this section we provide a classical proof of Brauer's famous characterization of the characters of a finite group G. Definition 2.8.1 The exponent exp( G) of the group G -/= 1 is the least integer > 1 such that gm= 1 for all g E G.

m

Lemma 2.8.2 Let G be a finite group. Then:

(a) The set cf(G, C) of all complex valued class functions 'lj; : G --+ (C is a commutative CC-algebra with identity element le E Irric( G) with respect to pointwise addition and multiplication. (b) cf(G, C) is a normed algebra with respect to the inner product

[a, /3]

1 "~ a(g)jJ(g), = Taf for all a, jJ

E cf(G, C).

gEG

Proof.

(a) This follows immediately from

(a+ jJ)(g) = a(g) + jJ(g) E (a· jJ)(g) = a(g)jJ(g) E

(C (C

for all a, jJ E cf(G, C) and g E G. (b) This is obvious by Definition 2.3.22 and Lemma 2.3.23.

2.8 Brauer's characterization of characters

91

Definition 2.8.3 The set k(G)

Char(G)

= {0

E

cf(G,C) I 0 =

L

ZiXi,

where

Xi E

Irric(G) and

Zi E

Z}

i=l

is a subalgebra of cf(G, C) with the same identity element la E Irrc(G). It is called the ring of generalized characters or the character ring of G.

Lemma 2.8.4 LetH be a subgroup ofG. Let( E cf(G,C) and'lj; E cf(H,C). Then ( ,'lj;G = ((IH ·'l/J)G.

In particular, the set of all induced class functions 'lj;G, 'ljJ E cf (H, C), forms an ideal in the CC-algebra cf(G, C) of class functions of G. Proof.

((, 'lj;G)(g)

=

((g)'lj;G(g)

=

((g)

(itl L

'l/Jo(y-lgy))

yEG

=

1 IHI

L ((y-lgy)'lj;o(y-lgy)

yEG

=

1 IHI

L (1H(y-lgy)'I/Jo(y-lgy) = ((IH. 'lj;)G(g) yEG

by Definitions 2.5.1 and 2.6.2.

Definition 2.8.5 Let p be a prime. A subgroup H of the finite group G is called p-elementary, if H is a direct product H = A x B, where B is a p-subgroup and A is a cyclic subgroup of order prime to p. Let £p

= {H

:::; G I H is p-elementary subgroup of G}.

The set of elementary subgroups of G is the union £

=

LJ £p, where P denotes the set of all prime divisors of IGI. pEP

Throughout the remainder of this section the following notations are used. Let G be a finite group of exponent m and E a primitive m-th root of unity. Then S = Z[E] is a subring of CC containing Z. Suppose that them-th cyclotomic polynomial m(X) has degreed. Then Eo = 1, E, ... , Ed is a basis of the free Z-module S.

92

Group representations and character theory

For any subring Z :S R :S (('. let:

where£ denotes the set of all elementary subgroups Ej of G. By Lemma 2.8.4 Vz(G) is an ideal in the character ring Char(G) of G. Lemma 2.8.6 Let p be a prime divisor of IGI. Let E = (a) x B be a pelementary subgroup of G, where a is a p-regular element of G. Then there is an integer valued class function 0 E Vs (E) such that 0 (ab) = I(a) I and B(aib) = 0 for all b EB and ai # a. Proof. Let n be the order of a, and w a primitive n-th root of unity. Then w is some power of E, and Z[w] :S S = Z[c:]. Then irreducible characters Ai of E with kernel Ker(>.i) =Bare defined by Ai(a) = wi for i = 1, 2, ... , n. Let 0 be the class function of E defined by B(ab) = n and B(aib) = 0 for all b E B and ai -f=. a. In particular, 0 vanishes on all b E B. Hence 0 is a class n

function of (a). Thus 0 =

I: ci>.i for some Ci

E .- 1 (g)g is a well-defined element of the group algebra AG. For each h E M the following equations hold. he,x

=

1 "~ .>. -1 (g)hg IMJ gEM

1 IMI

L A-l(gh)..\(h)gh 1 >.(h) IMI L A-l(gh)gh gEM

gEM

>.(h)e,x.

Therefore, (e.x) 2 = (1/IM)) EhEM >.- 1 (h)he,x = (1/IMI) EhEM e,x = e,x. As e,x E AM ::; AZ(G), the idempotent e,x of AG belongs to the center ZAG of the group algebra AG.

2.9 Projective representations and central extensions

105

We now claim that e.\AG is isomorphic to the twisted group algebra A.\'YG of O with 2-cocycle >.,EM. Certainly A= {g* = (1,g) E G' I g E O} is a transversal of M in G'. Since he.\ = >.(h)e>- E AM for every h E M, we have AMe.\ = Ae.\. Therefore, AGe.\ = :z=gEG ffig* AMe.\ = :z=gEG ffiAg*e>,. Let £ = {b 9 = g*e>- I g E O}. Then £ is an A-vector space basis of the algebra AGe.\. Furthermore, by Lemma 2.9.7 for all g, h E O the following equations hold:

Hence e.\AG ~ A.\'YO. (b) Let a EM. By Lemma 2.9.11 there is a>. E Hom(M,A*) such that ,*(>.)=>.,=a. Thus e>-AG ~ A>-'YO ~ A,,O. This completes the proof. Theorem 2.9.14 (Schur) Let O be a finite group and A an algebraically closed field of characteristic p ~ 0. Then the following statements hold:

(a) The Schur multiplicator M is a finite abelian group such that the order of each element M divides IOI.

(b) There is a central extension 1 -+ M -+ G -+ 0 -+ 1 of O such that every projective representation of O can be lifted to a representation of G over A. (c) If A has characteristic p > 0, then (p, IMI) = 1. Proof. (a) By Proposition 2.9.12 B 2 (0, A*) has a complement Min Z 2 (0, A*) such that M ~ H 2 ( 0, A*) is a finite abelian group. Furthermore, each element a E H 2 (0, A*) has an order which divides IOI. (b) In particular, M satisfies the hypothesis of Theorem 2.9.13. Hence there is a central extension 1 -+ M -+ G -+ 0 -+ 1. By Lemma 2.9.7 each finite-dimensional projective representation of Odefines a factor set a E Z 2 ( 0, A*), and it corresponds uniquely to an AaOmodule, where AaO denotes the twisted group algebra of O over A with respect to a. Furthermore, Lemma 2.9.8 states that AaO ~ A130 for a, (3 E Z 2 (0, A*) if and only if a= (3 mod (B 2 (0, A*)). Therefore Theorem 2.9.13 asserts that each projective representation of O can be lifted to a representation of the group algebra AG. (c) If p > 0 then pf IMI by Lemma 2.9.11.This completes the proof. The following theorem is a restatement of several results of Chapter 11 of I. M. Isaacs book [79]. Also its proof follows Isaacs' presentation closely. It will

106

Group representations and character theory

be applied later on to determine the structure of an induced module v 0 , where Vis a simple module of a normal subgroup Hof G such that Tc(V) = G. Theorem 2.9.15 Let A be an algebraically closed field with characteristic p 2:: 0. Let H .('£ 9 EG a 9 g) = a 1 E P, where a 1 is the coefficient of 1 E G, has no non-zero one sided ideals X of PG in its kernel Ker>.. Furthermore, >.(ab~ ba) = 0 for all a, b E PG. Therefore, f(x, y) = >.(xy) , x, y E PG , is a non-degenerate, associative and symmetric bilinear form on PG. (b) Define T :pc PG---+ Homp(PGpc, P) by T(y)x

= f(x, y) for all x, y

E PG.

Then Tis an P-isomorphism which is an PG-module homomorphism because f is non-degenerate and associative. (c) By (a) there is a non-degenerate P-linear map A : PG ---+ P. Let U = soc(P) be the socle of the indecomposable projective right PG-module P. By Theorem 3.3.13(d) there is a primitive idempotent O cf=. e = e 2 E PG such that P = ePG up to PG-module isomorphism. Thus 0 cf=. >.(U)

= >.(eU) = >.(Ue).

Hence U e cf=. 0 and the simple PG-module P / P J is a composition factor of U by Theorem 3.3.13(e). Another application of Theorem 3.3.13(a) and (c) yields that each indecomposable direct summand V of FG has a simple head V /VJ. Therefore, (b) implies that each direct indecomposable summand of the left PG-module pcPG ~ Homp(FGFc,F) has a simple socle. As (FG)* ~ PG and* preserves direct sums and composition lengths by Lemma 3.4.5, it follows that each indecomposable direct summand of FG has a simple socle. Therefore soc(P) ~ P/PJ. This completes the proof. Definition 3.3.18 An idempotent e cf=. 0 of a semi-perfect ring S is centrally primitive (block idempotent) if e is a primitive idempotent of the center Z(S) of S. The ideal B = eS is a block ideal of S. If B = S then Sis an indecomposable ring. Lemma 3.3.19 Every semi-perfect ring S is a direct sum of finitely many, uniquely determined block ideals Bi = eiS, i = 1, 2, ... , s. The set { ei I i = 1, 2, ... , s} is the set of all centrally primitive idempotents of S; they are orthogonal.

3.3 Semi-perfect rings

135

Proof. Let J be the Jacobson radical of S. Suppose that {xj I j = l, 2, ... , t} is a set of orthogonal centrally primitive idempotents. Then { Xj = Xj + J I j = 1, 2, 3, ... , t} is an orthogonal set of central idempotents. As S = S/ J is Artinian, t is bounded by the number l(S) of simple S-modules. Therefore every decomposition of S into block ideals has only finitely many direct summands. Suppose that S = L~=l EBeiS = I::";, 1 EBfJS are two different decompositions of S into block ideals. Then for every ei there is at least one ]j such that x = edJ -/- 0. As ei and fj are idempotents of the center Z(S) of S,

x2

= (edJ)2 = (ei) 2 (]j) 2 = edj = x

E

eiS

n fjS n Z(S).

Therefore, ei = eix + ei ( ei - x) = x + (ei - x) is an orthogonal decomposition of ei into central idempotents x and (ei - x). Since ei is centrally primitive, it follows that ei = x. Similarly one can show that fj = x. Hence ei = f 1 . A straightforward inductive argument now completes the proof. Definition 3.3.20 An indecomposable S-module M of a semi-perfect ring S belongs to a block B = eS of S if Me -/- 0. Since the centrally primitive idempotents are orthogonal, it follows that an indecomposable S-module M belongs to a unique block B of S. Definition 3.3.21 The primitive idempotents e and S are linked, if there is a sequence

f

of the semi-perfect ring

of primitive idempotents ei E S such that eiS and ei+ 1 S have a common composition factor for i = 1, 2, ... , n - l. Theorem 3.3.22 Let e and f be two primitive idempotents of the semi-perfect ring S with Jacobson radical J. Then the following assertions are equivalent:

(a) e and f are linked. (b) eS/ eJ and f S / f J belong to the same block B of S.

(c) e and f are elements of the same block ideal B of S. Proof. Suppose that e and f are linked. As the relation "to be linked" is an equivalence relation, we may assume that eS and f S have a common composition factor M. Let B = gS be the block of M. Then Mg= M implies eSg-/- 0-/- f Sg. Hence e, f EB. The equivalence of (b) and (c) is clear. It suffices to show that (a) is a consequence of (c). By Lemma 3.3.19 we may assume that Sis an indecomposable ring. Let 1 = 91 + 92 + · · · + 9n

136

Modular representation theory

be a decomposition of 1 into a sum of orthogonal primitive idempotents 9i E S such that g1 = e, each gi is linked with g1 for i = 1, 2, ... , k but gj is not linked with 91 for j = k + 1, k + 2, ... , n. If T = 1 giS and U = LJ=k+l g1 S then

I:7=

S

=T

EB U

TU

and

= 0 = UT

by Theorem 3.3.13(e). Since Sis indecomposable, U = 0. Hence f = g1r 1 + g2r2 + · · ·+ gkrk, ri E S. In particular gd I- 0 for at least one i E {1, 2, ... , k }. Thus giS f =/. 0. Hence giS and f S have a common composition factor by Theorem 3.3.13(e), which shows that e and fare linked. Theorem 3.3.23 Let R be a complete discrete rank one valuation ring with maximal ideal J(R) = nR and let S be a module finite R-algebra. If p: S--+ S = S/1rS is the canonical ring epimorphism with kernel Ker p = 1rS then the following assertions hold:

(a) The map a--+ p(a) defines a bijection between all central idempotents of S and all central idempotents of S.

·

(b) e is a block idempotent of S if and only if p( e) is a block idempotent of

s.

(c) The map B

--+ p(B) defines a bijection between all block ideals B of S and all block ideals B of S.

Proof. Certainly it suffices to show (a). Let O =/. x E S be a central idempotent of S. By Lemma 3.1.20 Sis semi-perfect. Hence there exists e = e2 ES such that x = e + 1r S. As x is central in S, it follows p(eS(l - e))

Thus eS(l - e)

~

= xS(l -

x)

= x(l -

x)S

= o.

1rS, which implies

eS(l - e)

= neS(l -

e).

But eS(l - e) is a finitely generated R-module. Thus eS(l - e) = 0 by Nakayama's Lemma 2.1.16. Similarly (1 - e)Se = 0. Hence e is a central idempotent of S. Suppose e1 and e2 are two central idempotents satisfying p(e1) = p(e2). Then p(e1e2) = p(e1)p(e2) = p(e1) and e = e1 - e1e2 E 1rS. Since e = e2, we obtain eS e1e2 = e2.

=

neS. Thus e

=

0 by Lemma 2.1.16 and e1

=

3.4 Group lattices and Heller modules

3.4

137

Group lattices and Heller modules

In order to be able to use the results of ordinary representation theory of finite groups G in modular representation theory the following concepts and notations are introduced. Definition 3.4.1 Let (F, R, S) be a p-modular system for G. A finitely generated RG-module M which is free as an R-module is called RC-lattice. The category of all RC-lattices is denoted by £(RG). If M E £(RG) then M=M®RF~M/M1r. Notation 3.4.2 Let (F, R, S) be a p-modular system for the finite group G. Let A E {F, R}. £(RG) £(AG) = { Mod(FG)

if if

A=R A=F

where £(RG) and £(FG) denote the categories of all RC-lattices and finitely generated PG-modules, respectively.

Definition 3.4.3 If A E {F,R,S} then representations r;,: G---. GL(n,A) correspond uniquely to isomorphism classes of n-dimensional AG-modules or lattices M. Given r;,, then let M be then-dimensional A-vector space or free A-module An on which GL(n, A) acts by matrix multiplication. For each m E M and g E G define mg= mr;,(g). Conversely, if Mis an n-dimensional AG-module or lattice then M affords the representation r;,M: G---. GL(n,A) defined by then x n matrices r;,M(g) corresponding to the A-linear maps: m---. mg for all m EM and g E G. Definition 3.4.4 Let (F, R, S) be a p-modular system for the finite group G and let A E {F,R,S}. For every ME £(AG), M' = HomA(M,A) is a left AG-module via XCY:m---.CY(mx)EA for all x E AG, CY E M' and m E M. The antiautomorphism g ---. g- 1 of G induces an antiautomorphism of AG which allows us to define a right AG-module structure on M' by

for all g E G, CY E M' and m E M. This right AG-module M* is called the dual of M. The AG-module M E £(AG) is self-dual if M ~ M*.

138

Modular representation theory

Lemma 3.4.5 The functor* : M---+ M* is a contravariant functor of L'.(AG) into itself. It is exact and has the following properties: (a) M**

~

M for all ME L'.(AG).

(b) * commutes with direct sums.

(c) (AG)*~ AG. ( d) * preserves projectivity of modules.

(e) If His a subgroup of G and U

E L'.(AH) then

(U ®AH AG)*~ U* ®AH

AG. Proof.

This is routine.

Definition 3.4.6 Let X, Y E L'.(AG). Then HomA(X, Y) is a right AGmodule with respect to the conjugate action of G defined by a 9 (x) = [a(xg- 1 )]g E Y for all x E X,g E G, and a E HomA(X, Y).

In particular, HomA(X, Y) E L'.(AG). Lemma 3.4.7 For each pair U, ME L'.(AG) the following statements hold:

(a) HomA(U, M)

~

U* ®AM as right AG-modules.

(b) HomAc(U, M) ~ HomAc(AAc, U*®AM), where AAc denotes the trivial AG-module. Proof. (a) By hypothesis U is a free right A-module. Hence U = I:7= 1 EBuiA. Let {u;: I 1 :::; i :::; k} be the dual basis of U*. Then U* = I:7= 1 EBu'!: A and u;: (Uj) = 6ij · Now each element t of U* ® A M has a unique representation t = I:7= 1 u;: ® vi, for some Vi EM. Define cp: U* ®AM---+ HomA(U, M) by means of [cp (u? ® vi)] (u) = viu'!:(u) for all u EU.

Suppose that

cp(I:7= 1 u;: ® vi) = 0.

Then

Hence Kercp = 0. Now let h E HomA(U,M). Then

Therefore, h =

cp(I:7= 1 u;: ® h( ui))

and cp is also surjective.

3.4 Group lattices and Heller modules

139

(b) Since HomAc(X, Y) ~ Inv 0 HomA(X, Y) for every pair of right AGmodules X and Y, assertion (a) implies that

HomAc(U, M) = Invc(U* @AM)

~

HomAc(AAc, U* @AM).

This completes the proof. Remark 3.4.8 Let (F, R, B) be a p-modular system for the finite group G.

If p : RG --+ FG is the canonical ring epimorphism from the group ring RG onto the group algebra FG with kernel Ker p = 1r RG then the following diagram

o-RG-BG=B@RRG

lp FG

! 0

links the ordinary representations of G over B and the modular representations of Gover F. In order to make this link more precise the following definitions and results are stated. Definition 3.4.9 Let (F, R, B) be a p-modular system for the finite group G and let T be a finitely generated BG-module. Then an RC-submodule M of. Tis an R-form of T if T ~ B @RM as BG-modules and Mis an RC-lattice. The FG-module f' = F @RM is called a reduction of the constants of the BG-module T. Lemma 3.4.10 Every finitely generated BG-module T has at least one R-form M.

Proof. Let {t 1 , t 2 , ... , tk} be a basis of the B-vector space T. Then k

M= LLRtig gEG i=l

is an R-form of T. The R-forms of BG-modules T are not uniquely determined by T as the following example shows.

140

Modular representation theory

Example 3.4.11 Let G =< x > be the cyclic group of order 2, and let (F, R, S) be a 2-modular system for G. The BG-module T = SG has M 1 = RG and M2 = e1R EB e2R as non-isomorphic R-forms, where e 1 = (1/2)(1 + x), e2 = (1/2)(1 - x) E SG.

Definition 3.4.12 Let L be an RC-submodule of the RC-module M. Then Lis called R-pure if L = clL = {m EM I me EL for some O i=- c ER}. Since finitely generated torsion free R-modules are free, it follows that L is an R-pure RC-submodule of an RC-lattice M if and only if O---+ L ---+ M---+ M / L ---+ 0 is an exact sequence of RC-lattices. Throughout the remaining part of this section (F, R, S) is a p-modular system for the finite group G. Lemma 3.4.13 Let M be an RC-lattice and MS= M®RS. Then there is a

one-to-one correspondence between the BG-submodules of MS and the R-pure RC-submodules of M. Proof. Let X i=- 0 be an BG-submodule of MS. Then X 1 = X R-pure RC-submodule because

clX1

= {m

E M I mr E X1

for some

nM

is an

O i=- r E R} = X1

and X = X 1 &JR S. Conversely, if Y is an R-pure RC-submodule of M then = (Y ® R S) n M = clY.

Y

f> be an indecomposable projective RC-module and let W be an RC-lattice. Then:

Lemma 3.4.14 Let

(a) HomRc(F, W) is an R-form of Homsc(F &JR S, W ®RS) (b) dimp(Hompc(P, W)) W®RF.

= dims(Homsc(F &JR S, W &JR S)), where W =

Proof. Assertion (a) is trivial. Since P ~ P/ F1r, it is easy to verify that

f>

is a projective RC-module and

Proposition 3.4.15 Let M be an RC-lattice such that M ®RS = T 1 EB T2 , T; i=- 0 for i = l, 2. Then there are R-pure RC-submodules M 1 and M 2 such that:

3.4 Group lattices and Heller modules

141

(b) M1 ®RS~ T1 ~ (M/M2) ®RS. (c) M2 ®RS~ T2 ~ (M/M1) ®RS.

(d) O ---+ M; ---+ M for i = l, 2.

---+

M/ M;

---+

0 is a short exact sequence of PG-modules

(e) If M is an indecomposable projective RC-module, and if T1 is a simple BG-module such that M1 is multiplicity free, then Homsc(T1, T2)

= 0.

Proof. The assertions (a) to (d) follow immediately from Lemma 3.4.13. Assertion (e) is a consequence of Lemma 3.4.14. Lemma 3.4.16 (Schanuel's Lemma) Let S be a ring and O ---+ K ---+ P ---+ M ---+ 0, 0 ---+ K' ---+ P' ---+ M ---+ 0 be two short exact sequences of S-modules where P and P' are projective. Then

PEBK'~P'EBK. Proof. Let X = {(p,p') E P EB P' I a(p) = ,6(p')} where a and ,6 are the given epimorphisms P ---+ M ---+ 0 and P' ---+ M ---+ 0, respectively. Then X is an S-submodule of P EB P'. Let 1r: X---+ P be defined by 1r(p,p1 ) = p for all (p,p') EX. Let p E P. As a and ,6 are surjective there is an element p 1 E P' such that a(p) = ,6(p'). Hence 1r is an epimorphism from X onto P with kernel Ker(1r) = {(O,p') E P EB P' \ ,6(p') = O} ~ Ker(,6) = K'. Since Pis projective the sequence O ---+ K' ---+ X ---+ P ---+ 0 splits by Theorem 3.3.4. Thus X ~ P EB K'. Similarly X ~ P' EB K. This completes the proof. Definition 3.4.17 Let S be a semi-perfect Noetherian ring and V a finitely generated right S-module. Then V has a projective cover P with minimal projective resolution O +--- V ,;,__ P +--- DV +--- 0, where DV = KerT < P. By Schanuel's Lemma the S-module DV is uniquely determined up to S-module isomorphism; it is the Heller module of V. The mapping n: V---+ DV is called the Heller operator. Let D1V = DV and niy = D(Di~l V) for i = 2, 3, .... Then [tiy is called i-th Heller module of V. If there is an integer n =2': 0 such that V ~ nn+l V then V is called periodic. The smallest integer n with V ~ nn+i V is the period of V. Let P; be a projective cover of [tiV, i = 0, 1, 2, ... where n°v = V. Then 0 +--- V +--- P 0 +--- P 1 +--- P 2 +--- ..• is a minimal projective resolution of V. By Schanuel's Lemma the projective S-modules P;, i = 0, 1, 2, ... are uniquely determined by V up to S-module isomorphism. Definition 3.4.18 An AG-module M E £(AG) is called projective-free if M does not contain a non-zero projective AG-submodule.

142

Modular representation theory

Lemma 3.4.19

(a) O(V1 EB V2)

~

OV1 EB OVi, ¼ E £(AG).

(b) nv is projective-free for every VE £(AG).

(c) nv = 0 if and only if V is projective. Proof. Assertion (a) is trivial. (b) Suppose that O -=/- Q :S OV is a projective submodule. Let O -----+ OV -----+ P -----+ V -----+ 0 be a minimal projective resolution of V. Since OV is a small submodule of P, Q is contained in every maximal submodule of P. In particular, Q is not a direct summand of P. However, 0 -----+ Q -----+ P implies by Lemma 3.4.5 that P* ~ Q* EB X because Q is projective. Hence Q is a direct summand of P, a contradiction. (c) This follows immediately from (b). Lemma 3.4.20 Let O -----+ W -----+ Q -----+ V -----+ 0 be a short exact sequence where VE £(AG) and Q is a projective AG-module. Then W ~ nv EB X, where X is a projective AG-module.

Proof. Let O-----+ nv-----+ P-----+ V-----+ 0 be a minimal projective resolution of V. Then Schanuel's Lemma 3.4.16 implies

P EB w

~

Q EB nv.

Therefore, the assertion holds by Lemma 3.4.19 and the Krull-Schmidt Theorem 3.2.4. The following result is known as Heller's lemma, see [69]. Proposition 3.4.21 If V E £(AG) is a projective-free AG-module then

v ~ n[n(v*)*J. Proof. Let V*. Then

O -----+

O(V*) -----+ P -----+ V* -----+ 0 be a minimal projective resolution of 0 t - O(V*)*

t-

P*

t--

Vt-- 0

is a projective resolution by Lemma 3.4.5. Now Lemma 3.4.20 completes the proof, because V is projective-free. Corollary 3.4.22 If V E £(AG) is an indecomposable projective-free AGmodule then so is nv.

Proof. Suppose nv = W1 EB W2. Then Wi is projective-free by Lemma 3.4.19 (b) for i = 1, 2. Applying Lemmas 3.4.5, Lemma 3.4.19 and Proposition 3.4.21 it follows that: V* ~ O[O(V)*] ~ O(Wn EB O(Wn. Hence Vis not indecomposable by Lemma 3.4.5, a contradiction.

3.4 Group lattices and Heller modules

143

Corollary 3.4.23 Let U, V E £(AG) be projective-free AG-modules. OU ~ OV if and only if U ~ V. Therefore, 0- 1 exists and 0- 1 (U) = [O(U*)]*.

Then

Proof. If OU ~ OV then U* ~ O[O(U)*] ~ O[O(V)*] ~ V* by Proposition 3.4.21. Hence U ~ V by Lemma 3.4.5. The second assertion now follows from Proposition 3.4.21. Remark 3.4.24 The Corollaries 3.4.22 and 3.4.23 immediately imply that the isomorphism classes of indecomposable projective-free AG-modules V E .C(AG) are the disjoint union of the Heller orbits {Oi(V) I i EN}. Lemma 3.4.25 If V E .C(RG) then

OV ~ OV.

Proof. Since every V E .C(RG) is a free R-module, tensoring with ®RF preserves minimal projective resolutions because J(FG) = J(RG)f'rrRG. Now J. G. Thompson's R-form lemma [140] can be stated as Theorem 3.4.26 (J. G. Thompson) !JV E .C(RG) is a projective-free RCmodule then V = V ®RF is a projective-free PG-module.

Proof. Proposition 3.4.21 and Lemma 3.4.25 yield the following isomorphisms:

V ~ O[O(V*)*]

~ O[O(V*)*].

Thus Vis projective-free by Lemma 3.4.19(b). Corollary 3.4.27 Let (F, R, S) be a splitting p-modular system for the finite group G. Let .. of the group algebra FG of the finite group G over the field F is the triple B -----+ e -----+ >.., where B is a block ideal of FG, e is its centrally primitive idempotent and >.. is a representative of the F-equivalence class of the linear character

>..: ZFG-----+ eZFG/eJ(ZFG) of the center ZFG of FG. Definition 3.6.20 Let M be an indecomposable AG-module with endomorphism ring E = EndAo(M). Then there is an A-algebra homomorphism (:ZAG-----+ E which is defined by ((z)(m) = mz for all m EM and z E ZAG. Let J(E) be the Jacobson radical of the local ring E, and let T : E-----+ A 1 = E / J(E) be the canonical residue class A-algebra epimorphism. Then A 1 is an extension field of A. Therefore

is a linear character of ZAG which is uniquely determined by M. The character AM is called the centrally linear character of M. The following result is due to W. Hamernik [58].

160

Modular representation theory

Proposition 3.6.21 The indecomposable AG-module M belongs to the block B - e - A of G if and only if the centrally linear character AM of M is equivalent to the centrally linear character A of B.

Proof. Suppose that M belongs to B. Then M = Me. Hence ((e) = lM E EndAc(M) = E. Thus 1 = T(lM) = T((e) E A1 = E/J(E). As A1 is an extension of A, the centrally linear character AM = T( of M is equivalent to A by Remark 3.6.14 and Proposition 3.6.16. Conversely, if A and AM are equivalent, then AM(e) = 1. Since E = EndAc(M) is a local ring, ((e) rfc J(E). Hence ((e) = lM, Thus M = Me and M belongs to B.

3.7

Defect groups

Let (F, R, S) be a p-modular system for the finite group G. In this section, A. Rosenberg's definition (see [131]) of the defect groups b(B) of a block B e - A of the group algebra FG of the finite group G over the field F of characteristic p > 0 is given, and it is shown that they are uniquely determined by B - e - A up to conjugacy by elements g of G. Lemma 3.7.1 (Brauer) Let (F, R, S) be a splitting p-modular system for the finite group G. Let B = (B, B, Bs = B &JR S) be a block of G, and let k(B) be the number of non-isomorphic simple BG-modules of Bs. If Z(B) denotes the center of the block ideal B = eFG = B &JR F, then

k(B)

= 0 for the finite-dimensional F -algebra R with Jacobson radical J ( R). Let S be the F-subspace of R generated by all commutators ab - ba, a, b E R, and let T(R) = {a ER I aPm ES forsome integerm}. Then the following statements hold.

(a) T(R) is an F-subspace of R containing S. (b) J(R)

~

T(R).

(c) dimp(R/T(R)) equals the numbers ofnon-isomorphicsimpleFG-modules. Proof. (a) This assertion holds by Lemma 3.9.2. (b) As J(R) is nilpotent, J(R) ~ T(R). (c) Lets be the number of simple (non-isomorphic) R-modules. From (a) and (b) it follows at once that T(R)/ J(R) = T(R/ J(R)). Since F is a splitting field for R/ J(R), the ring R/ J Corollary 2.1.29. Hence

Co,!

EB:=l (F)n,

by

T(R/J(R)) = E9T[(F)n,l·

i=l

Since R/T(R) ""'R/ J(R)/T(R)/ J(R), it suffices to show that dimp (T[(F)n,]) =

n; - l.

Hence we may assume that R is a ring of n x n matrices over F. Let { eij I 1 :::; i,j :::; n} be a full set of matrix units of R. Then eiieij - e 1 jei 1 ES for i # j, and O # eii - ejj = ei1e1i - e1iei1 E S for i # l. Hence n 2 - 1 :::; dimp S:::; dimp T(R). As ef~ = e 11 for all m, the element e 11 ¢:. T(R). Thus dimp T(R) = n 2 - 1, which completes the proof. Theorem 3.11.2 (Brauer) Let G be a finite group and G be a field of characteristic p > 0 such that F is a splitting field for the group algebra FG. Then the numbers of isomorphism classes of simple FG-modules equals the number c of p-regular conjugacy classes of G.

182

Modular representation theory

Proof. Let {Ci = gf I i = 1, 2, ... , k} be the set of all conjugacy classes of G. Let S be the subspace of FG generated by all commutators ab - ba, where a, b ER= FG. Clearly, the F-vector space Sis generated by all commutators hk - kh, where h, k E G. Furthermore, hk - kh = hk - h- 1 (hk)h. Thus hk and kh belong to the same conjugacy class Ci of G, and their coefficients 1 and -1 add up to 0. Suppose that x = LgEG r 9 g ES for some r 9 E F. Then X

=

I:7=1 LgECinsup(x) rgg, I:

and

r 9 =0

fori=l,2, ... ,k.

gECinsup(x)

Let T(R) = {a E R I aP' E S for some t ::CO: O}. By Lemma 2.2.4 there exist for every element g E G exactly one p-regular element gP and exactly one p-singular element 9p' such that g = 9p9p' = 9p' 9p · Let prn be the order of gp. Let n ::CO: 0 be any natural number. Then by Lemma 3.9.2 (g - 9p' )P

m+n

=(9p9p')

pm+n

- 9p'

pm+n

=0 mod (S).

Thus g = g~ mod (T(R)) by Lemma 3.11.1. Therefore, if r 1 , r 2 , ... , re is a system of representatives of the c p-regular conjugacy classes Cj = rf, 1 ~ j ~ c of G, then {rj + T(R) I j = 1, 2, ... , c} is a system of generators of the F-vector space R/T(R). Hence it suffices to show that they are independent over F. Otherwise, there are aj E F which are not all zero such that C

a=

L ajrj E T(R). j=l

Thus aPw E S for some integer w by Lemma 3.11.1. Let

IGI

= paq where

(p, q) = 1. Then by the proof of Theorem 3.9.3 there is an integer x such that t px = l mod (q). Choose t ::CO: w such that r1J = rj for all j = 1, 2, ... , c. Then Lemma 3.9.2 implies that

L, L aP,, rP = aP r · mod (S) C

= -

C

JJ-

JJ

j=l



j=l

Since the rj are representatives of all distinct p-regular conjugacy classes t

Cj of G, it follows from ( *) and ( **) that a1J = 0, and so aj = 0 for every j. This completes the proof.

Definition 3.11.3 Let G be a finite group of order IG\ = paq, where p is a prime and (p, q) = 1. Let G 0 be the set of p-regular elements of the finite group G. Let (F, R, S) be a splitting p-modular system for G. Let U(S) and

3.11 Modular characters of finite groups

183

U(F) be the group of q-th roots of unity in the multiplicative groups S* and F* of the fields S and F, respectively. Then both groups are cyclic of order q, and the natural map p: R----+ F induces an isomorphism between U(S) and U(F), because (p, q) = 1 and the polynomial Xq - 1 E F[X] has no multiple roots. Let µ : G ----+ GLn(F) be a modular representation afforded by the FGmodule M of dimension n = dimp(M). Let x E G be of order m. Then x = XpXp' = xp,xp, where Xp is a p-element and Xp' E G is p-regular. As Fis a splitting field for G there is a matrix PE GLn(F) such that the restriction µl(x) has the following matrix

)

*

E GLn(F),

An

where the eigenvalues Ai of µ(x) are m-th roots of unity. Since 1 is the only pa-th root of unity in F* the matrix p- 1 µ(x)P factors as p- 1 µ(x )P

= (P- 1 µ(xp)P) (P- 1 µ(xp' )P), where 1

J

E GLn(P).

* Hence

µ(x) = µ(xp,) = A1

+ A2 +···+An,

where each Ai is a q-th root of unity of F. Therefore all Ai E U(F), and so Ai = p(Ei) for some q-th roots Ei of unity belonging to U(S). Each Ei is uniquely determined by µi, because p : U ( S) ----+ U ( F) is an isomorphism. For each p-regular element x E G define

Then the function µM : x E G 0 afforded by the FG-module M.

f---+

U(S)

.,;,. Since the support sup(e) consists of p-regular elements by Theorem 3.9.3 the assertion of this corollary follows immediately.

3.12

Blocks of defect zero

In this section we characterize the irreducible ordinary characters x of a finite group G which belong to a p-block of defect zero. We also give a ring theoretical description of the corresponding block ideals of such a block of G. Throughout (F, R, S) denotes a splitting p-modular system for the finite group G. In [109] the author has shown the following subsidiary result. Lemma 3.12.1 Let B +---> e be a block of FG with defect group o(B) =c D-/=1. Then the fallowing assertions hold:

(a) The set { ed I d E D} is linearly independent over F. (b) The set {e(l - d) 11-/=- d ED} is linearly independent over F. (c) If 1-/=- d ED has order o(d) = pm then e(l - d)Pm-l-/=- 0. Proof. (a) Suppose LdED fded = 0 for some fd E F. Since D is a group, we may assume Ji = -1 if not all f d = 0. Hence e = Li#dED f ded. Thus for every x E sup( e) there is a y E sup( e) and a 1 -/=- d E D such that x = yd. Since D is a defect group of B, we may assume that x E Cc(D). Hence y = xd- 1 = d- 1 x. However, Theorem 3.9.3 asserts that x and y are p-regular. Thus d = 1 by Lemma 2.2.4. This contradiction completes the proof of (a). (b) This follows easily from (a), and so does (c).

Theorem 3.12.2 (Brauer) Let B = (B, B, Bs) be a block of G with defect group o(B) =c D. Then the following assertions are equivalent:

(a) B is a simple Artinian ring. (b) The center Z(B) of B is a field.

(c) D = 1. (d)

fJ

~ (R)n for some natural number n 2: 1.

(e) B s is a simple Artinian ring.

188

Modular representation theory

Proof. If (d) holds then Bs = B@RS ~ (S)n. Hence Bs is a simple Artinian ring. Suppose that (e) holds, then the center Z(Bs) ~ S because Sis a splitting field for the simple Artinian algebra Bs. Hence . is an AH-module automorphism of W ® l and so W ® l ~ µ>.(W ® 1). Let C =Kerµ. Then VH ~ >.(W ® l) E0 C and W ~ >.(W ® l) is isomorphic to a direct summand of VH. This completes the proof. The following result is due to J. A. Green [50]. Theorem 3.13.15 Keep the Notation 3.13.1. Let P be a p-subgroup of G and H = Nc(P). Let A E {F,R} and a- be the Brauer correspondence between the blocks B +-+ e of G with defect groups t5(B) =c P and the blocks b +-+ &(e) of H with defect group t5 ( b) = P. Let f be the Green correspondence between the non-projective indecomposable AG-modules U with vertex vx(U) E A and the non-projective indecomposable AH -modules W with vertex vx(W) E A. Then the Brauer correspondence a- and the Green correspondence f are related in the following sense: U belongs to B if and only if f (U) belongs to b = &(B). Proof. Let Ube an indecomposable B-module with vertex vx(U) = D E A. Then Theorem 3.13.6 implies that UH= f(U)EBY-projective where W = f(U) is an indecomposable AH-module with vertex vx(W) =HD. Let e1, e2, ... , e 8 be the set of all block idempotents of AG. Then 1 = ~:=lei and so 1 = 1 &(ei) by Theorem 3.8.13 and Corollary 3.8.15. Since the non-zero idempotents &( ei) of ZAH are orthogonal, ther~ is a unique index i E {1, 2, ... , s} such that W = W&(ei) because Wis indecomposable. Now Lemma 3.13.14 yields the existence of an AG-module V satisfying V = Vei, VI we and WI VH. By Theorem 3.2.4 of Krull-Schmidt we may assume that V is indecomposable because W is indecomposable. Hence V has vertex vx(V) =c vx(W) =HD. Therefore, Vis isomorphic to the Green

~:=

3.13 Green correspondence

199

correspondent U = g(W) of W = f(U) by Theorem 3.13.6. As U = Ue it follows that e = ei and so f(U)u(e) = f(U). Another application of Theorem 3.13.6 now completes the proof. Theorem 3.13.16 Let (F, R, S) be a splitting p-modular system for the finite group G of order IGI = paq, (p, q) = 1. Let B be a block of G with defect group b(B) =G D and defect d(B) = d. Let v(n) be the highest power of p dividing the integer n. Then the following assertions hold:

(a) B contains a simple FG-module M with v(dimF M) vx(M) =GD.

= a-

d and vertex

(b) B contains a simple BG-module L with v(dims L) =a~ d. Proof. (a) Let H = NG(D) and let b = O'(B) be the Brauer correspondent of B in H. By Corollary 3.10.14 every simple FH-module N has vertex vx(N) = D. Let H = H/D. Then N considered as FH-module has vertex vx(N) =ii 1 by Lemma 3.10.11. Thus N is a projective simple FH-module. Therefore, N is liftable to a simple SH-module N 1 = N f +--> µ be a block of the group algebra FU. Suppose that bv is defined. If either one of b0 or (bv)G is defined then so is the other and (bV)G = b0 .

If C is a conjugacy class of G then µ 0 ( C) = µ ( L.gECnU g) by Definition 3.16.1. Clearly, µ 0 is an F-linear map, and (µV)G = µ 0 as F-linear maps. By hypothesis µv is an F-algebra homomorphism from ZFV into an extension field E of F. Therefore µ 0 is an F-algebra homomorphism if and only if (µ V)G is an F-algebra homomorphism. This completes the proof. Brauer's second and third main theorems will be derived from the following proposition which is also due to him [14]. Proof.

Proposition 3.16.3 Let F be a field of characteristic p > 0. Let D be a psubgroup of the finite group G. Let U be a subgroup of G such that DCc(D) ::;; U::;; Nc(D). Let av be the Brauer homomorphism from ZFG into ZFCa(D). Then the following assertions hold:

(a) b0 is defined for each block b +--> f

+-->

µ of U.

(b) µ 0 = µav is the linear character of the block B = b0 and B has a defect group 5(B) 2: D.

+-->

e

+--> ,\

of G,

Let C1 = C n Cc(D) and µ(C: 1) + µ(C: 2) by Definition 3.16.1. As D is normal in U Proposition 3.8. 7 implies that µ( C:2) = 0. Hence Proof.

Let C be any conjugacy class of G.

C2 = C n (U ~ Cc(D). Then µ 0 (C) µ 0 (C)

=

= µ(C:1) = µav(C)

by Definition 3.8.1. Therefore µ 0 is an F-algebra homomorphism from ZFG into the field T = JZFU/fJ(ZFU) by Theorem 3.8.2. Thus (a) and the first part of (b) hold. Let 5(B) be a defect group of the block B = b0 of G. Since µav(e) = µ(!) = 1 Lemma 3.8.10 implies that D ::;; 0 5(B). This completes the proof. Now we can give Hamernik's and the author's proof [59] for Brauer's Third Main Theorem [16], its p. 157. Theorem 3.16.4 (Brauer's Third Main Theorem) Let (F, R, S) be a pmodular system for the finite group G. Let U be a subgroup of G having a p-block b with defect group 5(b) = D such that Cc(D)::;; U. Then the following statements hold:

(a) B = b0 exists and is a block of G with a defect group 5 (B) 2: D. (b) B = b0 is the principal block Bo of G if and only if b is the principal block bo of U.

214

Modular representation theory

Proof. By hypothesis D and Ce(D) are subgroups of U. Hence K = DCe(D) :S U. Since b +-+ f +-+ /L is a block of U with defect group D, its Brauer cc:,rrespondent b +-+ J +-+ µ in N = Nu(D) has defect group c5(b) = D and b = bu by Theorem_ 3.8.13. Furthermore, K = DCu(D) = DCN(D) : D. Thus Ce(D1) S Ce(D) SH and B 1 is the principal block of H with defect group D1. As IP : D1I < IP: DI we can apply induction and see that B = Bf existing by (a) is the principal block of G. Conversely, let B = be = be be the principal block of G. By Proposition 3.16.3 its centrally linear character A = µe = µan where an is the Brauer homomorphism from ZPG into ZPCe(D) with respect to D. Let M = lFe be the trivial PG-module, and let AM be its centrally linear character. By Proposition 3.6.21 AM is equivalent to A= µan. Clearly, M is also the trivial PN-module. Since the idempotents an(e) and J belong to the center of the group algebra P N it follows that

1 = A(e) = µan(e) = µ[Jan(e)] = AM[Jan(e)], and M J a( e) =/- 0. In particular, M J =/- 0 and the block principal block of N. This completes the proof.

b +-+ J

+-+

µ is the

3.16 Brauer's second and third main theorems

215

The proof of the second main theorem given here rests on Nagao's lemma

[119]. Lemma 3.16.5 (Nagao) Let (F,R,S) be a p-modular system for the finite group G. Let {! : RG ---+ FG be the canonical epimorphism with kernel Ker(Q) = nRG. Let D be a p-subgroup of G. Let H be a subgroup of G such that DCc(D ~ H ~ Nc(D)). Let O'D : ZFG ---+ ZFH be the Brauer homomorphism with respect to D. Let V be an RC-lattice such that V = Ve. Let s,µ be the set of all p-subgroups S of H and let 6 = {S E S,lJ I D i.H S}. Then the fallowing statements hold:

(a) For each block idempotent e of RG with O'D(!(e) -=/- 0 there is a uniquely determined central idempotent O'o(e) -=/- 0 of RH such that Q[O'o(e)] O'D(!(e). (b) There is an 6-projective RH -lattice W such that

Proof. (a) By Theorem 3.8.2 O'v{!(e)-=/- 0 is a central idempotent of the group algebra FH. It can be lifted to a uniquely determined central idempotent O'o(e)-=/- 0 of RH by Theorem 3.3.23. (b) Let f = e - O'o(e)e ERG. Hence

f2

=

e-

20'o(e)e + O'o(e)e

because e E ZRG. If

f =

= f, and

O'o(e)f

= 0=

foo(e)

0, then V = Ve implies that W = 0 and

Otherwise, the equations

e=

O'o(e)e + f and [O'o(e)e]f

=

f[O'o(e)e]

describe an orthogonal decomposition of the idempotent

= 0

e ERG.

Thus

is an H-invariant direct decomposition of the RH-lattice "\!'iH· Therefore f is the identity endomorphism of the RH-lattice W = ("\!'iH )f. Let M -=/- 0 be any indecomposable direct summand of W with centrally linear character AM. Then M f = M implies that there is a class sum 6 in the central support of f E ZRH such that AM(C) -=/- 0. Now Lemma 3.10.7 states that the vertex vx(M) of M is H-conjugate to a subgroup of the defect group J(C) of C. Clearly J(C) E 6 because D i.H J(C) by Theorem 3.8.2 and its proof. Hence vx(M) E 6. Therefore Mis an 6-projective RH-lattice. This completes the proof.

216

Modular representation theory

Definition 3.16.6 Let (F, R, S) be a splitting p-modular system for the finite group G. Let B be a p-block of G with defect group D. Let Xi, x2 , ... , Xk be the irreducible ordinary characters of G. Let x E D be of order pm, and ?/J1,'1/J2, ... ,'1/Js be the irreducible ordinary characters of Cc(x). Then s

Xii Ca (x)

=

L nir?/Jrr=l

~et y E Cc(x) be any p-regular element. Then s

r=l

r=l

for some pm-th root of unity Er, because x E Z[Cc(x)]. Let cp 1 , cp 2 , •.. , 'PR(x) be the irreducible Brauer characters of Cc(x) and let {drj I 1 S r S s = k(Cc(x)), l S j S R(x) = R(Cc(x)} be the set of decomposition numbers of Cc (x). Then

R(x) '1/Jr(Y)

=

L drj'PJ(y) for all p-regular elements y E Cc(x).

j=l

The uniquely determined algebraic integers s

dfj

=

LnirErdrj ER, l S j S R(x), 1 Si S

S

r=l

are called the generalized decomposition numbers of G with respect to the p-element x E D. Theorem 3.16. 7 (Brauer's Second Main Theorem) Let x be a p-element of the finite group G. Let (F, R, S) be a splitting p-modular system for G. Let Xi be an irreducible ordinary character of G. Let cp'J be an irreducible modular character of Cc(x) belonging to the p-block b of Cc(x). If the generalized decomposition number d'fj =I= 0, then be exists and Xi belongs to the block B = be of G.

Proof. Let P = (x; be of order pm. Let H = Cc(x) = Cc(P). Let D be a defect group of the block b of H. Then PS D by Corollary 3.8.8 because Pis normal in H. Hence Cc(D) SH. Therefore be exists by Theorem 3.16.4(a). Let ebe the block idempotent of the block B of G containing the irreducible character Xi of G. Let Jp be the Brauer homomorphism from ZFG into ZF H with respect to P. As P S D Lemma 3.8.10 states that JpQ(e) =I= 0. By Lemma 3.16.5 there is a uniquely determined central idempotent Jo(e) =I= 0 of

3.16 Brauer's second and third main theorems

217

RH such that g[CT0 (e)] = O"pg(e). Let$ be the set of all p-subgroups S of H. Let 6 = {S E $ I D iH S}. Then Lemma 3.16.5 asserts that there is an 6-projective RH-lattice W such that

Let 7j; 1 , 7j; 2 , ... , 7/Js be the set of inequivalent ordinary irreducible characters of H. Then using the notation of Definition 3.16.6 we have s

l(x)

XifH = Lnir7/Jr, 7/Jr = Ldrjcpj, r=l

j=l

and 7/Jr(x) = Er7/Jr(l) for some pm-th root of unity because pm is the order of x. As r=l

there is at least one r E {1, 2, ... , s} such that nirErdrj c/ 0. In particular, Er c/ 0 and drj c/ 0. Let Wt be an indecomposable direct summand of the RH-lattice W. Let Xt be the irreducible character of H afforded by Wt. Then Xt(x) = 0 by Theorem 3.10.9 because Wt is 6-projective. Hence Xr is an irreducible constituent of the character of H afforded by the RH-lattice (VjH) O"o( e). As drj c/ 0 the modular irreducible character cp'j of H belongs to an irreducible constituent M of the FH-module

Since cp'j belongs to the block b .+ T for some character T =/. 0 of B. By (e) there is also a short exact sequence of indecomposable RC-lattices o

-----+

gMi -

A --+ ngMi-1 --+ o.

Another application of Corollary 3.4.27 yields the existence of an indecomposable RC-lattice W of B such that W = W 0R F has a simple head isomorphic to the simple PG-module ½. Furthermore, the character >. + ni-1 of B is afforded by W. Hence there exists a short sequence of RC-lattices 0 --+ Y --+ W -----+ D,gMi-l -----+ 0 such that >. is afforded by Y. Now Lemma 3.17.11 yields the existence of a projective RN-lattice Q such that

J(W) EB Q ~ J(r!gMi-i) o J(Y) ~ (r!Mi_i) o J(Y). Therefore

J(W) EB Q ~ (r!Mi_i) o J(Y).

(**)

Applying now the previous arguments it follows that the Loewy lengths of the uniserial FN-modules J(W), D,Mi-l and J(Y) arep-2, p- l andp-1, respectively. Furthermore, Q =/. 0, and J(Y) ~ D,Mi by (**) and Definition 3.17.1. Therefore >. is afforded by f!gMi. Thus >. = ri, and 1 = [>., ri]

= [ni, ri] = o.

This contradiction shows that r i is either irreducible or its non-trivial irreducible constituent >. is an exceptional character of B.

3.17 Blocks of defect one

237

Since [ri, Di] = 0 it follows now that one of the characters ri and Di is irreducible in any case. Furthermore, if t = p - l, then all ri and Di are irreducible for O :S: i :S: t - l. Therefore we may assume that t-/- p-l. From Proposition 3.17.9 it follows that B has r = (p - 1)/t > 1 distinct exceptional irreducible characters Xj· Suppose that for some i E {O, 1, ... , t - 1} ri is not irreducible. Then [ri, xk] = 1 for some k E {1, 2, ... , r} by the above arguments and Lemma 3.17.5. Since O"(i) = ri + S1i and [ri,Di] = 0 it follows that [O"(i),Xk] = 1. By Proposition 3.17.9 the r exceptional characters Xj of B have the same values on each p-regular element g of G. As O"(i)(x) = 0 for all p-singular elements x E G by Theorem 3.10.9 it follows that the inner products [O"(i),Xj] = 1 for all j = 1, 2, ... , r. Hence [ri, XJ] = 1 for all j = 1, 2, ... , r, because [ri, Xk] = 0. The same argument also works if [Di, Xk] = 1 for some k E {1, 2, ... , r }. This completes the proof. Theorem 3.17.13 (R. Brauer) Let (F, R, S) be a splitting p-modular system for the finite group G with a cyclic Sylow p-subgroup D of prime order IDI =p > 2. Let B be a block of G with defect group D, inertial index t and Brauer correspondent B 1 in N = Nc(D). Then the following statements hold: (a) There are exactly t characters 1Pi E Irrs(Bi) such that D :s; Ker('I/Ji) for 1 :S: i :S: t.

(b) The block B has t irreducible modular characters txj(l) > xa 2 (IWIIHl) 2 > a2 I~~ 1 (IWIIHl)2. Thus a2 (IBI - l)IWI < IBI. This is impossible because IWI ~ 2. Therefore a= 0, and E = 1 because (xn)IN is a character of N. Suppose that f3n(l) = d is not zero. Then (f3n)IB = dlB because B is normal in N. Thus [(Xn)IB, lB] = d > 0. Therefore all irreducible characters Xn of G occur in 1~ with multiplicity d by Frobenius reciprocity. Since a = 0 equation(***) implies that [0f,Xn] = bk,n for all l :S: n,k :S: x. By Proposition 2.7.10 each irreducible character 1/Jn = 0;! for some irreducible character 0n of B which is uniquely determined up to conjugation in W. Since a = 0 and E = 1 another application of (* * *) yields that [0;;, = 1/J;/,, Xk] = 6n,k for all 1 :S: n, k :S: x. Therefore the generalized character 1~ - 1/J;;, of G contains at least the x irreducible characters 1~, Xk with 1 :S: k :S: x and k cf. n with positive multiplicity. Hence [1 ~ -1/J;;,, 1~ -1/J;;,] ~ x. Let {y1 = 1, Y2, ... , Yr} be a system of representatives of the N - B double cosets in G. Then B n Bf = 1 for all 1 :S: i :S: r with i cf. 1 by Lemma 4.3.6. Let p denote the regular character of B. Then Theorem 2.6.9 states that

(l~- (1/J;;,))IB = ((l1¥)G - (1/J;/,))IB

= (11'{)1B + r11'{ (l)p - (1/Jn)IBI - TVJn(l)p = (11'{)1B - (1/Jn)IB

256

Group order formulas and structure theorem

because 1:!¥(1)

=

IWI =

'lf!n(l). Now Frobenius reciprocity implies that

This contradiction proves that d = 0. Thus (Xn)IN = 'lf!n and Xn(l) = IWI for all 1 :s; n :s; x. Let g =/- 1 be any 2-singular element of G. Then Proposition 2.7.13 states that Xn(g) = X1(g) E Z for 1 :S: n :S: x. Now Theorem 2.3.26 implies that X

> IHI ~ ICa(g)I

=

X

L

x(g)x(g- 1) >

L Xn(g)

2

=

xx1(g) 2 •

n=l

xElrrc(G)

Hence xn (g) = 0 for 1 :S: n :s; x and each 2-singular element 1 =/- g E G. Therefore each character Xn is of 2-defect zero, and the order ISi of a Sylow 2-subgroup S of G divides the degree Xn(l) = IWI by Theorem 3.12.4. Hence S is either cyclic or a generalized quaternion group by Lemma 4.3.6 and Theorem 1.2.6. This contradiction to the hypothesis shows that x :S: IHI. Thus (b) holds.

4.4

Brauer's group order formula

In this section R. Brauer's group order formula [16], its p. 228, is derived from Brauer's Second Main Theorem 3.16.7. It will find several applications in Chapter 9. Theorem 4.4.1 (Brauer) Let G be a finite group. Lett be a p-element and let z be an involution of G. Let U be a subgroup of G containing the extended centralizer C 0(t) oft in G. Let B be a p-block of G with defect group D such that t 0 n D =/- 0. Let RB(U) be the set of all blocks B of U such that B = iJU. Then for each p-regular element y E Ca(t) the following equation holds:

E.1_

L

IHl2 xEirrc(B)

x(z)2x[(ty)- 1 ])

x(l)

= IUI

L

L LL

BE'Rs(U),f,Elrrc(B) u

v

'¢(zu)'¢(zv)'¢[(ty)- 1 ] , ICa(zu)IICa(zv)l'¢(1)

where Zu and Zv range independently over a complete set of representatives of the U -conjugacy classes of involutions comprising z 0 n U. In particular, if ty is strongly real, then both expressions of the equation are non-zero, and all of their character values are real numbers.

4.4 Brauer's group order formula

Proof.

257

As in the proof of Lemma 4.1.3 let

L

aty=

xEirrc(G)

and let

,t,Elrrc(U) u

Y

1)

L LL v,(zu)v,(zv)V'l(ty)-ll .

1u1

bt =

x(z)2x(y- 1 r x(l)

ICa(zu)IICu(zv)lv,(l)

v

lfty is strongly real there is a pair (r, s) of involutions such that rs= ty. Hence aty = bty-=/- 0 by Lemma 4.1.3, and x[(ty)- 1 ] = x(ty) and v,[(ty)- 1 ] = v,(ty). Since the values of irreducible characters on involutions are rational integers all character values in the expressions for aty and btY are real numbers. Thus the last statement of the assertion holds. For each v, E Irrc(U) let z0

l

n U = LJ z;{,

and

u=l

'°" v,(zu) h2 = '~ °" ICu(zv)I" 'lj!(zv) = ~ ICu(zu)I' £

hl ,t,

Then bty -- IUI

£

,t,

hl,t, h2,t, ,t,[(ty)-1] ,t,(l) ·

"L.. ,µElrrc(U)

Let Bl(Ca(t)) be the set of all p-blocks b of Ca(t). Let IBr(b) be the set of all modular irreducible characters c.pt of b E Bl(Ca(t)). Let d~,'P' and d~,'P' be the generalized decomposition numbers of c.pt E IBr(b) and x E Irrc(G) arid v, E Irrc(U), respectively. Then Theorem 3.16.7 and Definition 3.16.6 imply that

'~ °"

IGI a(ty)-1 = IHl2

'~ °" ( '~ °"

bEBl(Ca(t)) ,p'EIBr(b)

L

b(ty)-1 = IUI

bEBl(Ca (t))

2

x(z) t ) c.p t( y) (i)dx,'P'

xEirrc(G) X

L ( L h~~~ ~'P'P') c.p\y). V,( )

'P' EIBr(b)

,j,Elrrc(U)

As a(ty)-1 = b(ty)-1 holds for all p-regular elements y E Ca(t) and since the modular characters c.pt of Ca(t) are linear independent, c.pt appears with the same coefficient on both sides of the equation. Hence IGI IHl2

'~ °"

.

xEirrc(G)

x(z) 2 X

~'P' -_IUI '~ °"

(1)

'~°"

BE'R.B (U) ,µEirrc(B)

Let Irrc(B) = {x, X2, ... , Xk(B)}- Let RB(t) be the set of roots b of B in Ca(t). Then by Theorem 3.16.7 d~'P'

= 0 for all c.pt

E

IBr(b) such that b (/. RB(t).

258

Group order formulas and structure theorem

Again by the linear independence of the r_pt it follows that d~'P' = O for all r_pt E IBr(b) such that b ej_ RB(t). Another application of Theorem 3.16.7 yields that bu exists and 1/J E Irrie(bu) whenever d~'P' -=J 0. As B = be for all

=

b E RB(t) and (bu)e

IG 1 IH 12

be it follows that

x(z) 2 x(l) dx'P'

"""'

~

=

xEirr S. (b) Whenever S has a unique maximal elementary abelian subgroup A of maximal order, then A is a weakly closed maximal non-cyclic elementary abelian characteristic subgroup of S. The Sylow 2-subgroups S of any alternating group G = A 4 k with k ::>- 2 have such a subgroup A of order 22 k, see [93], its Lemma 2.3. Furthermore, INo(A): NH(A)I =/- 1. (c) If A is not weakly closed in S then Lemma 1.4.3 and Theorem 4.7.3 help to decide whether IN0 (A) : NH(A)I =J l. This inequality is true for all sporadic simple groups G, see the table in Section 7.5. Corollary 4.8.10 Let S be a Sylow 2-subgroup of a finite simple group G such that S is neither dihedral nor semi-dihedral, and that G does not contain a strongly embedded subgroup. Let H be the centralizer Co(z) of a 2-central involution z.

(a) If for some of the maximal non-cyclic elementary abelian characteristic subgroups A of S the subgroup U conjugacy class of involutions, then G

= (H, No(A); of G has a unique

= (Co(z), No(A);.

(b) Otherwise the orders of U and G can be calculated by Theorem

(c) If

4. 2.1.

IGI =/- IUI,

then ((lu) 0 is a non-trivial faithful permutation representation of the simple group G of known degree.

Proof. Since U has a unique conjugacy class of involutions a similar argument as in the proof of the above Theorem 4.8.5 shows that U is a strongly embedded subgroup if it is different from G. The remaining statements follow easily. This contradiction completes the proof of (a). The remaining statements (b) and (c) follow easily from Theorems 4.8.5 and 4.2.1.

276

Group order formulas and structure theorem

Remark 4.8.11 If case (c) of Corollary 4.8.10 occurs, then Theorem 5.2.18 of the next chapter implies that G can be described by generators and relations.

Definition 4.8.12 Let G be a finite group with a 2-central involution z and centralizer H = Ca(z) satisfying G #- O(G)H. Then a finite group X which is not isomorphic to G is called H -satellite of G if X has a 2-central involution t with centralizer Cx(t) ~ H and X #-

O(X)Cx(t). Remark 4.8.13 A list of known simple groups of H-type having nonisomorphic simple satellites is given in the introduction. For details and proofs see Chapter 8. There we also provide examples showing that all cases of Corollary 4.8.10 occur.

5

Permutation representations

This chapter provides several theoretical results and practical deterministic algorithms which suffice to calculate a presentation and the concrete character table of a finite permutation group G acting transitively on O = {1, 2, ... , n }, where n is the index of the stabilizer H = Staba(l) of 1 E O in G. Let G = Uf= 1 HxiH, x 1 = 1, be the double coset decomposition of G with respect to H. In Section 5.1 the relations between the G-orbits Ai of G on 0 2 and the H-orbits of Hon Oare described. In Section 5.2 standard results and concepts of computational permutation group theory are presented. In particular, C. Sims' membership test and the Schreier-Sims method for the construction of a base and strong generating system of a transitive permutation group G are dealt with. They are used to construct a presentation of such a group G in terms of finitely many generators and defining relations, see Theorem 5.2.18. In order to get the concrete chracter table of G one needs representatives of all conjugacy classes gf, 1 :s; j :s; k( G). Let S = {x 1 , x 2 , ... , X£} be a given set of generators of the finite transitive permutation group G. In Section 5.3 M. Kratzer's deterministic Algorithm 5.3.18 is described which provides a uniquely determined system of representatives for the conjugacy classes of G as short words each of which being a power of a particular length-lexicographically lowest base word with respect to the generators Xi E S. This deterministic algorithm produces always the same representatives of the conjugacy classes of a given group G whenever it is run on a computer. In Section 5.4 it is shown that the d adjacency matrices Ai E Matn(Z) of the double cosets HxiH of Hin G form a C-vector space basis of the endomorphism ring It= Endica(CO) of the complex permutation module CO of G. The adjacency matrices Ai are useful for explicit calculations with endomorphisms a E Endica(CO). However, they are also theoretically important. In Section 5.5 for each double coset HxiH, 1 :s; i :s; d, an intersection matrix Di E Mata(Z) is defined. In Proposition 5.5.5 it is shown that the map Ai ------, Di, 1 :s; i :s; d, induces an anti-isomorphism between It = Endica(CO) and the intersection algebra B = (Di I 1 :s; i :s; d) :s; Mata(C). Since this algebra is generated by a few integral matrices of fairly small size its structure can be studied in detail. Let r be the dimension of the center Z(B) of B. Then B has r centrally primitive idempotents es. Each es is a uniquely d

determined linear combination es

= I: CsiDi

of the intersection matrices Di.

i=l

Theorem 5.5.6 of M. Weller and the author [118] provides an explicit formula

278

Permutation representations

for the complex coefficients Csi of each e8 • Using now the anti-isomorphism between Q: = Endrcc(CD) and Bone obtains an explicit formula for the corresponding centrally primitive idempotents E 8 of the endomorphism ring Q: as linear combinations of the adjacency matrices Ai. In Section 5.6 these results are used for the proof of the recent character formula of M. Weller and the author [118], see Theorem 5.6.5. It determines all character values Xs (g) for all irreducible constituents Xs of the permutation character (lH )G and all elements g E G. This result generalizes Frobenius' classical character formula, see Corollary 5.6.6. ' Section 5. 7 then provides an explicit algorithm for the calculation of all character values Xs (g) of any irreducible constituent Xs of the permutation character (lH )G and all g E G. In general, not each irreducible character x E Irrrc( G) appears in (lH ) 0 . Therefore several classical algorithms and methods for completing the concrete character table of G are mentioned in Section 5.8. The results of these sections will be applied in the existence and the uniqueness proofs of the simple groups dealt with in the last five chapters.

5.1

Permutation groups

In this section some well-known definitions and results on permutation groups are collected. Throughout n is a positive integer, and D = {1, 2, ... , n }. The group of all permutations of D is the symmetric group So = Sn of degree n. Its order

ISnl = n!. Definition 5.1.1

(a) A permutation group on Dis a subgroup G of Sn,

(b) Let a E D be fixed. Then its orbit is the set a 0 stabilizer in G is the subgroup Stabc(a) =Ga= {g E GI a 9

= {a 9 I g

E G}, and its

= a} ofG.

(c) G is a transitive permutation group on D, if D

= a 0 for each a

E D.

Definition 5.1.2 Let G be a transitive permutation group on D, and a E D. The orbits w 0 " of the stabilizer Ga of a in D are called suborbits of G with respect to a E D. Theorem 5.1.3 Let G be a permutation group on D, and a E D. Then the length of the orbit a 0 is

5.1 Permutation groups

279

Proof. Let {g1 = 1, g 2 , ... , gt} be a right transversal of the stabilizer Ga of a in G. Then the points a 9i E1a 0 correspond uniquely to the cosets Gagi of Ga in G. Definition 5.1.4 Let G be a transitive permutation group on acts on 0 2 = n X n = {(a,,B) I a,,B En} by

n. Then G

(a,,8)9 = (a 9,,B9 ) E O for all (a,,B) E 0 2 and g E G. Certainly, A1 = {(a,a) E 0 2 I a E O} is a G-orbit of 0 2 . Let Ai= (ai,,Bi) 0 with (ai, ,Bi) E 0 2 and ai =I- ,Bi for i = 2, 3, ... d be the other orbits of G in 0 2 . Then the number d of orbits of Gin 0 2 is called the rank of G. Lemma 5.1.5 Let G be a transitive permutation group on

n.

Then:

(a) The set Ai(ai) ={,BE n I (ai,,B) E Ai} is a Gai-orbit in O for each G-orbit Ai= (ai, ,Bi) 0 of n 2 • (b) There is a one-to-one correspondence between the G-orbits Ai= (ai, ,Bi) 0 in 0 2 and the Gai -orbits in 0. Proof. (a) Let A = (a, ,8) 0 be a G-orbit in 0 2 . Then (a, 1') E 0 2 belongs to A if and only if (a,1') = (a,,8) 9 for someg E G. Thus a= a 9 , and 1' = ,89 , which is equivalent to 1' E Conversely, if 1' = ,89 for some g E Ga, then

,e

0 a

and g E Ga.

(a,1') = (a,,8 9 ) = (a 9,,B9 ) = (a,,8) 9 E (a,,8) 0 = A. Therefore A(a) = {1' En I (a, 1') EA}= ,e0 a inn. (b) This follows immediately from (a). Lemma 5.1.6 Let H be a subgroup of the finite group G. Let

n=

(lH )0

=

l, be a

d

be its transitive permutation module of G. Let G

= LJ

HxiH, x 1

i=l

double coset decomposition of G with respect to H. Then there is a one-to-one correspondence between the double cosets H xiH of H in G and the G-orbits Ai = (H, H xi) 0 of G in 0 2 . d

Proof.

Let G =

LJ

HxiH, x1 = l, be a double coset decomposition of Hin

i=l

G. Let {y1 = 1, Y2, ... , Yn} be a right transversal of H in G. Then H is the stabilizer of the transitive permutation group G on n = (lH)G = {Hyj I j = 1,2, ... n}. Therefore the double cosets HxiH are the H-orbits Ai inn. Hence the assertion follows from Lemma 5.1.5.

280

Permutation representations

Definition 5.1. 7 Let H be a subgroup of the finite group G with index IG : d

HI = n. Let G =

LJ

i=l

HxiH, x 1 = 1, be a double coset decomposition of G

with respect to H. Let n = (lH ) 0 , and let Ai = (H, H xi)G, l :S: i :S: d, be the orbits of G in !1 2 = f! X n. Then: (a) The length IAil= IHI/IH n Hx; I = ki of Ai is called the subdegree of the G-orbit Ai. '(b) The adjacency matrix Ai = ((Ai)(a,,6)) E Matn(Z) of the G-orbit Ai is defined by: if (a, /3) E Ai (Ai)(a,!3) = { otherwise

~

for all (a,/3) E !12 . Remark 5.1.8 The adjacency matrix A 1 of the G-orbit A1 = (H, H) 0 of G in !12 is the identity matrix in Matn(Z). Lemma 5.1.9 Keep the notation and hypothesis of Definition 5.1. 7. The following assertions hold:

(a) The d adjacency matrices Ai, 1 :S: i :S: d, of the G-orbits Ai in !1 2 are linearly independent over Z. (b) Each column and each row of the adjacency matrix Ai of the G-orbit Ai = (H, H xi)G in !1 2 has ki = IH : Hx; n HI entries equal to l. d

(c) J =

I: Ai

is the (n x n)-matrix (juv) of Matn(Z) with juv = l for all i=l 1 :S: u, v :S: n = IG: HI.

Proof. (a) Let Ai be the adjacency matrix of the G-orbit Ai in !1 2 • As G is transitive on n, we may assume that Ai = (1, /3i)G. From Lemma 5.1.5 and Definition 5.1.7 it follows that for /3 En we have

(Ai)(l,,6) = 1 if and only if /3 E (/3i)H. Since !Ail = l(/3i)HI = IH: Hx; n HI = ki, the first row of A has precisely ki entries equal to 1, and n - ki entries equal to zero. If Aj is any other adjacency matrix than Ai, then its first row has all its entries equal to zero in those columns, where the first row of Ai has entries equal to 1. Thus (a) holds. (b) Since G is transitive on n for each a E n there is a g E G such that a= 19 . Therefore Ai = (a,Hxig) 0 . Hence there are ki = IH : Hx; n HI

5.2 Orbits, stabilizers and group order

281

entries equal to 1 in the a-th row of Ai. By symmetry the same results hold for the first and ,6-th column bf Ai. Hence (b) holds. (c) Is immediate by (b).

5.2

Orbits, stabilizers and group order

Throughout this section G denotes a finite permutation group acting on the set f! = {1, 2, ... , n} of points. Let S = {g1, 92, ... , 9t} be a fixed set of generators of G.

Definition 5.2.1 The labeled, directed graph r of the permutation group G on n is defined by: (a) The vertices of rare the points a off!. (b) For a pair (a, ,6) E 0 2 there is a directed labeled edge a _!_, ,6, if and only if there is a generator s E S of G mapping a to ,6. The edges of r are bidirectional by defining the edge a whenever a _!_, ,6 belongs to r.

,6 of

r

Definition 5.2.2 Two points a, ,6 E f! belong to the same connected component of r, if there is a path

in r from a to ,6, where all i = 1, 2, ... , t}.

9i;

E S U

s- 1

= {g?, ... , gl'

I

Ei

±1, for

Remark 5.2.3 Let a E f!. Then ,6 E f! belongs to the orbit aG if and only if a and ,6 belong to the same connected component of the graph r of the permutation group G. Definition 5.2.4 Let Ube a subgroup of G = (g 1 ,g2 , ... ,gt)- A transversal T of U in G is called a Schreier transversal if Un T = {1}, and whenever

Lemma 5.2.5 Any subgroup U of finite index in a finitely generated group G has a Schreier transversal.

282

Permutation representations

Proof. Suppose that G is generated by S = {s 1, s 2 , ..• , Sn}. Let F be the free group on the set S. Then F has normal subgroup R such that F / R = G, and a subgroup U1 containing R satisfying Ui/R = U. Hence IF: U1I = IG: UI. Therefore it may be assumed that U is a subgroup of finite index in the free group G. In particular, each element g E G has a uniquely determined standard factorization g = x 1x 2 ... Xm with respect to S by Remark 1.1.4, and its length l(g) = m. For any coset U x of U in G let l (U x) be the minimal length of all the elements v E U x. The Schreier transversal T is constructed by induction on the lengths of the cosets of U. If l(U x) = 0 then x E T n U = l. So we include x = l into T. Suppose that for all cosets of U in G with length l(Ux):::; n - l a Schreier representative s E T has been constructed. Let y E U x be of minimal length in Ux. Then y = y 1y 2 ... Yn for suitably chosen YJ ES. Since l(U Y1Y2 ... Yn-1) :::; n - l induction asserts that T n U Y1Y2 ... Yn-1 = s and l (Us) = m :::; n-1. Furthermore, s has a standard factorization s = v1v2 ... Vm such that all partial factors v1 v2 ... Vk of s belong to T. Let t = SYn· Then l(t) :S: l(s) + 1 :S: n because Ut = U syn = Uy1y2 .. · Yn-lYn =Uy= U x. Hence t E Ux. Thus l(t) = n and m = n - l. In particular, t = V1V2 .. ,Vn-lYn is a Schreier representative of the coset Ux. So we may put t into T. This completes the proof. Definition 5.2.6 Let Ube a subgroup of finite index in the group G generated by S = {s 1, s 2 , .•. , sn}. Let a E D such that U = Stabc(a). Then for each /3 E a 0 there is a path

of a 0 by Lemma 5.2.5. The r-tuple v(/3)

=

(sip Si2"

.. '

sd

E

r

(Su s- 1

is called the Schreier vector of {3.

Remark 5.2. 7 Let a E D, and determines a group element 9/3

such that

/3 =

=

/3

E a0

.

Then the Schreier vector v(/3) of

Si1 Si2 ... Si,.

/3

E G

ag~ .

The following definitions are due to C. C. Sims, see [134] and [133].

Definition 5.2.8 An ordered sequence B = [a 1, a 2 , •.. ak] of points °'i E D, 1 :::; i :::; k, is called a base for the permutation group G on D if the identity element 1 E G is the only permutation of G that fixes all points a1, a2, ... , °'k of B.

5.2 Orbits, stabilizers and group order

283

Definition 5.2.9 A generating set S = {g1,g2, ... ,gt} of the permutation group G on O is called a strong generating set for G relative to the base B = [a1, a2, ... , ak], if

(Sn Ga11 ,a 2,... ,aJ

= Ga 1,a 2,... ,a,

for all i

= 1,

2, ... , k.

In particular, a strong generating set S of G contains a set of generating systems for all stabilizers Ga 1,a 2,... ,a,. Remark 5.2.10 For each base B of G there is a descending chain of stabilizers

and IaiGal,"'2,···•"'i-1 I = IG a1,a2, ... ,a,_1 ; G a1,a2, ... ,a, I £or i. = 1 , 2 , ... , 'k . Therefore, knowing the stabilizers Ga 1,a 2,... ,a, implies knowing all the indices

Therefore the group order IGI

=

IG: Ga1 IIGa1 : Ga1,a2 I - · - IGa1,a2, ... ,ak-l : Ga1,a2, ... ,ak I

can be calculated. The following theorem is due to C. C. Sims. Theorem 5.2.11 (Membership Test) Let B = [a 1 , a 2 , ... , ak] be a base of the transitive permutation group G on 0. Let S = {g1, g2, ... , gt} be a strong generating set for G relative to the base B. Then there is a deterministic test for deciding whether a given permutation p E Sn belongs to the permutation group G. If the test has a positive answer, then p can be written as a word in terms of the strong generators Bi E S.

Proof.

Calculate the descending chain

of stabilizers and a transversal Ti of Ga 1,a 2,... ,a,+ 1 in Ga 1,a 2 , ... ,a, for i 1, 2, ... , k - 1. Let r a 1 be the stabilizer of a 1 in r. Then G a 1 = r a 1 n G. 1. Step: Calculate af. As G is transitive on 0, there is an h 1 E G such that af = aJ 1 Hence ph 11 E f a 1 , and p E G if and only if ph 11 E Ga 1. Suppose that ph 11 E Ga 1 . 2. Step: Calculate a~ h-1 -1

h-1 1

and a~ for all x E T1 .

Then either a~ 1 x belongs tor a 1 a 2 for some x = h 2 E X 1 or p r:J. G. In the first case it follows that ph11 h 21 E Ga 1a 2 -

284

Permutation representations

Iterating this procedure, it follows at some step that p integer i = 1, 2, ... , k - l there is an hi+l EI'; such that 1 1 Ph 1- h2

·· ·

1 G or that for each

1 hi E G a1,a2, ... ,a;+1 h i+l·

1 E G a 1 ,a 2 , ... ,°'k 1 h- 1 . . . h. 1ar, p hI n part ICU k 1 2

. 1·ies t h at p = 1, wh.ICh imp hkhk-1, ... , h2h1 E G. Since h1 E G = (S), and hi E Ga 1 ,a 2 , ... ,a;_ 1 = (Sn Ga 1 ,a 2 , ... ,a;_ 1 ) for i = 1, 2, ... , k, it also follows that pis a product of strong generators gj E S. T-his completes the proof.

Remark 5.2.12 (Schreier-Sims Methoa) Schreier's Theorem 1.1.8 and Sims' Theorem 5.2.11 may be used to construct a base B and a strong generating set S for a finite transitive permutation group acting on 0. Let X be a given set of generators of G, and a E 0. Suppose that H is a subgroup of the stabilizer Stabc(a) in G such that a base B 1 and a strong set of generators S1 have been calculated for H. Since 0 = ac Remark 5.2.7 states that for each (3 E O one can choose a 9f3 E G such that ag/3 = (3 and 9f3 -/=- g'Y for all"( E O which are distinct from (3. Hence T = {gf3lf3 E O} is a transversal of the stabilizer Stabc(a) and there is a map A : G--'>T defined by .X(g) = 9{3 if and only if ag = ag/3. Hence A(gf3x) = 9f3x for each pair x EX and (3 E O because ag/3x = (3x. Now Theorem 1.1.8 implies that Stabc(a) is generated by S = {gf3x(gf3x )- 1 1x EX, (3 E O}. Using Theorem 5.2.11 it can be tested whether for a pair (3 E aG and x EX its Schreier generator gf3x(gf3x )- 1 of Stabc(a) belongs to Hor not. If 9f3X(gf3x)- 1 1 H, then replace H by (H,gf3x(gf3x)- 1 ) and B1 by [B1,f3]. By Theorem 1.1.8 this procedure terminates after a finite number of steps and provides all the Schreier generators of Stabc(a). Together with the strong generating set S of H, they build a strong generating set S of the stabilizer. The Schreier-Sims method has been implemented on the computer algebra systems MAGMA and GAP.

Remark 5.2.13 [44] The Schreier-Sims method provides a finite number of tests that are sufficient to answer whether a subgroup H of the stabilizer Stabc(a) of a point a E O equals the whole stabilizer. Unfortunately, even in the case where H = Stabc(a) Theorem 1.1.8 states that the number of tests equals the length of the orbit ac times the number of generators of G. In the existence proofs of some large sporadic simple groups laGI is very large. Therefore the set of generators of H = Stabc (a) has to be reduced. This can be done by Gollan's double coset trick [45] stated as Proposition 5.2.14 (Gollan) Let G be a finite transitive permutation group on O = {1,2, ... ,n}. Let x E G and Ha subgroup ofG such that G

= (H,x),

5.2 Orbits, stabilizers and group order

where H :=:; Staba (a) for some fixed a E ri E G. Let K such that

=

H

n Hx-i

n. Let a 0 =

285

8

LJ ('Yi) H,

i=l

'Yi = ar' for

and for each i E {1, 2, ... s} choose i5i,j E ('"yi)H

u(Ji,j)K, j=l Si

("!i)H

=

l'·i

and Oi,j = for suitable ki,j E H. Suppose that the following two conditions are satisfied:

8

(b) riki,jX E

LJ

HriH for all l :=:;

i=l Then the stabilizer Staba (a)

i:::; s and

l :::; j :=:; si.

= H.

8

= LJ ("!i)H,

each 'Yi = ar, for some ri E G. We claim that i=l r 1 , r 2 , •.. , rs are (H, H)--double coset representatives of G. Otherwise, r i E HrjH for some i-/- j. Hence ri = h1rjh2 for suitable h1, h 2 EH:=:; Staba(a). Thus 'Yi= = ah1r;h2 = ar;h2 = 1J2 E 1f' Proof.

Since a 0

on

a contradiction. Hence the double cosets HriH, 1 :=:; i :::; s, are pairwise disjoint. Now Theorem 5.1.3 and Lemma 5.1.5 imply that Y = Staba(a) = H if 8

and only if G

= LJ

HriH. i=l For each 'Y E a 0 there is a unique suborbit 'Yf such that 'Y E 'Yf. By Remark 5.2. 7 for each 'Y E ,yf there is an element h7 E H satisfying 'Y = Furthermore, we may assume that h7 E H can be chosen in such a way that h 7 -/- h 7 ,, whenever 'Y -/- 'Y'. Therefore

l" .

by Theorem 5.1.3. If h EH n Hr,, then h

= h~·

for some h1 EH. Hence

because H :=:; Staba(a). Therefore H n Hr, :=:; StabH('Yi) for i = 1, 2 ... , s. Since [StabH('Yi)t-- 1 :=:; H for all 1 :=:; i :=:; s by (a) it follows that StabH('Yi) =

286

Permutation representations

HnHr, foralli=l,2, ... ,s. Thus

I LJ Hrih,I ,E,,H As 'Yf! =

s,

LJ (Jij )K for

=

IHl2 IH n Hr, I =

.

IHriHI for 1 :Si :S s.

all i = 1, 2, ... , s, condition (b) implies that H rkH x

j=l s

LJ

:s;

s

HriH for all 1 :Sr :S s. Therefore HxH :S

i=l '

LJ

i=l

HriH. As G = (H,x) it

s

follows that G =

LJ

HriH which completes the proof.

i=l Remark 5.2.15 With the notation of Proposition 5.2.14. (a) In practical applications of the double coset trick it will be important to make a clever choice of the additional generator x of G = (H, x), because the number of tests to verify condition (b) of Proposition 5.2.14 depends on the index IH : Kl. Therefore x should be chosen so that the subgroup K = H n Hx- 1 of H is as large as possible. (b) For calculations with very large permutation representations Gollan [44] and Weller [150] have written standalone programs for checking conditions (a) and (b) of Proposition 5.2.14. They first calculate a base inn - a and strong generating system of the subgroup H. Thus they can check conditions (a) and (b) of Proposition 5.2.14 by means of Theorem 5.2.11. Remark 5.2.16 Even when H -/- Stabc(a) Proposition 5.2.14 is helpful to find the stabilizer of a in the permutation group G. Suppose that condition (a) of Proposition 5.2.14 is violated. Then there is a g E [StabH('Yi)t;- 1 for some 1 :S i :S s such that g ff_ H. Now 'Yi = ar' -1

implies that ar' ='Yi="(;, gr, = agr,, and so g E Stabc(a). If condition (b) of Proposition 5.2.14 is violated then this algorithm produces a new double coset representative r ofH in G = (H,x). In particular, both conditions (a) and (b) of Proposition 5.2.14 can be verified by means of a finite number of applications of the algorithm mentioned above. Corollary 5.2.17 Let G = (H, x) be a transitive permutation group on n = {1, 2, ... , n} such that H :S Stabc(a) for some a E n. Let Bi = [,81, ,82, ... , ,Bk] be a base of H in n - {a} and let { s 1 , s 2 , ... , St} be a set of strong generators of H with respect to B 1 . Suppose that the conditions (a) and (b} of Proposition 5.2.14 can be verified. Then the fallowing assertions hold:

(a) JGI = IHln.

5.2 Orbits, stabilizers and group order

(b) B = {B1, a} and S set of G.

287

= {s1, s2, ... , St, x} are a base and strong generating

(a) This follows immediately from Proposition 5.2.14 and TheoProof. rem 5.1.3 because H = Staba(a). (b) By hypothesis S is a strong set of generators of H with respect to a base B 1 . Now the argument of Remark 5.2.12 completes the proof because H = Staba(a). The following theorem is related to an unpublished result of C. Sims, see [21], its p. 140 and Theorem 6.2 of [76], its p. 211. Theorem 5.2.18 Let G be a finite transitive permutation group on D = {1, 2, ... , n}. Let H = Staba(a) for some fixed a E 0. Let x E G - H such that G = (H, x). Let (SI R 1 ) be a presentation of H with sets Sand R 1 of generators and defining relations, respectively. Then there are d elements ri E G and points ii = ar-; E D such that: d

(a)

n = LJ (,i)H

is a decomposition of fl into H-orbits.

i=l d

(b) G

= LJ H riH is a double cos et decomposition of G with respect to H. i=l

(c) Ki= StabH(,i)

= H n H,.., for all i =

1, 2, ... , d.

(d) If Si= {Yi,l,Yi,2,···,Yi,tJ denotes a set of generators of Ki for each i = 1, 2, ... , d, then for each pair (i, ki) with 1 :::; ki :::; ti there is a uniquely determined element ui,k; E H such that

(Yi,k; r'

-1

= Ui,k;

for all

1 :::; ki :::; ti,

and

1 :::; i :::; d.

(e) For each i = 1, 2, ... , d let Ti = {ki,l, ki, 2 , ... , ki,q,} be a system of elq,

ements ki,Ji E H such that HriH =

LJ

Hriki,qiK is an (H-K)-double

j;=l

coset decomposition of HriH. Let 6i,J; = Then for each pair (i,ji) with 1 :::; Ji :::; qi and 1 :::; i :::; d there are a unique double coset representative rp, and uniquely determined elements kP,Jp E Tp, Vi,Ji E K, and wi,J; E H such that

(f) Let F be the free group on X = {S, x}. Write each element Yi,ki' Ui,ki' ki,j' Vi,ji 'Wi,ji E H

288

Permutation representations

as a fixed word in the generators s E S for all 1 :::; ki :::; ti, 1 :::; Ji :::; qi, and 1 :::; i :::; d. Each double coset representative ri can also be written as a fixed word in the generators x and s E S. Since F is free on X each of these elements can uniquely be lifted to a corresponding element of F. Lift also the set R1 of defining relations of H to its corresponding subset Ri of F. Define the sets

Then (XI

Ri UR~ U R;) is a presentation of the permutation group G.

Proof. (a) Since H is a subgroup of G the G-set D decomposes in finitely many pairwise disjoint fl-orbits "'ff, 1 :::; i :::; d. As G is transitive on D each "'/i = ar' for a suitable ri E G. Since H = Stab 0 (a) we may choose r 1 = 1. (c) Clearly, HnHr,:::; StabHbi) = HnHr;. Conversely, if h E StabHbi), then ar; ="'Ii= "'ff= ar;h implies that hr;_, EH, and so h E Hr; n H. (b) Each H-orbit "'ff of D = ex° corresponds to a double coset HriH of G with respect to the subgroup H, and aHr;H = ar;H = "'ff. Hence Theorem 5.1.3 and (c) imply that IStabH("'/i)I = IH: (HnHr;)I = lr;-'

!1;HI

1

l~;}fl. Therefore d

d

d

!DI= Ll"'ffl = I:IH: (HnHr;)I = i=l

i=l

L

IH~HI_

i=l

I I

d

Hence IGI = IH!IDI =

I:

IHriHI. By (a) the double cosets HriH are

i=l

pairwise disjoint. Thus (b) holds. (d) This assertion is an immediate consequence of (c). q;

(e) Since HriH =

LJ

Hriki,q,K is an (H -K)-double coset decomposition

j;=l

of HriH it follows that

u q;

,.,,Hr;H =,;;fl= uc ,.

(s:,,J·,.)K, u,.

h were

s: ui,j; ="'fiki,j; .

j,=l

Let ki,j, E Ti. Then the point ar;k;,,;x of D belongs to a unique H-orbit

"'!;!

q;

of D by (a). Since bi)H =

LJ j;=l

((\JJK there is a unique representative

5.3 Conjugacy classes of permutation groups

kP,Jr E Tp such that

representative implies that

Vi,j,

x E

(r:"· 1" )K.

of StabK(JP,Jr) in

K

289

Therefore there is a unique coset

such that

x

=

1 :"' 1"vi,Ji,

which

(f) Let N be the normal subgroup of F generated by Ri UR; UR;. Let G* = F/N, S* = {s* = sNI s E S} and x* = xN. Then G* = (S*,x*). Since G = (S, x) satisfies all the relations of Ri UR; UR; by hypotheses, (d) and (e) it follows from the universal mapping property of the free group F that there is an epimorphism c.p: G* -, G induced by c.p(s*) = s E S :S: H :S: G for all s* E S* and c.p(x*) = x E G. As H = (SI R 1 ) is a presentation of H the restriction of the map c.p to H* yields an isomorphism between H* and H. Furthermore, the surjective map c.p : G* -, G induces a G* -action on n by /3 9 * = {3'P( 9 *) for all /3 E D and g* E G*. Hence 13s* = {3'P(s*) = /3 8 for all s E S. In particular, the actions of H* and H on n can be identified. Let U* = Staba, (a). Then H* :S: U*, and m

a0*

=

LJ (ii)H*,

where

Ii

=

ar;

=

for all

acp(r;J

r7, l

:S: i :S: d.

i=l

Let K* = H* n (H*)(x*)-l. Then c.p(K*)

= u

=K

=

H

n HX-l, and

qi

K*

Ii

K

= Ii

(Ji,ji) K

wh ere

s:

ui,ji

ki



= Ii ' '

for some

ki,ji

E Ti.

]i=l

Since c.p restricted to H* is an isomorphism it follows from (d) and (e) that H* and U* satisfy the conditions (a) and (b) of Proposition 5.2.14. Hence U* = H*, and IG*I = IHIIDI = IGI by Corollary 5.2.17. Therefore c.p is an isomorphism. This completes the proof.

5.3

Conjugacy classes of permutation groups

Throughout this section G denotes a finite permutation group operating on D = {1, 2, ... , n }. It is easy to see that two permutations u and T of the symmetric group S0 of degree n are in the same conjugacy class of So if and only if they have the same cycle type. This follows immediately from the following definition.

290

Permutation representations

Definition 5.3.1 For each u E Sn and i E D the sequence i, u(i), u 2 (i), ... terminates. Hence there is smallest integer a( i) such that ua( i) ( i) = i. The a( i)-tuple a(i)-1(.)) (i,. (J" (.) i ' ... '(J" i is called the cycle of u containing i E D. Since D is a finite disjoint union of cycles of u the points of D can be renumbered such that k

La(i) = n. i=l

A 1-cycle of u is called a fixed point. In general, the cycle type of a permutation u E Sn is denoted by

where mi denotes the number of cycles of u E Sn of length i. Definition 5.3.2 A not necessarily strictly decreasing sequence A1

> A2 >

s

• • · :;:> As of integers As is called a partition of n, if .

I: Ai = n.

~1

Proposition 5.3.3 The number k(Sn) of conjugacy classes of the symmetric group Sn of degree n equals the number 1r(n) of partitions of n.

Proof. By reordering the cycle lengths of the cycle types of the permutations u of Sn obtains a bijection between the cycle types ( 1mi, 2m 2 , ••• , kmk) and the partitions of n, because k

n = Limi. i=l

The integer i does not occur in the sequence of the corresponding partition of n if mi= 0. Lemma 5.3.4 is

(a) Each permutation 1r E Sn is a product of its cycles, that

where ii, i 2 , •.• , ik are the starting points of the u-orbits, and ti, t 2 , •.. , tk are their cyclic lengths. (b) If u E Sn, then u1ru- 1 = (u(i1), ... , u(itJ)(u(i2), ... , u(it 2 ) )

Proof.

Both assertions can be checked easily.

•••

(u(ik), ... , u(itk )).

5.3 Conjugacy classes of permutation groups

Proposition 5.3.5 If u E So has cyclic type (1 mi, 2m 2 ,

••• ,

291

kmk), then:

(a) ICsn(u)I = m1!2m 2 m2! ... kmkmk!, (b) lu 8 "1 = ISo: Csn(u)I =

m 1 !2m2m~/. .. kmkmk! ·

Proof. (a) By Lemma 5.3.4 a permutation 1r E Csn (u) can either permute the cycles of length i among themselves or perform a cyclic permutation on each of its cycles or it can be a product of such permutations. Since there are mi! possibilities to permute cycles of length i, and there are imi possibilities to produce cyclic rotations, assertion (a) holds. (b) This follows from (a) by Theorem 5.1.3.

For general permutation groups G operating on D = {1, 2, ... , n} such a beautiful characterization of their conjugacy classes does not exist. Nevertheless the computer algebra systems MAGMA and GAP have built-in functions to compile a set of representatives for the conjugacy classes of a permutation group G as long as the size IDI = n of D is not too large. Calculating the centralizer Cc(g) of an element g E G is done as follows. Let S = {g1, g2, ... , gt} be a strong generating set of G relative to a given base B = [a 1 , a 2 , ... , ak] of G on D. Then calculate the descending chain of stabilizers

In order to search for elements in the centralizer Cc(g) of some element g E G or in the normalizers Nc(A) of some subgroup A of G, one needs an order on the elements of the group. The following definition is taken from [21], its p. 99.

Definition 5.3.6 Order the points of D such that the base points a 1 , a 2 , ... , °'k of the base B of G are the first k points of D. Then order the base images BG in their lexicographical order as follows. Let g, h E G. Then:

(a) B 9

< Bh

if [af,ag, ... ,af]

= [a?,aq, ... ,a?] and af+ 1 < a?+ 1 ,

(b) g < h if g and h have the same images on the first i points of D and (i + 1)9 < (i + l)h. Remark 5.3. 7 Since B is a base for G it follows that g B 9 < Bh for g, h E G.

< h if and only if

Remark 5.3.8 Given g E G. Then a base and strong generating set of Cc(g) = {x E G I xg = gx} is computed by searching through the tree of all base images of elements of G. Observe that any base B of G is a base for the subgroup C = Cc(g). Let Ci = C n Ga 1 ,a 2 , ... ,a, for i = 1, 2, ... , k.

292

Permutation representations

Suppose a strong generating set of Ci = Ca(g) n Ga 1 ,a 2 , ... ,a, has already been found. Then searching in Ga 1 ,a 2 , ... ,a,_ 1 nGa 1 ,a 2 , ... ,a, for elements of c i - l one needs only to consider the elements x that are first in xCi. In terms of base images this means that 'Yi must be the first point in its orbit of Ci, since 'Yi = (3f for all elements y below b1, "(2, ... , ,'i]. Recently M. Kratzer [96] has given a deterministic algorithm which constructs an almost uniquely determined system of representatives for the conjugacy classes of a finite permutation group G on 0. It also can be applied 'Yhen O is large. The following definitions and results are due to M. Kratzer.

Definition 5.3.9 Let cc(G) be the set of all conjugacy classes of the finite group G. A partial order relation-< on cc(G) is defined by x 0 -< ya if and only if

l(x)I < l(Y)I or l(x)I = IMI and lx 0 1< IY0 1,

where x, y E G.

Definition 5.3.10 Let G be a finite group. Let {x1,x2, ... ,xk} be a fixed set of generators for G. The_ infinitely deep k-nary tree C(G) in which the root vertex is marked by the identity la E G and the k successors of each vertex are marked successively by x1, x 2 , ... , Xk is called the complete word tree of G:

There is a canonical one-to-one correspondence between vertices in C(G) and words in generators of G: For any vertex v of C(G) let Wv denote the incremental product of the vertex markers occurring along the direct path from the root vertex to v in C(G). Conversely, starting from the root vertex and reading a given word w = w(x 1, x 2 , ••• , Xk) in generators of G like a sequence of directions guides one to the unique vertex Vw in C(G) such that

5.3 Conjugacy classes of permutation groups

293

Definition 5.3.11 Let S = {x 1 , x 2 , ... , xk} be a generating set for the finite group G. Let W( G) be the set of all words in these generators. Let w E W(G). A words E W(G) is called a cyclic shift ofwordw ifthere are subwords u and v of s such that s = uv and w = vu. Lemma 5.3.12 Let w E W(G). Then any cyclic shift of w is conjugated to win G. Proof. Lets be an arbitrary cyclic shift of w. According to Definition 5.3.11 there exist subwords u, v of s such that s = uv and w =vu= u- 1 uvu = su.

Definition 5.3.13 For 1 :S: i :S: k the infinitely deep (k - i + 1)-nary tree 'T(xi) in which the root vertex is marked by xi and the k - i + 1 sucessors of each vertex are marked successively by Xi, Xi+ 1 , ... ,xk is called the Xi-word tree. Furthermore, the infinitely deep tree R( G) in which the root vertex is marked by the identity element la E G and the sequence of subtrees at depth 1 is equal to 'T(x 1 ), 'T(x 2 ), ... , 'T(xk) is called the shift-reduced word tree of G, where each 'T(xe) is the shift reduced subtree consisting of all shift reduced words w beginning with Xe such that no Xi is a subword of w for all i = 1, 2, ... , f, - 1. In particular, 'T(xi) is the subtree of all shift reduced words w beginning with x 1 .

Again we identify a vertex v of R( G) with the incremental product Wv of the vertex markers occurring along the direct path from the root vertex to v in R(G). Lemma 5.3.14 Let w E W( G). Then the following statements hold:

(a) There exists a unique vertex u in R( G) such that Wu is the lexicographically lowest cyclic shift of w.

(b) If a vertex v of R( G) corresponds to a cyclic shift of Wu then Wv and Wu begin with the same generator of G. Proof. (a) Suppose xe is the lexicographically lowest generator occurring in w. Then the lexicographically lowest cyclic shift of w must begin with xe and correspond to a vertex u in the subtree 'T(xe) of R(G). Since R(G) is a subgraph of C(G) the existence of a bijection from W( G) onto C(G) proves the uniqueness of u. (b) Wu is the lexicographically lowest cyclic shift of both w and Wv. Hence, the assertion follows by applying part (a) to the word Wv instead of w.

294

Permutation representations

Definition 5.3.15 The length-lexicographical order on the set W( G) of words of Gin the generators x E {x1,x2, ... ,xk} of G is defined as follows. Let w,x E W(G). Then w i;;; v if and only if

lwl < lvl or lwl = lvl and w :S: v

in the lexicographical order. Proposition 5.3.16 A breadth-first-search on the shift-reduced word tree R( G) ot G traverses the subset { Wv E W( G) I v is a vertex of R( G)} C W( G) in increasing order with respect to i;;;. Along with cyclic shifts the following class of words can be neglected. Definition 5.3.17 For 1 :S: i :S: k set ni := l(xi)I to be the order of generator Xi E G. A word w E W( G) collapses if any trivial power x~i is a subword of w. Given a collapsing word w there is always a prefix panda suffix s of w such that IPsl < lwl and w = ps as elements of G. According to Proposition 5.3.16 the length-lexicographically lowest cyclic shift of ps will already be tested when BFS arrives at w. Therefore we may skip collapsing words. Algorithm 5.3.18 (Kratzer) INPUT G= begin v := R(G) .Root Vertex( );

W:={la}; C := l; m:=IGI; while c Im do v := R( G). SuccessorBFS ( v ) ;

# # # #

(x1,xz, ... ,xk);

actual vertex set of class representatives found so far termination checksum c = I::wEW lw 0 1 checksum limit

i := G.IndexinitiaLGenerator( Wv ); if Wv does not collapse and Wv is not a power of any of its prefixes and Wv r;;; s for all cyclic shifts s of Wv beginning with generator Xi then d := l(wv)I; if not G. IsConjugate ( Wv, w ) for all w E W having order d then W:=WU{wv}; c:=c+G.CLassLength( Wv ); E:={2,3, ... ,d-l}; while c I m and E I 0 do q := minE; if not G. IsConjugate( w~, w ) for all w E W having order gcdfd,q) then W:=WU{w~}; c := c+ G.CLassLength( w~ ); E := E-{q}; else

5.4 Endomorphism rings of permutation modules

295

E:=E-{nq: nEN}; end if; end while; end if; end if; end while; end; OUTPUT W;

Theorem 5.3.19 (Kratzer) Let S = {x1, x2, ... , xk} be a set of generators of the finite group G. Algorithm 5.3.18 sit terminates and returns a uniquely determined system of representatives for the conjugacy classes of the input group G. This system consists of words w E W(G) each of which being a power of the particular length-lexicographically lowest baseword with respect to the generators Xi E S. Proof.

By design the equation c=

L

lwcl

wEW

is an invariant of Algorithm sit. Whereas the value of c can never decrease it also increases if and only if a new class representative has been found. As mentioned above Lemmas 5.3.12 and 5.3.14(a) ensure that each conjugacy class of G has a non-empty intersection with the set { Wv I v is a vertex of R( G) } . Thus, after finitely many iterations the checksum c reaches its theoretical upper bound m = IGI. Therefore the constructed set Wis a complete system of representatives for the conjugacy classes of G and the algorithm terminates. The asserted minimality of the words in W is an immediate consequence of Proposition 5.3.16 and the additionally used power heuristics.

5.4

Endomorphism rings of permutation modules

Thoughout this section H denotes a fixed subgroup of the finite group G with index n = IG : HI, and n = {g1 = 1, 92, ... , 9n} is a fixed right transversal of H in G. Furthermore, (lH ) 0 denotes the permutation module of G over the ring Z of integers with stabilizer H. Sometimes, the elements of n are also denoted by Greek letters a, /3, ... etc. As before 8ij denotes the Kronecker 8-symbol.

Definition 5.4.1 Let G be a transitive permutation group on n. The permutation matrix of element g E G is the (n x n)-matrix M(g) E Matn(Z) whose entries M (g) (a, /3) are defined by:

M(g)(a,/3) =

8a.n{3

for each pair (a,/3) E 0 2 .

296

Permutation representations

The endomorphism ring of the permutation module (lH )0 is defined by End;w[(1H) 0 ] = {A

E

Matn(Z) I AM(g) = M(g)A for all g

E

G}.

Let F be any field, and FD the FG-module defined by the permutation group action of G on D. Then FD ~ (lH )0 ®z F is called the permutation module of G over F with stabilizer H. Again, each g E G determines a permutation matrix M(g) E Matn(F), and EndFa(FD) = {A E Matn(F) I AM(g) = M(g)A for all g E G} is the endomorphism ring of the permutation module FD. Remark 5.4.2 Let G be a transitive permutation group on D = {1, 2, ... , n} with stabilizer Stabc(l) = H. Let T = {g1 = 1,g2 , ... ,gn} be a transveral of Hin G such that l 9i = i for all 1 ::; i::; n. Let M(g) be the permutation matrix of the element g E G. Then it suffices to calculate the entries M(g)(l,j) of the first row of M (g), because for each 2 ::; i ::; n the entry M (g) (i, j) of M(g) is obtained by M(g)(i,j) = M(g)(l9i,j) = M(g)(l,j9i- 1 ). Theorem 5.4.3 Let H be a subgroup of G of index n

= IG : HI. Let G =

d

LJ

HxiH, x 1

= 1, be a double coset decomposition of G with respect to H.

i=l

Let F be any field, and FD = (lH )0 ®z F be the permutation module of G over F with stabilizer H. Then the endomorphism ring EndFc(FD) is addimensional F-vector space with basis { Ai I i = 1, 2, ... , d} consisting of the d adjacency matrices Ai E Matn (F).

For each g E G let M(g) E Matn(Z) be its permutation matrix. Then Oa9f3 for all (a,(3) E D2 . Let X = (xa,{3) E Matn(Z). Then X E End;w[(lH ) 0 ] if and only if XM(g) = M(g)X for each g E G. This is equivalent to: Proof.

M(g)(a,(3) =

X

= M(g)XM(g)- 1 for all g

E

G.

Now for each (a,(3) E D2 the (a,(3)-entry of M(g)XM(g)- 1 is given by M(g )X M(g )- 1 (a, /3)

=

L

Oc,9 µ,Xµ,v0v 9 -1 f3

=

Xag {39,

µ,,v

because Ov 9 -1 f3 = Ovf39 for each g E G. Therefore X = (xa13) belongs to Endzc[(1H) 0 ] if and only if Xaf3 = Xa"/3" for all g E G. Hence Definition 5.1.7 implies that each of the d adjacency matrices Ai, i = 1, 2, ... , d, commutes

5.4 Endomorphism rings of permutation modules

297

with all permutation matrices M(g), g E G. Thus all Ai E Endw[(lH ) 0 ]. Furthermore, they are linearly independent over Z by Lemma 5.1.9. Since Fn = (lH )0 18iz F it follows that the corresponding matrices Ai E Matn(F) are linearly independent over F. Now Theorem 2.6.12 implies that {A I i = 1, 2, ... , d} is an F-vector space basis of the endomorphism ring Endp(Fn), because d equals the number of H - H double cosets of G. Remark 5.4.4 Theorem 5.4.3 helps considerably in implementing the adjacency matrices Ai of the double cosets HxiH of G, 1 :S i :S d. If ki = IH : H n Hxi I, then by Definition 5.1.7 the first row of Ai has exactly ki ones and n - ki zeroes, where n = 10.I = IG: HI. Furthermore, the ones appear in the i-1

i-1

i

s=l

s=l

s=l

I: ks+ 1, I: ks+ 2, ... , I: k

ki columns j of Ai indexed by:

8 •

Therefore the

first row of Ai is easy to program. The other rows of Ai are now obtained automatically by Remark 5.4.2, because Ai E Endca(CCn) by Theorem 5.4.3. Lemma 5.4.5 Let H be a subgroup of G with index n = d

LJ

IG: HI.

Let G =

HxiH, x1 = 1, be a double coset decomposition of G with respect to H. For

i=l

each i E {1, 2, ... , d} let 1r(i) be the index of the unique double coset Hx1r(i)H containing x; 1 . Let Ai be the adjacency matrix of the G-orbit Ai == (H, H xi)G of 0, 2 • Then the following assertions hold:

(a)

1T:

i------, 1r(i) is a permutation of {1, 2, ... , d}.

(b)

A1r(i)

=

(H,Hx1r(i))G is the G-orbit of0.2 containing (Hxi,H) E

(c) (Ha,Hb)

0. 2 .

E Ai if and only if (Hb,Ha) E A1r(i)·

(d) The adjacency matrix A1r(i) of the G-orbit

A1r(i)

is the transpose (Ai)T

of Ai.

(e) The G-orbits

Ai

and

A1r(i)

have equal subdegrees ki

=

k1r(i)·

(f) The trace t r (A 1r(i). A-) - { 0kilG : HI J -

if i = j otherwise

for all 1 :S i,j :S d. d

Proof.

(a) As G

= LJ

H xiH is a disjoint union of the double cosets of G, for

i=l

each i E {1, 2, ... , d} there is a uniquely determined double coset H X1r(i)H of G which contains x; 1 . Therefore the map _1T : i ------, 1r( i) is a permutation on {1,2, ... ,d}.

298

Permutation representations

(b) By definition of 7r( i) there are elements h 1 , h 2 E H such that

xi 1 = h1Xn(i)h2. Hence (Hxi, H)G = (H, Hxi 1)x,c = (H, Hh1xnJi)h2)c = (H, Hxn(i))c. (c) Suppose that (Ha, Hb) E Ai= (H, Hxi) . Then Ha= Hg and Hb = HxiG for some g E G. Now (Hb,Ha) = (Hxig,Hg) = (Hxi,H)c E A;;(il by (b). The converse is shown similarly. (d) This assertion follows immediately from (c) and Definition 5.1. 7. (e) Certainly IHxiHI = 1Hxi 1HI for all i = 1, 2, ... , d. Hence IAnciJI I -1 I I IAil kn(i) = IG: HI = Hxi H = HxiH = IG: HI = ki by Definition 5.1.7. (f) Lemma 5.1.9, Definition 5.1.7 and assertion (d) imply that

tr(An(i)AJ) = ; : A;-+ Aut(D) by :(a) Then:

(a) Ai

=

= 'Pi(D)} Jori=

= tp:; 1 atpi

1,2.

for all a E Ai.

:(A:) is a subgroup of Aut(D) for i

= l, 2.

(b) There is a bijection between the isomorphism classes of amalgams U1

,t,1 +-----

D

,t,2 -----+

U2 having . t he same type as U1

'P1 +-----

D

'P2 -----+

U2

and the A1 - A 2 double cosets of Aut(D). In particular, if q is the number of A1 - A2 double cosets of Aut(D), then there are q non-isomorphic free products U1 *D U2 of U1 and U2 with amalgamated subgroup D. Proof.

Assertion (a) is trivial.

(b) By Lemma 7.1.7 for each amalgam U1 ;£_}:_ D ~ U2 of the same type as U1 ;!'.!._ D ~ U2 there are automorphisms ai E Aut(Ui), 'Y, 8 E Aut(D) such that

is commutative. In fact, 8 = /31 and 'Y 7.1.7.

= (/31/32 )- 1 in the notation of Lemma

340

Methods for constructing finite simple groups

Hence it suffices to classify all the isomorphism classes of the amalgams U1 $.!- D ~ U2 with 'YE Aut(D). Let 7 1,72 E Aut(D). Then

if and only if there is a triple of automorphisms a- 1 E Aut(U1), (3 E Aut(D) and a-2 E Aut(U2) such that the following two equations hold: o:1 0 be a prime not dividing IHI IEw I which is smaller than m = IHl 2 - 1 and which is minimal among the primes p not yet dealt with. Let F = Fp E J be a finite splitting field of characteristic p > 0 of the alternating group

Am. Then do the following steps:

(a) Construct the faithful multiplicity-free semisimple F H-module V corresponding to v and the faithful F Ew -module W corresponding tow.

(b) Identify H and Ew with their isomorphic images in GLn(F) under their representations afforded by V and W, respectively. Determine then by means of Corollary 7.2.4 a double coset decomposition

u 8

CaLn(F)(D)

=

CaLn(F)(H)TiCGLn(F)(Ew)

i=l

of the centralizer CaLn(F)(D) of D in GLn(F). For each double coset representative Ti, let Gi

=

(H, Ti-i EwTi), 1 :S: i :S: s.

Then Cai (z) ::=: H for all i.

362

Methods for constructing finite simple groups

(c) For each i E { 1, 2, ... s} compute the orders of some suitable elements of Gi and use this information to check whether a Sylow 2-subgroup of Gi may be isomorphic to S. If the Sylow 2-subgroups of all constructed groups Gi are not isomorphic to S, then the algorithm ends. Now let Gip = (H, r- 1 EipT) be any of the groups Gi having fulfilled the Sylow 2-group test of Step 5(c). Then the canonical n-dimensional vector space M = Fn is an irreducible FGip-mOdule with multiplicity-free restriction MIH ~ V. Step 6 Since MIH has a proper non-zero F H-submodule U one can construct a permutation representation 1r : Gip ---+ Sm into the symmetric group on m letters with stabilizer fI 2: H and a strong base and generating set for 1r( Gip). Step 7 Apply the double coset trick of Proposition 5.2.14 and check that has stabilizer fI = H. If so, then Jr : Gip ---+ Sm is faithful, and

1r

IGipl = IHlm. In particular, Gip has a 2-central involution t with Cc"' (t) ~ H. Furthermore, one can now calculate a presentation of Gip by means of Theorem 5.2.18. Step 8 Using the faithful permutation representation 1r : Gip ---+ Sm of Gip and Kratzer's Algorithm 5.3.18 compute a complete set of matrix representatives of all the conjugacy classes of Gip. Step 9 Compute a concrete character table of Gip by means of the Algorithm 5. 7.1, Theorem 2.6.5 and the LLL-algorithm 5.8.4. In particular, it can be decided whether G = Gip is simple. If not, then the concrete character table of Gip provides matrix generators of a proper normal subgroup of Gip. Step 10 Whenever a previous step cannot be successfully performed choose another maximal elementary abelian characteristic subgroup A E 2( and apply Step 2, etc. Remark 7.4.9 Using the concept of r-modular compatible pairs described in Definition 7.2.9 it is sometimes possible to weaken the hypothesis of Step 5 of Algorithm 7.4.8 on the characteristic p > 0 of the finite field F. Suppose that p divides IHI [Ew I, but Kratzer's Algorithm 7.3.10 applied to the p-modular characters of the triple (H, D, Eip) produces a p-modular compatible pair (v, w) with multiplicity-free restrictions vlD = wlD· Then all the further steps of Algorithm 7.4.8 can be performed for any such modular compatible pair. This generalization of Algorithm 7.4.8 has found several applications, see the table in Section 7.5.

7.4 Michler's algorithm constructing simple groups from given centralizers

363

Remark 7.4.10 The hypothesis of Algorithm 7.4.8 that the centralizer CH(A) of the chosen maximal elementary abelian characteristic subgroup A of the Sylow 2-subgroup of H be a solvable group is generous. For all the simple alternating groups An Kondo has shown that CH(A) = A, see [93]. The same result holds for all sporadic simple groups G distinct from the O'Nan group, and the Higman-Sims group. In the latter cases one always can show that CH(A) is an abelian 2-group, see [117] and Lemma 10.1.2, respectively.

Remark 7.4.11 In the following chapters only successful applications of Algorithm 7.4.8 will be presented. However, one can only construct a simple group G with a given centralizer H = C 0 (z) of a 2-central involution z by it, if all its steps can be performed successfully. In all the successful applications of the algorithm it turned out that a Sylow 2-subgroup S of H has exactly one maximal elementary abelian characteristic subgroup A with D = NH(A)-/- H.

Remark 7.4.12 In the classical literature on the sporadic simple groups, the centralizer H = Cc(z) is often described by the isomorphism types of its largest normal subgroup P = 0 2 (H) of 2-power order and the simple composition factors of H = H/P. However, P and H neither determine the group structure of H nor that of the target groups G uniquely up to isomorphism. It is well known and easy to check that any Sylow 2-subgroup S of the alternating group G = A 10 has a center Z(S) = (z) of order 2, and that H = Cc(z) is isomorphic to the wreathed product (z) I S4 of the cyclic group (z) by the symmetric group S 4 of degree 4. Therefore H 9" B : S 4 , where B 9" 24 is a normal elementary abelian 2-group of order 16 in H. In [83] Janko showed that there is another non-isomorphic split extension H 1 of B by S 4 with center Z(H) = (z), and that H 1 is isomorphic to the centralizer Cc 1 (z) of a 2-central involution z of the Mathieu group G 1 = Jvl22 , which will be defined in Chapter 8. Therefore in this book all the input groups H of Algorithm 7.4.8 are described by generators of H and their defining relations. Then they are uniquely determined up to isomorphism.

Remark 7.4.13 In Section 7.5 several steps of Algorithm 7.4.8 are generalized from the prime 2 to arbitrary primes p. Sometimes Kratzer's Algorithm 7.3.10 fails to verify condition (c) of Step 4 of Algorithm 7.4.8 but the analogous conditions of Algorithm 7.5.2 described below can be satisfied for odd prime. This alternative has proved to be useful in several applications.

364

Methods for constructing finite simple groups

7.5

Uniqueness criterion

The results of this section are due to the author [112]. The following sufficient uniqueness criterion will be applied in the uniqueness proofs dealt with in the last four chapters of the book. Theorem 7.5.1 Let H be a finite group with center Z(H) of even order. Suppose that some finite simple group Q5 of H -type has been constructed with the following properties:

(1) For some prime p > 0 Q5 has one conjugacy class of p-central elements Q of order p.

(2) Let Q be a central element of order p of a fixed Sylow p-subgroup SlJ of Q5. Let IJ1 = N r8 ( (Q)). There is a non-cyclic elementary abelian normal subgroup 2t of SlJ containing Q such that '.D = N'.l't(2t) is maximal among the normalizers N'.l't(SB) -=I=- IJ1 of non-cyclic elementary abelian normal subgroups SB of SlJ with Q E SB, and '.D is uniquely determined up to isomorphism by these properties.

(3) Q5 = (IJ'l, 9J1), where 9J1 = N® (2t). (4) The automorphism group Aut('.D) of '.D has only one IJ11 - 9J11 double coset, where IJ11 = {'y E Aut(IJ'l) I 'Y('.D) = '.D}, and 9J11 = {µ E Aut(9J1 I µ('.D) = '.D}. (5) There is a smallest integer n > l and a finite splitting field F for Q5 with characteristic r > 0 not dividing IIJ'll l9J11 such that the the free product IJ1 *:n 9J1 of the groups m= N r8 ( (Q)) and 9J1 = N r8 (2t) with amalgamated subgroup '.D = IJ1 n 9J1 has up to algebraic conjugacy or duality exactly one representation Ii : IJ1 *:n 9J1-----+ GLn(F) with faithful restrictions to IJ1 and 9J1 of degree n such that lli(IJ'l *:n 9J1) I divides Il!5 I. Then Q5 is (up to isomorphism) the unique finite simple group of H -type provided it can be shown that each finite simple group G of H -type has the following properties:

(a) G has one conjugacy class of p-central elements of order p. (b) IGI

= 11!51,

(c) G has a faithful irreducible FG-module W of dimension dimp(W)

= n.

7.5 Uniqueness criterion

365

(d) A Sylow p-subgroup P of G contains a central element g of order p and a non-cyclic elementary abelian normal subgroup A such that

L

= Nc((g))

~

N®((= (!J.., 1 ) of L6 (2) is maximal with respect to satisfying the following conditions:

(a) b..={cpE/zcp=zEB}. 1>.2,···,>.k-l>.k) is an elementary abelian normal subgroup of H of order IM I = 22 k-l.

(d) ,C, = (T/1, (1, ... , T/k-1, (k-1, T/k) is a subgroup of H satisfying the following defining relations:

[T/s, T/t] = 1 forl :::: s, t:::: k, (T/j(j) 3 = 1 = ((J+1T/J+i) 3 for 1:::: j:::: k - 1, [(i,(j] = 1 forl:::: i,j:::: k-1, [1ri, 1rJ] = [1ri, T/J] = [>.i, >.J] = 1, forl:::: i,j:::: k,

>.7' =

Ai1fi forl :::: i:::: k, [T/i, Aj] = [>.i, 7rj] = 1, for i

# j,

(A 1r·)(i = A·+1 forl < i 1· < k-1 [(i, Aj] = [(i, Aj7rj] = 1 for i # j. 1,

In particular, ,C,

~

1,

1,

-

'

-

'

S 2 k, the symmetric group of degree 2k.

(e) S = (1r1, T/1, ... , nk, T/k, >.1>.2, ... , Ak-1Ak, a1, ... , ak-1) is a Sylow 2-subgroup of H and

A4k.

(f) Q = (a 1 , ... , D"k-i) is isomorphic to the symmetric group Sk, because (ajaJ+ 1) 3 = 1 for 1:::: j:::: k - 1,

[an as] = 1 for all 1 :::: r, s :::: k - 1 such that Ir - sl > 1. (g) V = (1r1 , T)i, ... , 7rk, T/k) is the only elementary abelian subgroup of order 2 2 k in S and CH(V)

= V.

(h) D = NH(V) = V: P, where P = (>.1>.2, ... ,Ak-1Ak,a1, ... ,ak-1) is a semi-direct product of the elementary abelian subgroup B = (>. 1>. 2, ... , IE\ = 2k-l and the symmetric group Q.

Ak-1Ak) of order

(i) The representatives of the H-classes of involutions are: 1fs,t

= T/1 ... T/s · 1fs+l ... 1fs+t for O < s + t:::: k and T/k,O

= T/1 · · · T/k

· 7fk ·

(j) Each involution of H is H-conjugate to an involution of V. Proof. All these statements have been proved in [92] and [93] by routine arguments. Lemma 8.6.2 Let S be a Sylow 2-subgroup of a finite simple group G such that for some power 22 k > 2 there is exactly one elementary abelian subgroup A of order 2 2 k in S. Suppose that there is a involution z in the center of S such that each involution of H = Cc(z) is H-conjugate to an involution in A. Then Nc(A) > NH(A).

420

Finite simple groups with proper satellites

Proof. Since G is simple Theorem 4.7.3 states that there is a g E G - H such that zg E S. Thus zgh E A for some h E H by hypothesis. As A is abelian A:::; Ca(zgh) = Hgh_ Hence Ah-ig-i ::; H. Let T be a Sylow 2-subgroup of H containing Ah-ig-i. Then TY= S for some y EH by Sylow's Theorem. Since Ah-ig-iy and A are elementary abelian subgroups of order 22 k in S they are equal by hypothesis. Thus x = h- 1 g- 1 y E Na(A) - NH(A). This completes the proof. The following result is due to D. Held [66]. His proof depends on a deep theorem due to W. J. Wong [153]. Therefore a new proof is given by the avthor. It also shows that GL 4 (2) ~ As,

Theorem 8.6.3 (Held) Let 7-i be the centralizer of a 2-central involution in the alternating group As. Then any finite simple group G having a 2-central involution z such that C 0 (z) ~ 1{ is isomorphic to either As or A 9 . Proof. Let G be any finite simple group with a 2-central involution z such that H := Ca(z) ~ 1{, where 1{ is defined in Proposition 8.6.1 and k = 2. Then His generated by involutions Pi, Yi, m and c corresponding to the generators 'Tri, T/i, A1A2 and (1 of 1{, respectively, where 1 ::; i ::; 2. Let s = (Y1Y2)c. Then Proposition 8.6.1 implies that H has a Sylow 2-subgroup S which is generated by the elements Pi, Yi, m and s. Furthermore, S has the following set of defining relations:

[pi,Pj] = [Yi,Yj] = [Pi,Yj] = 1 for 1::; i,j s; 2, Yi= Y2, [m, s] = 1, Pi = Pi, yf' = PiYi for 1 ::; i ::; 2.

P1 = P2,

Let V = (Pi,Yi}, where 1 ::; i::; 2. Using MAGMA on the group 1{ it has been checked that V is the unique maximal elementary abelian subgroup of Sand that D = NH(V) = S. Let B = {p1 ,y 1 ,p2 ,y 1 } be a fixed basis of the GF(2)-vector space V. Let 77(m) and 77(s) be the matrices in ® = GL 4 (2) corresponding to the conjugate actions on V of the elements m and s of P with respect to B, respectively. Then the above relations imply that: 77(m) = (

.(i ) ,

1

... 1

77(s) = (

i:.~i ). .1 ..

Let~= (77(m),77(s)). By Proposition 8.6.1 each involution of His conjugate in H to some involution of V. Hence Lemma 8.6.2 implies that Na(V) > NH(V). Using Step 2 of Algorithm 7.4.8 and MAGMA it follows that there is a unique subgroup of® containing~ with maximal odd index l : ~I = 9. Furthermore, is generated by ~ and the two matrices: b1

=

( .1 . ) 11.. .. 1.

... 1

,

b2

=

(1 ... ) .1 ..

... 1 .. 11

·

8.6 Simple satellites of the alternating groups

421

Proposition 8.6.1 also states that D splits over V with complement P = (m, s). Using MAGMA again it has been checked that H 1 (P, V) = 0. Thus Gaschutz's Theorem 1.4.15 implies that E = Nc(V) splits over Vas well with a complement K containing P, and all complements of Vin E are conjugate. Thus E is the unique split extension of V by up to isomorphism, and it satisfies the above relations of S and the following ones:

= Yi, yfi = PiYi for 1 ::::: i ::::: 2 and [bi,Pj] = [bi, Yj] = 1 for all i =/. j, [m, b 1 ] = b1, [m, b2] = b2, bf= h

p~i

~ E for all simple groups G of Ji-type. By Proposition 8.6.l(i) H has six conjugacy classes. They are represented by Y1, P2, Y1Y2, Y2P2, P1P2, Y1Y2P2- Let G1 = (H, Nc(A)). Then z = p 1 p 2 and p 2 are representatives of the two conjugacy classes of involutions of G 1 because y1 = Y2, y~ 2 = P2Y2, p~2 = Y2 and

In particular, Nc(V)

-y1 y 2, Zb1b2 --pb1Pb2 1 2 -

zb1(b 2 )2 _Pbi(b2) 2 Pb1(b2) 2 -

Let Q:: be the centralizer in Ca 1 (p2) = (x1, Y1, c1,P2, Y2, m) by Theorem 4.8.5. Since Ca(P2) ~ Ca 1 (b2) = (x1, Y1, b1, b2) and 2 does not divide [Ca(b2): Ca 1 (b2)[ we first determine Ca(b2). From the above relations it follows that U = Ca(b2)/(b2) is a finite group with a self centralizing noncyclic Sylow 2-subgroup of order 4. Hence il is a simple group isomorphic to PSL 2 (5) by Theorem 1.6.2. Using MAGMA it has been checked that C®(b 2) ~ (p 1, 1) 1, 01). Furthermore, the following relations hold: Pib, = !Ji, !Jib, = Pi!Ji, [m, bi]= bi, and Pim= !Ji for 1 Si S 2, Pis= P2, P2 m = tJ2, [01, P2] = [01, tJ2] = 1, [01 5 ] = b2. From these relations it follows that

Therefore we have to find an element f of order 5 in ~ such that (ftJ 1 ) 3 = 1. Applying MAGMA we see that a solution for this computational task is provided by the following matrix

.1.1 ) . f = ( 1111 11.. 1. ..

8.6 Simple satellites of the alternating groups

423

In particular, there is an element fin Cc(b2) of order 5 such that 'l/J(f) = f, and Cc(b2) = (f,p1, Y1, b1) x (b2). Since p 2 = P2 it follows now that Cc(P2) = (f, p 1, y 1, b1, m, P2, Y2) contains a subgroup of index 2 which isomorphic to the direct product of A5 ""'(f, Y1,P1) and the Klein 4-group (P2, Y2). By Proposition 8.6.1 Cc(z) ""'Hand Cc(P1) ""'CA 9 (1r1). Furthermore, the fusion of the involutions of Cc(z) and Cc(P1) is the same as the one of the involutions of the corresponding centralizers of Ag in Ag. Hence Theorem 4.2.1 implies that IGI = IAgl. Since G is assumed to be simple and IC : G1I = 9 it follows that G has a faithful permutation representation of degree 9. Thus G""' Ag. This completes the proof. Definition 8.6.4 In the general linear group GL2n(2) let A= (a;J) be the anti-diagonal matrix of order 2, where

if i + j = 2n + 1 otherwise. Then the symplectic group Sp 2n(2) is up to conjugation the subgroup Sp2n(2) ={XE GL2n(2) I xT AX= A} of GL2n(2),

where

xr

denotes the transpose of the matrix X, see [24], its p. 3.

The following result is the essential part of Yamaki's Theorem [155]. The proof given here is due to R. Staszewski, M. Weller and the author. Proposition 8.6.5 Let H be the subgroup of GL 6 (2) generated by the following matrices:

"~ £,

~

( (

11 .... .1 .... .. 1 ... . 1. 1 .. 111.11 . 1 ... 1 11 .... .1 .... . 11 ... . 1. 1 .. . 11111 .. .. . 1



PF (

) ,, ~ ( "~

(

Let z = P1P2P3 and Bk assertions hold:

(a) H

11 .... .1 .... . 11 ... .. .1 .. . 1. 111 ..... 1 . . 1 .. 1 .1 .... ..1 ... 1 . 11 . 1 . 1 .. 1 . 1. 1 ... 1 ..... .1 ... .. 1 ... .. . 1 .. . 1 .. 1. 1 .... 1

=

)

p,

~

(

) ,, ~ ( ) >v (

(YkYk+i)Zk fork

= (P1 ,P2,p3, i\, R.2, Y1, Y2, Y3, z1, z2)

1 ..... . 1 .... . 11 ... . 1. 1 .. 11111 . . 1 ... 1 1 .. 1. 1 . 1 .... .. 11. 1 ... 1 .. . 1 .. 1. ..... 1 1 ..... .1 .... ..1 ... ..11 .. . 1 .. 1. .1

= 1, 2.

) ~ ) '"~ ( ) £,

(

11 .... .1 .... ..1 ... . 1. 1 .. .11.11 ..... 1 1 ..... .1 .... 1 .. 1 .. 1 . 1 ... . 1 .. 1 . 1.11.1

) )

Then the following

has center Z(H)

= (z).

424

Finite simple groups with proper satellites

(b) M = (P1,P2,p3,i\,C2) is an elementary abelian normal subgroup of Hof order !Ml= 25 .

(c) L = (Y1, z1, Y2, z2, y3) is a subgroup of H such that L

~

S 6, the symmetric

group of degree 6. (d) H

=M

(e) H

~

: L is a semi-direct product of M and L.

C A 12 (a), where a is the 2-central involution a= (12)(34)(56)(78)(9 10)(1112) of the alternating group A 12 .

(f) Each subgroup ¼ = (Pi, Yi) is a Klein four group for i = l, 2, 3. (g) V = V1 x Vi x Vi is the only elementary abelian normal subgroup of the Sylow 2-subgroup S = (V,C1,C2, s1) of Hof order !VI= 2 6 . (h) D

= NH(V) = V: R

~ 26

: S4,

where R

(i) The matrix .11.1. . 1 . 1 . 1

y= (

d: 1 i:

l

= (£1, £2, s1, s2)

~

S4.

EI'=GL5(2)

111.l . . . 1.11

has order 7, belongs to Nr(V), and satisfies the following conditions: (i 1 ) G

= (H, y) :Sr= GL 6 (2)

is a simple group of order !GI= 29 -3 4 -5-7.

(i2)

K=(R,y)~L3(2).

(i3)

E=Nc(V)=V:K, andEnH=D.

(j) G = (H,E)

~ Sp5(2).

(k) Representatives gi and the corresponding centralizer orders ICc(gi) I of the 30 conjugacy classes gf of G are given in the following table:

I

i la 2a 2b

2c 2d 3a 3b

3c 4a 4b 4c 4d 4e 5a 6a

I 1 (y 4 z1yz1y"z1)° (y"z1yz1)4 Z1 (y" z1y" zi) 0 (yz1)4 (y"z1)" (y"z1y"z1)" (y" z1 YZ1 )" (yz1)° (y"z1yz1y"z1yz1)" (y"z1yz1)" y 0 z1yL,z1

y" z1 yz1 yz1

y z1yL,z1yL,z1yL,z1 0

ICc(9i)I 2" . 3 4 • 5. 7 2 8 • 3" · 5 2" · 3" 2" · 3 2 .3 2 4 • 3° . 5 2". 3 4 2" . 3° 21 · 3 2° · 3 2°. 3 2(

2" 2·3·5 2" · 3·

I l

i 6b

6c 6d

6e 61 6g

7a 8a 8b

9a lOa

12a 12b 12c 15a

I

9i

y 0 z1yz1 y" z1 y 4 z1 y"z1y"z1 (yz1)" y" z1 yz1 yz1 yz1

yqz1y 0 z1yL,z1yL,z1 y 4 z1yL,z1 y y"z1yz1 y 0 z1yz1 y"z1 y 4 z1yz1y"z1 yz1 y" z1 yz1 y" z1 yz1 y"z1yz1yz1yz1 y" z1 y" z1 yz1

I ICc (gi)j 24 • 2" · 24 2" · 2· · 2"

3" 3" • 3 3" 3" ·3 7 24 2" 3" 2-5 2° · 3 2" · 3 2" · 3 3.5

8.6 Simple satellites of the alternating groups

425

(1) The character table of the symplectic group G = Sp5(2) is given in the Atlas /29}, its p. 47. (a) Let A be the involutory anti-diagonal matrix of Proof. defined in Definition 8.6.4. Then

r

c = (X E r I xr AX = A) is a symplectic subgroup of r. By Carter [24], its p. 3 it has order

IGI = 29 . 34 . 5 . 7.

i

It has been checked by means of MAGMA that z

=

(

1. 1. .... . . .. 1 .. ... 1 . . 1 .. 1 .. .. .

·. . . . 1

= P1P2P3

generates the center Z(H) of H

= Cc(z) = (P1,P2,p3,J!.1,C2,Y1,Y2,Y3,z1,z2)

:S:: G.

Since r is a simple linear group having a parabolic subgroup P ~ 25 : GL 5 (2), it has a faithful permutation representation of degree 63 with stabilizer P. Therefore there is an injective permutation representation 1r : G -, S 63 of G into the symmetric group of degree 63. (e) Let H = CA 12 (a) = (1r1,1r2,1r3,>.1,>.2,1J1,1J2,1J3,(1,(2), where all its generators are defined in Proposition 8.6.1. Using MAGMA again an explicit group isomorphism T between H = CA 12 (a) and 1r(H):::; S53 has been found. It induces an isomorphism /3 : H -, H by means of the following map: /3(1ri) = Pi for 1 :S:: i :S:: 3,

/3(1Ji) = Yi for 1,:::; i:::; 3,

/3(>.j) = J!.j for j = 1, 2, /J((j) = Zj for j = 1, 2.

In particular, assertion (e) holds. The statements (a)-(d) and (f)-(h) have been checked by means of MAGMA as well. (i) By Proposition 8.6.1 S = (P1,P2,p3,Y1,Y2,Y3,J!.1,J!.2,s1,s2) is a Sylow 2-subgroup of G with maximal elementary abelian normal subgroup V = (P1,P2,p3,Y1,Y2,Y3) of order 26. Using the permutation representation 1r : G-, S53 again it has been checked that 1r(D) = N.rr(H)(1r(V)) ~ V: S4 , where S4 ~ 1r(R) = (1r(J!.1), 1r(J!.2), 1r(s1), 1r(s2)). Furthermore, N.1r(G)(1r(V)) = (1r(V)): (1r(R),1r(s)), where s E G is the matrix of order 7 stated in assertion (i). Taking inverse images, it follows that L = (S4 , s) ~ L 3 (2), and E = N.c(V) = V : L. Therefore (i) holds.

426

Finite simple groups with proper satellites

(j) Again identify r with its image 1r(f) :S: S63 under the faithful permutation representation 1r. Using MAGMA it has been checked that

n(G) = n((H,y)) =(XE 1r(f) I n(X)T1r(A)1r(X) = n(A)). Therefore G = (H,g) ~ Sp5(2). (k) The representatives gi of the conjugacy classes gf of G = (H, g) (y,z1) and the orders ICc(gi)I have been determined by means of Kratzer's Algorithm and MAGMA. , , (1) The character table of G has been computed by MAGMA using the faithful permutation representation 7r of degree 63. It agrees with the one of Sp 6 (2) in the Atlas [29], its p. 47. This completes the proof.

Theorem 8.6.6 (Yamaki) The alternating groups A 12 , A 13 and the sympletic group Sp 6 (2) are H-satellites, where H = CA 12 (a) is the centralizer of the 2-central involution a of A12 defined in Proposition 8.6.1. Proof. Let G be any finite simple group with a 2-central involution z such that H := Cc(z) ~ Ji, where 1i is defined in Proposition 8.6.1 and k = 3. Then H is generated by inyolutions Pi, Yi, and mj, Cj corresponding to the generators ?ri, 'f/i, µj = AjAj+i and (j of Ji, respectively, where 1 :S: i :S: 3 and 1 :S: j :S: 2. Let Sj = (YjYJ+l)c; for 1 :S: j :S: 2. Proposition 8.6.1 implies also that H has a Sylow 2-subgroup S which is generated by the elements Pi, Yi, mj and Sj. Furthermore, S has the following set of defining relations: [pi,Pj] = [Yi,Yj] = [pi,Yj] = 1 for 1 :S: i,j :S: 3, [mi,mj] = 1 for 1 :S: i,j :S: 2, pfi = Pi+l, yfi = Yi+l, for 1 :S: i :S: 2, [pi, Sj] = [Yi, Sj] = 1 for i =/. j , my = mjpj for 1 :S: j :S: 2, [mi, Yj] = [mi,Pj] = 1 for i =/. j, (s1s2) 3 = 1.

Let V = (Pi, Yi), where 1 :S: i :S: 3. Using MAGMA on the group 1i it has been checked that V is the unique maximal elementary abelian subgroup of S. By Proposition 8.6.1 D = NH(V) is a semi-direct product of V by P = (m1, m2, s1, s2) ~ 22 : S3. Let B = {p1, Y1,P2, Y1, p3, y3} be a fixed basis of the GF(2)-vector space V. Let ry(mj) and ry(sj) be the matrices in Q5 = GL4(2) corresponding to the conjugate actions on V of the elements mj and Sj of P with respect to B for 1 :S: j :S: 2, respectively. Let ~ = (ry(mj ), ry(sj)) for 1 :S: j :S: 2. By Proposition 8.6.1 each involution of His conjugate in H to some involution of V. Hence Lemma 8.6.2 implies that Nc(V) > NH(V). Using Step 2 of Algorithm 7.4.8 and MAGMA it

8.6 Simple satellites of the alternating groups

427

follows that there are two subgroups 1, 2 of (5 containing fl with maximal odd indices l1 : fll = 27 and l2 : fll = 7. In the case of 2 an application of all further steps of Algorithm 7.4.8 leads to the simple group G3 ~ Sp5(2) of H-type by Proposition 8.6.5. In the case of 1 it follows that there are three matrices bi, l :S: i :S: 3 of order 3 in GL 6 (2) such that the following additional relations hold:

Pt; = Yi, yfi = PiYi for 1 :S: i :S: 3, and [bi,Pj] = [bi, Yj] = 1 for all i =/= j, [mi, bi] = bi, [mi, bi+1] = bi+l for all 1 :S: i :S: 2, [mi, bj] = 1, [bi, sj] = 1 for all j =I= i, i + l, bti = bi+ 1 for all 1 :S: i :S: 2. Proposition 8.6.1 also states that D splits over V with complement P ~ fl. Thus Gaschiitz's Theorem 1.4.15 implies that E = N 0 (V) splits over V as well with a complement K containing P. Thus E is the split extension of V by 1. Let G1 = (H,Na(A)). Let as= Pl···Ps for 1 :S: s :S: 3. By Proposition 8.6.l(i) the following ten elements of V represent all conjugacy classes of involutions of H:

Ps,t

= Y1 · · · Ys

· Ps+1 ... Ps+t for O < s

+ t :S: 3 and Y3,o

= Y1 ... y3 · p3.

Now the above relations imply that b1 ···b,

as+t

b1

b

= P1 · · · Ps' · Ps+1 · · · Ps+t = Y1 · · · Ys · Ps+1 · · · Ps+t = Ps,t·

Similarly it follows that a~ 1 b2 b~ = Y3,0· Therefore the three involutions as, 1 :S: s :S: 3, are a complete set of representatives of all involutions of G 1 because their centralizers in G 1 have pairwise distinct orders. This follows from Ca 1 (a3) = Ca(z) =Hand the following equations: Ca 1 (a2) = [(P1,Y1,P2,Y2,m1,s1,z1) x (p3,y3,c3)]: (m1m2), Ca 1 (a1)

=

[(P1,Y1) x (P2,Y2,p3,y3,m2,s2,z2,c2,c3)]: (m1).

(*) (**)

Suppose that there is a proper normal subgroup X =I= l in G 1. If IXI were even, then it would contain an involution. Suppose p 1 E X. Then P2 = pf1,p3 = p; 2 EX which implies that z,p 1p 2 EX. Hence all involutions of G 1 belong to X. Suppose that z E X. From the above fusion argument it follows that

428

Finite simple groups with proper satellites

Hence p3 E X, which implies again that all involutions of G 1 belong to X. Suppose that P1P2 E X. Then Y1Y2 = (P1P2/ 1 b2 ,

(P1P2/ 1 Cb 2 ) 2 = Y1Y2P2 EX.

Hence p 2 EX. Therefore all involutions of G 1 belong to X. Thus X contains all involutions of G 1 in any case. In particular, Vis a subgroup of X, and so is Na 1 (V). By Proposition 8.6.1 we know that His generated by involutions. Hence G 1 2: X 2: (H, Nc 1 (V)) = Q1. This contradiction to Xi= G1 shows that IXI is odd. Since z, Y1Y2Y3 and ZY1Y2Y3 are conjugate in G1 Theorem 4.2.2 implies that

where L = (z, y 1y2y3). By Proposition 8.6.1 Cx(z) :c:; O(Cc(z)) = 1. Hence X = 1, and G 1 is simple. Let T : H ---+ H be the isomorphism defined in the first paragraph of the proof. Then its restriction T 1 to V = Nr1 (V) has an extensionµ : N A 12 (V) ---+ E given by:

µ(f3i) = bi, where Pi= (4i- 3,4i - 2,4i -1) for 1 :c:; i :c:; 3. From the relations in A 12 and the above relations involving the elements bi it follows that µ is an isomorphism. In particular, the amalgams H ~ V ---+ N A 12 (V) and H ~ D ---+ E are of the same type. Since A12 = (NA 12 (V), H) and G 1 = (H, E) it follows that both groups are epimorphic images of the free product H *DE corresponding to the amalgam H ~ D---+ E. From the character tables of H, D, and E it follows that this amalgam has a compatible pair(x, 7/J) E mfchar(H) x mfchar(E) of degree 11 with irreducible constituents

and common restriction (with respect to µ 1 : D X!D = 7PID =

T

= T2 + T3 + T 6

---+

E)

E mfchar(D),

where the indices denote the degrees of the irreducible constituents and bold face indices denote faithful irreducible characters. Now Corollary 7.2.4 implies that the free product P = H *D E has an irreducible representation r;, of dimension 11 over a finite splitting field F for H and E of characteristic r not dividing IHIIEI. Since A12 = (NA 12 (V),H) and G1 = (H,E) it follows that both groups are epimorphic images of the free product H *D E corresponding to the amalgam H ~ D ---+ E. Therefore the isomorphisms T and µ induce isomorphisms from the simple groups A 12 and G 1 onto r;,(H *DE). Hence G1 ~ A12-

8.6 Simple satellites of the alternating groups

429

Let H 1 = (p 2, y2, p 3 , y 3 , m 2, s 2, z2). Then H 1 is the centralizer of the involution a2 = P2P3 of Y1 = (p2,Y2,P3,y3,m2,s2,z2,c2,c3) in Y1. Clearly, V1 = (p2 , y2, p3, y3 ) is the unique maximal elementary abelian subgroup of the Sylow 2-subgroup 8 1 = (Vi, m2, s2) of Y1. From the above relations follows that b2 , b3 E Ny1 (Vi). By the proof of Theorem 8.6.3 the generators of Y1 = (H1 , Ny1 (Vi)) satisfy the same relations as the corresponding generators of As. Hence Y 1 = (H1, b2, b3) is a simple subgroup of Y which is isomorphic to As. Furthermore, (*) implies that:

H1 = Cy1 (p2p3) = (p2,Y2,p3,y3,s2,m2,z2) ~ CA 8 (ct2) and C1 = Ca 1 (p1, yi) = (p1, Y1) x (p2, Y2,P3, Y3, m2, s2, z2, c2, c3). Suppose that G -=I- G1. Then Ca(a2) > Ca 1 (a2) or Ca(a1) > Ca 1 (a1) by Theorem 4.8.5. If Ca(P2P3) > Ca 1 (p2p3) then the arguments used in Theorem 8.6.3 imply that there is an element Ji E [Ca(p2p3)' / (P2,P3, Y2, y3, s2, z2)] of order 5 such that U1,P1, Y1, b1) ~ As and

Ca(P2P3)

= [(fi,P1,Y1,b1) x (p2,Y2,p3,y3,s2,m2z2)]: (m1m2).

Since Ji commutes with P2 and Y2 it also belongs to Ca(p2, Y2). Therefore Ca(p2,Y2) 81 = Ca(p1,Y1) > Ca 1 (p1,Y1). The Sylow 2-subgroup 82 = (p1, Y1,P2, Y2, p3, Y3, s2, m2) of C1 splits over U1 = (p1, Y1) by Proposition 8.6.1. Since IG: Gil is odd, Theorem 1.4.15 implies that C2 = Ca(p1,Y1) splits over U1 . Hence C 2 = U1 x Y for some subgroup Y of C2 containing Y1 such that IY : Y1 I -=I- 1 is odd. We now claim that Y is a simple group. As Y :s; Ca (p1, Y1) :s; Ca (P1) we have

Hence Cy(p2p3) = Cy1 (p2p3) = H1 ~ CA8 (ct2)Suppose that there is a proper normal subgroup X -=I- 1 in Y. If IXI were even, then it would contain an involution. Suppose p 2 E X. Then p3 = P? E X which implies that P2P3 E X. So in any case t = P2P3 E X because Y1 has exactly two conjugacy classes. They are represented by p 2 and t. From the above fusion argument it follows that

Y2Y3 = tb 2 b3 ,

tb 2 (b 2 ) 2 =

Y1Y2P2

E

X.

Hence p2 E X, which implies that all involutions of G 1 belong to X. In particular, V1 is a subgroup of X, and so is Ny(V1). By Proposition 8.6.1 we know that H1 is generated by involutions. Hence X 2: (H1, Na(V1)) = Y1. As Y1 ~ As is simple and IY : Y1 I is odd we get X = Y = Y1 because distinct conjugates of Y1 in X have trivial intersection. This contradiction to X -=I- Y

430

Finite simple groups with proper satellites

shows that /XI is odd. Since t, y 2 y 3 and ty2y 3 are conjugate in Y1 Theorem 4.2.2 implies that

jXj/Cx(L)/ 2 = /Cx(t)//Cx(Y2Y3)//Cx(ty2y3)/ = /Cx(t)/3, where L = (t,Y2Y3). Now Cx(t) S O(Cy(t)) = O(H1 ) = 1 implies that X = l. Thus Y is simple. Now(***) and Theorem 8.6.3 imply that Y ~ Ag, and Cc(a1) ~ [(P1, Y1) x Ag] : 2. In particular, also Cc(a1) > Cc 1 (a1). Conversely, if Cc (ai) > C01 ( a 1 ) then ( **) and Theorem 1.4.15 imply that C2 = Cc (P1, Y1) splits over U1. Hence C2 = U1 x Y for some subgroup Y of C 2 centaining Y1 such that jY: Y1]-/= 1 is odd. Hence Cc(P1, Y1) > Cc 1 (P1, Y1) = (P1,Y1) x Y1. As Cc(P1,Y1) = (P1,Y1) x Y it follows that Cy(a2) = Cy1 (a2) and that Y > Y1 does not have a normal subgroup of index 2. As above we conclude that Y ~ Ag and Cc(a1) ~ [(p1, y1) x Ag] : 2. By the proof of Theorem 8.6.3 we know that p 2 is centralized in Y by some element f of order 5. Thus Cc(a2) contains an element of order 5 not contained in Cc 1 (a 2). Since a2 and P2P3 are conjugate in G it follows that

Therefore the structure of the centralizers of the involutions of G are uniquely determined. By Proposition 8.6.1 Cc(z) ~ H, Cc(ai) ~ CA 13 (rr1) and Cc(a2) ~ CA 13 (1r 1 1r2). Furthermore, the fusion of the involutions of Cc(z), Cc(ai) and Cc (a 2 ) in G is the same as the one of the involutions in the corresponding centralizers of A 13 in A 13 . Hence Theorem 4.2.1 implies that JG/ = JA 13 j. Since G is assumed to be simple and j G : G 1 j = 13 it follows that G has a faithful permutation representation of degree 13. Thus G ~ A 13 . This completes the proof. For lack of space we can only state Kondo's Theorem [93]. Theorem 8.6.7 (Kondo) Let k ~ 4. Let 7-{ = CA 4 k (a) be the centralizer of the long involution a of the alternating group A 4 k defined in Proposition 8.6.1. Then a finite simple group G having a 2-central involution z such that Cc (z) ~ 7-i is either isomorphic to A 4 k or to A 4 k+l.

Its proof depends on the two previous results, Algorithm 7.4.8 and Theorem 4.8.5. It is done by induction on k. The crucial step is Kondo's highly original argument for the following result due to him. Proposition 8.6.8 Let k ~ 4. Let G be a finite simple group having a 2central involution z with centralizer H = Cc(z) ~ 7-i defined in Proposition 8.6.1. Then His generated by involutions Pi,Yi,mj and c1 corresponding to the generators 1ri, 7/i, ,,\1 ,,\H 1 and ( 1 of 7-i, respectively, where l S i S k and 1 S j S k - l. Let s1 = (Y1Y1+1)cj for l S j S k - l. Then the following statements hold:

8.6 Simple satellites of the alternating groups

431

= (p1, ... ,Pk, y 1, ... , Yk) is the unique elementary abelian subgroup of order 2 2k in the Sylow 2-subgroup S = (V, m1, ... , mk-1, c1, ... , Ck-1) of

(a) V

H.

= NH(V) = (V,m1, ... ,mk-1,s1, ... ,sk-1) has the following set of defining relations:

(b) D

[pi,Pj] = [Yi, Yi] = [pi, Yi] = 1 forl S i,j S k, pfi

[si, Sj]

=

[mi, mj] = 1, for l :S i,j :S k - l, = Pi+l, yfi = Yi+l, for l Si S k - l, [pi, Sj] = [Yi, Sj] = 1 for i-/= j, m}j = miPi for l :S j :S k - l, [mi,Yj] = [mi,Pj] = 1 Jori-/= j, (sjSj+1) 3 = 1 for l S j S k - l,

1 for all l S i,j :S k - l such that

Ii - jJ > 1.

(c) Na(V) > NH(V). (d) Na(V) is generated by D and k commuting elements bi of order 3 satisfying the following additional defining relations:

= Yi, yfi = PiYi for l S i S k and [bi,Pj] = [bi, Yi] = 1 for all i-/= j, = bi, [mi, bH1] = bi+1 for all l Si S k - l, [mi, bj] = 1 for allj-/= i, i + 1, bf; = bi+l for all l S i S k - l, [bi, Sj] = 1 for allj-/= i, i + l. pf;

[mi, bi]

(e) G has exactly k conjugacy classes of involutions. They are represented by {as= Pl .. ·Psll S S S k}. See Kondo [93]. For the sake of completeness we also state the following three results on the alternating groups without proof. Proof.

Theorem 8.6.9 (Held) Let H be the centralizer of a longest 2-central involution in the alternating group Aw. Then any finite simple group G having a 2-central involution z such that Ca(z) ~ H is isomorphic to Aw. Proof.

See Held [67].

432

Finite simple groups with proper satellites

Theorem 8.6.10 (Kondo) Let k ~ 3. Let H be the centralizer of a long 2-central involution a of the alternating group A 4 k+ 2 · Then a finite simple group G having a 2-central involution z such that Co(z) ~ H is isomorphic to A4k+2·

Proof.

See Kondo [93].

Theorem 8.6.11 (Kondo) Let k ~ 2. Let H be the centralizer of a long 2-central involution a of the alternating group A 4 k+3· Then a finite simple 9TOUp G having a 2-central involution z such that Co(z) ~ H is isomorphic to A4k+3·

9

Janko group J1

Janko's simple group J1 of order 175560 was the first sporadic simple group discovered in the 20th century. In his highly original paper [80] he aready used· the idea of constructing the character tables and small dimensional modular representations of the unknown target groups G from the given centralizer Ji = Cc(z) of 2-central involution z. This information enabled Z. Janko to prove the existence and uniqueness of J1 up to isomorphism. Thus he has given a model of studying finite simple groups from a representation theoretic point of view. Using Algorithm 7.4.8 M. Kratzer and the author gave in [98] a new existence proof for Janko's sporadic simple group J1. In [112] the author applied his uniqueness criterion stated in Theorem 7.5.1 to prove that the simple group J1 is uniquely determined by the centralizer of an involution. In this chapter both proofs are given.

9.1

Structure of the given centralizer

Definition 9.1.1 A finite group G having a 2-central involution z with centralizer H = Cc(z) ~ (z) x A 5 such that G has no subgroup of index 2 is said to be of J1 -type. Lemma 9.1.2 Let H

= (z) x

A 5 , z2

= 1 i- z.

Then:

(a) H has a faithful permutation representation of degree 7, and can be generated by two elements x and y satisfying the following relations: x5 = y6 = (xy)10 = (xy2)2 = [x, y3] = 1, and (xy)5 = y3.

(b) A= (z, a1, a 2 ) is a self-centralizing Sylow 2-subgroup of H, where z = y 3, a1

= xy 2,

a2

=

(x 2y) 2x.

(c) D = NH(A) = (A, d) ~ (z) x A4 , where d d 3 = 1, af = a2 and a~= a1a2.

=

y 2 x 2 satisfies the relations

(d) A system of representatives hi and the corresponding centralizer orders ICH(hi) I for the ten conjugacy classes hfl of H are given in the following

434

Janka group J1

table:

I

Class I la 2a 2b 2c 3a

1 (y)o xy• x4y (yy

5a

X

5b 6a

(x)" y xy (xy)o

10. 10b

I ICH(h;)I 2°. 3. 5 2°. 3. 5 2" 20 2·3 2·5 2-5 2-3 2-5 2·5

h;

I 2P I

3P

la la la la

la

la

2a 2b 2c la 5b 5a 2a 10b

10a

2a 2b 2c 3a la la 6a 2a 2a

3a 5b

5a 3 5b 5a

I 5P I

(e) The character table of H is given as: Class

la

2a

2b

2c

3a

5a

5b

6a

10a

Length

1

15

15

20

12

12

20

12

12

Repr.

1 z2

z

a1

za1

d

X

a2x

zd

zx

za2x

'Pl

1

1

1

1

1

1

1

1

1

1

'P2

1 -1

1

-1

1

1

1 -1

-1

-1

'P3

3 -3 -1

1

0

0

-a

- *°'

'P4

3--3 -1

1

0

0

-O
(H), K>(E)) = (X, Y, S) :S GL1(ll)

450

Janko group J1

generated by the matrices 5 5 9 8 9 8 9 8 8 9 ( 85 X= 6 8 8 8

59),

7 9 7 5 1 1 1

cg

6 3 5 3 3 2 3 3

Y=

6 3 3 2

)

5 9 1 1 1 9 7 5

. 8. 8 . . . ) .55.211 .66 .. 110 S= ( . 55.91010 10.33.55 1.33.66 1.88.55

is a finite simple group of order

l 1 with the irreducible characters Xi given in the table above. 1

75

62

58

71

49

7

64

69

56

•---+---+----e-----e-----e-----e----+----e-----e----•

Xi

X2 1

X1

75

Xl3

62

58

Xi4

X10

Xn

49

71

Xi2

X4

7

Xi5

64

69

X9

56

•---+---+----e-----e-----e-----e----+----e-----e----•

Xi

X2 1

X1

Xi3

X10

14

119

Xi5 49

27

106

Xn

Xi2

X4

64

69

7

Xi4

X9

56

•----e-----e-----e----+---+----e----+----e-----e----•

Xi

Xi3 1

Xi4

X10

14

119

X2

X7

27

106

Xn

49

Xi2

X4

64

69

7

Xl5

X9

56

•----e-----e-----e----+---+----e----+----e-----e----•

Xi

Xl3

X10

Xl5

X7

X2

Xn

X4

Xi2

Xi4

X9

In any case, each finite group G of Ji-type has an absolutely irreducible 11-modular character cp of degree c.p(l) = 7. Theorem 9.4.2 ( J anko) Up to isomorphism there exists only one finite group G of Ji -type with involution centralizer

Ca(z)

~

H = (z) x A5.

It is isomorphic to the finite simple group llJ of order IllJ I = 175560 described in Theorem 9. 3. 2. Proof. This is done by verification of the conditions of the uniqueness criterion given in Theorem 7.5.l. By Theorem 9.3.2 there exists a finite simple group llJ having an involution Z with C115(Z) ~ H ~ (z) x A5 such that lllJI = 175560, and J satisfies all the conditions (1) and (2), (3) and (5) of Theorem 7.5.1 for the primes p = 2, r = 11 and the minimal dimension n = 7. Furthermore, also condition (4) holds by Lemma 9.2.l(f). Theorem 9.2.3 asserts that all finite groups G of Ji-type have order IGI = 175560 = IllJ I. By Proposition 9.4.1 each such group G has an absolutely irreducible 11-modular representation of degree 7. Furthermore, Lemma 9.1.2 asserts that Na(A) ~ N 115 (~), where A and~ are Sylow 2-subgroups of G and llJ, respectively. Applying now Theorem 7.5.1 for the primes p = 2 and r = 11 we get G ~ llJ.

10

Higman-Sims group HS

In this chapter Previtali's and the author's existence and uniqueness proof [116] for the Higman-Sims group HS is given. The group HS and its automorphism group Aut(HS) are realized inside GL 22 (11), and a concrete character table of HS is constructed. For applications in Chapter 11 a presentation of Aut(HS) in terms of generators and relations is given as well. Originally, Higman and Sims [71] constructed their sporadic simple group HS as a rank 3 primitive permutation group of degree 100. The group HS was shown to be the unique simple group of its order by Parrott and Wong [124] and classified by the structure of the centralizer H = C0 (z) ~ (4 * 2 1 +4)S5 of a 2-central involution z of G = HS by Janko and Wong in [85]. Gorenstein and Harris in [48] were able to show that HS is the unique simple group whose Sylow 2-subgroup is isomorphic to a split extension of Z 4 x Z 4 x Z 4 by a dihedral group of order 8. The arguments given in these important articles are involved and do not generalize to other sporadic groups. Therefore a new existence and uniqueness proof for the Higman-Sims group was given by Previtali and the author using Algorithm 7.4.8 and Theorem 7.5.l. The proofs given here are more elementary.

10.1

Structure of the given centralizer

As mentioned by Janko and Wong [85] the given centralizer His an extension of a non-abelian group T ~ 4 * 2 1 +4 of order 64 by the symmetric group S5 . Cannon informed Previtali that the command ExtensionsOfSolubleGroups of MAGMA [12] produces 48 non-isomorphic extension groups Hi with 0 2 (Hi) ~ T, and Hi/ 0 2 (Hi) ~ S5 , where 0 2 (X) denotes the largest normal subgroup of 2-power order of the group X. In order to avoid ambiguities Previtali and the author gave in [116] a presentation of that extension H that could be shown to be isomorphic to the centralizer of a 2-central involution of the Higman-Sims group. This presentation is stated in the first lemma of this section. It gives rise to the following definition.

Definition 10.1.1 A finite simple group Xis said to be of HS-type if it possesses a 2-central involution z such that Cx(z) ~ H, where His the group of even order defined by generators and relations in Lemma 10.1.2 below.

Higman-Sims group HS

456

Lemma 10.1.2 Let H be the finite group of even order generated by elements r, s, and c subject to the following set R(H) of relations:

r3 = ss _= cs = 1, (r-1 s-1 )4 = (rc-1r-1 s-1 )2 = (r-1cs-2)2 = (r-1cs2)2 = 1,

rcr

sc- 2rs, scrcsr

=

= c2 ,

s 2cr

=

rcsrs.

Let f = srcsr- 1 sc, e = rscsc 2, t 2 = r- 1 c2scs, z2 = (rc- 1)3, u = ut t§, z = Z§, b1 = s- 1 rs- 1 r- 1 s- 1 rs, b2 = (sr)2, k = rsr- 1 sr- 1 s- 1 . Then f, u, t, z, b1 , b2, k are involutions, and u2, t2, and z 2 have order 4 and the fallowing assertions hold:

(=

(a) H has an outer automorphism"( defined by:

r'"Y

=

r, s'°Y

=

rsr 2s 2rsr 2s- 1 , c'°Y

=

rc 2 r- 1 c2r.

(b) S = (s,f,s 2,u2,t2 ,z2,u,t,z) is a Sylow 2-subgroup of H with center Z(S) = (z) such that ry(S) = S. Moreover s 4 = Z§ = z, U§ = u, t§ = t, and u 2 = t 2 = z 2 = 1. (c) P = 0 2(H) = (z2, t, u, b1, b2) = (z 2) * (t, u, b1, b2) Z(P) = (z). ( d) A

Sc'

4 * 21+4, with center

= (z, t, u) is the only elementary abelian normal subgroup of order 2 3

of S, and ry(u)

=

u, ry(t)

=

t, ry(z)

=

z.

(e) V = (z, t) is the unique normal Klein four subgroup of S. (f) CH(V)

= Cs(V) =

(!, s 2, u2, t2, z2, u, t, z) and Z(Cs(V))

= V.

(g) NH(V) = S. (h) C = CH(A) = Cs(A) = (u 2 , t 2, z2 ) is a homocyclic group isomorphic to Z4 X Z4 X Z4.

(i) C is the unique homocyclic rank 3 subgroup of order 64 in S. (j) Set D = NH(A), then A. Thus M is not normal in J'. This contradiction shows that M is not solvable. Hence it is a direct product of simple subnormal subgroups of J'. None of the three subgroups Xi of J is a direct product of non-cyclic isomorphic 2-subgroups. Hence Mis a simple group with a Sylow 2-subgroup W 2 of order 8 or 16. Suppose that IW2I = 16, then W2 = W1. Since M is a characteristic subgroup of J' and J' does not have a proper normal subgroup of odd index it follows that M = J'. If W1 ~ X 1 then M has a unique conjugacy class of involutions z, by Proposition 1.6.5. Thus Mis isomorphic to PSL 3(q) or U 3 (q) by Theorem 1.6.6, where q = pm for some odd prime p. Furthermore, ICa(z)I is divisible by p. However, z = Y1Y2 is the center of W1, and CM(z) = W1 has order 16, a contradiction. Suppose now that W 1 ~ X 2. Then M ~ PSL 2(q) for some odd prime power q > 3 or M ~ A 7 by the Gorenstein-Walter Theorem 1.6.4. If M ~ A1, then ICM(z)I = 24, which is a contradiction. In the other case IW1I = ICM(z)I E {q + 1,q - 1}. Hence q = 17, M ~ PSL2(17), and J ~ Aut(M) = PGL 2(17). Another application of MAGMA now yields that the Sylow 2-subgroups of PGL 2(17) are not isomorphic to the Sylow 2subgroup W of J. Hence we may assume that W1 = X 3. Since M does not have a normal subgroup of index 2 Lemma 1.4.3 implies that t is J' -conjugate to some involution of the maximal subgroup I of W1. The dihedral subgroup I of J' has five involutions: z = (ty1y3)2, Y2, Y1 = y;Y 1 Y3 , ty1y3y1 = ty1Y2Y3 = tzy3, (tzy 3)tY1Y3 = ty3. As Yi = z, and y1/_4 = tzy3 it follows that all five involutions of I are conjugate in J. Hence t and z are J-conjugate, a contradiction. Thus IW2I = 8, and W2 = I is a dihedral group of order 8. As W is indecomposable, and O(J) = 1 we get CJ(M) = l. Hence W is a Sylow 2-subgroup of Aut(M). Since Aut(A 7 ) ~ S7 has a Sylow 2subgroup of order 16 the Gorenstein-Walter Theorem 1.6.4 now asserts that M ~ PSL2(9) ~ A5. Hence J ~ Aut(A5). (e) This can easily checked by means of MAGMA. (f) A system of representatives ui of the conjugacy classes uf of U and the orders of their centralizers can be calculated by means of Kratzer's Algorithm 5.3.18. (g) This follows immediately from the power map information given in the table in Section 10.6.4. (h) Since T = CH(v) = H n U is a Sylow 2-subgroup of U the fusion of the conjugacy classes of 2-elements of T in U is uniquely determined by Sylow's Theorem. Using the faithful permutation representations of Hand U determined in Lemma 10.l.2(m) and assertion (e) the fusion patterns of the conjugacy classes of Tin Hand U can be calculated by means of MAGMA, respectively. Proposition 10.2.5 Let G be a finite simple group of HS-type having a 2central involution z with centralizer H = Ca(z). Let A = (z, t, u) be the

10.2 Fusion

465

uniquely determined maximal elementary abelian normal subgroup of the fixed Sylow 2-subgroup S of G defined in Lemma 10.1.2, and let E = Na(A). Then the following assertions hold:

(a) IGI = 29 · 32 · 53 · 7 · 11.

(b) G = (H, E). Proof. (a) By Propositions 10.2.2 and 10.2.4 each simple group G of HS-type has two conjugacy classes of involutions z0 and v 0 such that

IHI= ICa(z)I = 29 · 3 · 5 and IUI = ICa(v)I = 26 · 32 · 5. Using the faithful permutation representations of H and of U described in Lemma 10.1.2 and Proposition 10.2.4, respectively, and the fusion patterns of the conjugacy classes of H and U in G given in Propositions 10.2.2 and 10.2.4 an application of MAGMA yields that

r(z,w,z) = l{(x,y) E (z 0 nH) x (w 0 nH)lz E (xy)}I = 10720, r(z,w,w) = l{(x,y) E (z 0 n U) x (w 0 n U)lw E (xy) }I= 1755. Now Thompson's Theorem 4.2.1 states that IGI

r(z,w,z) · ICa(w)I +r(z,w,w) · ICa(z)I 29 . 32 . 53 . 7. 11.

(b) By Lemma 10.1.2 the Sylow 2-subgroup S of G is neither abelian nor metabelian. Therefore the Bender-Suzuki Theorem 1.7.3, implies that the simple group G cannot have a strongly embedded subgroup. As S is also neither dihedral nor semi-dihedral, Theorem 4.8.5 asserts that

G

=

(H,Ca(t)lt E J(H))

where I(H) denotes the set of all involutions t of H. By Proposition 10.2.3 G has two conjugacy classes of involutions. They are represented by z and v. Furthermore, by the proof of Proposition 10.2.3 for each t E I(H) there are elements h2, ... , hk EH and e1, e2, ... , ek EE such that

Let x = h1e1h2e2 · · · hkek. Then x E (H, E) and either [Ca(t)]"' = Ca(t"') = Ca(z) =Hor [Ca(t)]"' = Ca(t"') = Ca(v) = U. Hence G = (H,E,U). By Proposition 10.2.4 U = (v) x (Y1,Y2,Y3,y4,e), where Yi ED= (r,s):::; Hand v, e92 = srs 3 c 1 EH for g EE given in Proposition 10.2.2. Hence U:::; (H, E), which completes the proof.

466

Higman-Sims group HS

Proposition 10.2.6 Let G be a finite simple group of HS-type having a 2central involution z with centralizer H = Ca(z) = (r, s, c) defined in Lemma 10.1.2. Let A = (z, t, u) be the unique maximal elementary abelian normal subgroup of order 8 in the fixed Sylow 2-subgroup S of H, and let g be the element of order 3 in G of Proposition 10.2.2 satisfying z 9 = t, u 9 = u, and Na(A) = (r,s,g). Let v = (rs- 1) 4s 2, and U = Ca(v) = (v,y1,y3,y4,e) be as in Proposition 10.2.4 and k = y4y2(e- 1y4) 2ey4. Then the following statement holds:

, (a) G has a unique conjugacy class 3A of elements oforder 3. In particular r, g and e are conjugate in G.

(b) Ca(e)/(e) = K1 = (a,k) ~ 85 has the following set R(K1) of defining relations: k 3 = 1 = a4, ka 2 = k 2 and (ak) 6 = 1, h-1

where a= [(cr 2 ) 3 ]

1

2

9

for h1 = rs- 1cs 2 EH of order 6.

(c) No= Na(3A) = Na(e) = (Ca(e),y3) = (a,e,k,y3)

~ 83 x 85.

(d) G has five e-special conjugacy classes represented by the following classes of Na(e) given in the·table in Section 10.6.5: 3a, 6b, 6c, 12a, 15b.

(e) A system of representatives Xi of the 21 conjugacy classes x{'l0 of N 0 = Nc(3A) = (a, e, y3, k) and their centralizer orders ICN0 (xi) I are given in the table in Section 10.6.5.

(f) The fusion of the conjugacy classes of No = Nc(3A) into the conjugacy classes of G is given by the following table G 2A 2B No c d a b Hbdce

3A.

e

4A

b a ad

4c

6B

b

e b

g

Proof. (a) By Proposition 10.2.5 IGI = 29 · 32 · 53 • 7 · 11, hence U contains a Sylow 3-subgroup of G. Using the faithful permutation representation of U defined in Proposition 10.2.4 and MAGMA it can be checked that Cu(e) = (e) x (v) x W1, where W 1 = (k,p) with k = y4y2(e- 1y4)2ey4, p = tz satisfying the relations k 3 = 1 = p 2 and kP = k 2 . By MAGMA we got only one conjugacy class of 3 elements in U, hence in G. (b) Using the faithful permutation representation of H defined in Lemma 10.1.2 and MAGMA it follows that CH(r) = (r) x L 1 , where L 1 = (z1, z2) satisfies the relations ZI = 1 = Zi, and z; 1 = z~, for Z1 = cs- 1csr-ls- 1c and z2 = (cr 2 )3. Furthermore, the center of the dihedral group L1 is z. Hence L1 is a Sylow 2-subgroup of Cc(r). By (a) e E U = Cc(v) and rare conjugate in G. Furthermore, e9 = srs 3c- 1 E H by Proposition 10.2.4(d). Another application of MAGMA now yields that e9 h 1 = r for h 1 = rs- 1 cs- 2 E H

10.2 Fusion

467

h-1 2

of order 6. Thus L 11 9 is a Sylow 2-subgroup of C 1 = Cc(e) by (a). Since Cu(e) = (e) x (v) x (k,p), by Gaschiitz's Theorem 1.4.15 (e) has a complement K 1 in Cc(e) and up to isomorphism K1 2- (L2, v, k). Set a= z; 1192 . -12

As z,z 1 ,z2 E L1 the elements zh 1

9

=

h-1 2

h-12

tz, z 1 1

9

=

h-12

v and z2 1

9

=

a

belong to the Sylow 2-subgroup L2 = L 11 9 of K1, and k does not normalize the Klein four subgroup (v, z) of L2. Hence Ki/ O(K1) contains a normal subgroup K 2 S" PSL2(q) by the Gorenstein-Walter theorem. Now for any involution z1 E K2, ICK2 (z1)I E {4,8} n {q ± 1}. If q - l = 8, then K2 = PSL 2(9) has an elementary abelian Sylow 3-subgroup of order 9, but 3 divides jK1 I to the first power only. If q + l = 8, then in K2 = PSL2 (7) S" PSL3 (2) the involutions z and v are conjugated, another contradiction. Since k does not normalize (v, z), K 2 is not isomorphic to a symmetric group S 4 . Hence q > 3, and q = 5. Therefore K 2 S" PSL 2(5) S" A 5 , and K 2 contains an alternating group A 4 as subgroup. Applying then the Brauer-Wielandt formula as in Proposition 10.2.4 it follows that O(K1 ) = 1. Hence Cc(e) = (e) x K 1 and K1 = (a,k) S" S5, where a 4 = l = k 3 , (ak) 6 = 1 and ka 2 = k 2 . (c) Using the fact that eY 3 = e2 it follows from (b) that N 0 = N c ( e) = Cc(e)(y3) S" S3 x S5. (d) The 3A-special conjugacy classes of Gare easily read off from the table in Section 10.6.5. (e) The representatives of the conjugacy classes of N 0 = Nc(e) = (e, k, a, y3) have been calculated with Algorithm 5.3.18. (f) Observe that INc(r) n HI = 48. Hence this intersection contains the Sylow 2-subgroup of N 0 = N 0 (e) up to conjugation in G. Now apply the proof of Proposition 10.2.4(f) and the fusion follows. Lemma 10.2. 7 Let G be a finite simple group of HS-type having a 2-central involution z with centralizer H = C 0 (z). Then the following statements hold:

(a) G has a unique conjugacy class 7 A of elements s 1 of order 7, Cc(s 1 ) = (s 1), and Nc(s 1)/Cc(s1) is a cyclic group of order 6. (b) A system of representatives Yi and the corresponding centralizer orders ICN3 (Yi)I of the seven conjugacy classes of N3 = Nc(7A) = (s1,ti) is given in the following table:

I i I la 2a 3a 3b 6a 6b 7a

I

Yi

1 (t1)°

(ti)" (t1)4 t1 (t1) 0 s1

ICNq (yi)I

2.3.7 2·3 2·3 2·3 2·3 2·3 7

I 2P I 3P I 7P I la la la la 2a 2a 3b la 3a la 3b 3a 2a 6a 3a 3b 7a

2a 7a

6b

la

468

Higman-Sims group HS

(c) The character table of N3 = Nc(7A) is stated in the table in Section 10. 7.8.

(d) G has two conjugacy classes llA and llB of elements q1 and q2 of order 11, respectively. Furthermore Cc(qi) = (qi) and Nc(qi)/Cc(qi) is a cyclic group of order 5, for i = l, 2. Proof. Let GP denote a Sylow p--subgroup of G. As IGI = 29 · 32 · 53 · 7 · 11 by Proposition 10.2.5 it follows from Propositions 10.2.4( e) and 10.2.6( c) that ICc(Gn)ll5 3 · 7 · 11. On the other hand INc(Gu) : Cc(Gu)ll10. So INc(Gn)I = 2a · 5b · 7c · 11, where O ~ a, c ~ l and O ~ b ~ 4. Now IG : Nc(G 11 )1 = 1 (mod 11) forces a = c = 0 and b = l. By Burnside's Theorem 1.4.17 Nc(G 11 ) = Cc(G 11 ) would imply that G is 11-nilpotent, a contradiction, because G is simple. Thus INc(Gu)I = 5 · 11 and ICc(Gu)I = 11. Similarly ICc(G 7 )I 15 3 · 7 · 11. By Proposition 10.2.2(h) E possesses two conjugacy classes of elements of order 7. Hence !Ne( G 7 ) : Cc( G 7 ) I E {3, 6}. Thus INc(G1)I = i · 5a · 7 · 11 where i = 3,6, 0 ~ a ~ 3 and O ~ b ~ 1. The only triples (i, a, b) compatible with Sylow's Theorem are (6, 0, 0) or (3, 2, 1) as has been checked computationally. But the second alternative would imply that a 7-element commutes with an 11-element. So 1Nc(G7 )1 = 6 · 7 and ICc(G 7 )1 = 7. The remaining assertions are easily checked by means of MAGMA.

Proposition 10.2.8 Let G be a finite group of HS-type having a 2-central involution z with centralizer H = Cc ( z) = (r, s, c). Then the following assertions hold:

(a) CH(c) = (c) x (z 2 ), where z2 Sylow 2-subgroup (z2 ).

= (rs 2 ) 3

has order 4, and Cc(c) has cyclic

(b) NH(c) = (c) : S1 , where S1 = (a 1 , b1 ) is a Sylow 2-subgroup of order 16, and a 1 = s- 1 (cs)2r, b1 = sr- 1 csc satisfy the following set R(S1 ) of defining relations:

(c) (c) is the center of a Sylow 5-subgroup R of G which is contained in Nc(c), R is extraspecial of order 125 and exponent 5. It is generated by 2 elements x, y E P with the following set R(R) of defining relations: [x,y] = c2 , x 5 = y 5 = 1 and [c,y] = [c,x] = 1.

10.2 Fusion

469

(a1, b1 , x, y) has the following set

(d) N1 = Nc(c) = R : S1 = Nc(R) R(N1) of defining relations:

(e) A system of representatives ni and the corresponding centralizer orders ICN, (ni)I of the 19 conjugacy classes n;'' of the normalizer N 1 are given in the table in Section 10. 6. 6.

= Nc(5A)

(f) The character table of N1 = Nc(5A) is given in the table in Section 10. 'l.6.

(g) G has at least two conjugacy classes 5A and 5B of elements of order 5 represented by c and x, respectively. (h) N2 = Nc(5B) = Nc(c2) ~ (c2) : (!) x As = (c2, f, w, v), where c2 = (y4e) 2 and f = Y1Y4ey4y3e 2y4ey4 have orders 5 and 4 respectively, and As = (w, v) has defining relations v2 = ws = (wv ) 3 = 1.

(i) A system of representatives mi and the corresponding centralizer orders ICN2 (mi)I of the 25 conjugacy classes m;'2 of the normalizer N 2 Nc(5B) are given in the table in Section 10.6. 'l.

=

(j) The character table of N 2 = Nc(5B) is given in the table in Section 10. 'l. 'l. (k) Using the notations of the conjugacy classes of H = Cc(z), U = Cc(v), N1 = Nc(5A) and N2 = Nc(5B) as in the tables in Sections 10.6.1, 10. 6.4, 10. 6. 6 and 10. 6. 'l, respectively, their fusion patterns into the conjugacy classes of G are given by the following tables: G

N, H G N2

u

2A a a 2A a

f

2s b e

4c

4A a a

b

e

a

C

d

b

e

g

C

lOA a a

d

C

e

4c

Ss

C

d

a

e

a a

4A

2s b a

Sc

8A a a

C

b

f

20A b a

20s a b lOs

C

a

b

C

Proof. (a) As c E H has order 5, we calculate CH(c). By MAGMA it is a direct product (c) x (z2 ), where (z2 ) = Z(0 2 (H)) = Z(H') has order 4, and z~ = z. As seen before this implies that L = (z 2 ) E Syl 2 (Ca(c)). (b) This assertion has been checked computationally by means of MAGMA. (c) From (a) and Burnside's Theorem 1.4.17 it follows that M = C 0 (c) has a normal 2-complement R = O(M). Using MAGMA again we see that

4 70

Higman-Sims group HS

V1 = (z,q1) :S:: NH(c) is a Klein four group, where q1 = sr- 1 csc is conjugate to q1z in NH(c). By Brauer-Wielandt's Theorem 4.2.2

Now CR(z) = (c) and CR(V1) = 1 since z is the only involution in CH(c). On the other hand 251 IRI. So 51 ICR(q1)I = ICR(zq1)I and 1251 IRI. By Proposition 10.2.5 and Lemma 10.2.7 Risa {3, 5}-group. If 3 divides IRI then 91 IRI, too. Since Ca(e) c::c Z 3 x Ss this would force the existence of an element of order !5 in Ss, a contradiction. Thus IRI = 125, RE Syls(G), and ICa(c)I = 22 · 53 . Since 24 IINH(c)I and INa(c): Ca(c)ll4 we have INa(c)I = 24 · 53 . Before we can complete the proof of (c) we first have to prove (h). (h) Now observe that c2 = (y 4e) 2 EU has order 5. Applying MAGMA we see that Cu(c2) = (c 2) x (v, v2), where v2 = y1ey1y4(ey4)2e 2 is an involution. Furthermore Nu(c2) = ( (c2) : (!)) x (v, v2), where f = Y1y4ey4y3e 2y4ey4 has order 4, satisfies c{ = c~, and T = (v, v2) is a Klein four group. Suppose that z 0 n T contains an involution i. Then z = ix for some x E G, and d2h E H = Ca(z). Therefore c = c2 for some h E H by Lemma 10.l.2(n). Hence Txh :S:: Ca(c), a contradiction to (a). Hence all involutions of T are conjugate to v in G, that is T# = T n v 0 . Let X = Ca(c2). We now claim that T is a Sylow 2-subgroup of X. Suppose that T is a proper normal subgroup of a 2-subgroup Y of X. Then Z(Y) n T cf. l. Let i E Z(Y) n T#. Then v = ix, c2 EU, and so yx :S:: U. Now there is au E U such that c2 = c2, so yx :S:: Cu(c2) = Cu(c2)u and yx :S:: Tu, forcing T =YE Syl 2(X). Set X = X/(c 2). Now Cx(T) = Ca(v) n Ca(v2) nX = Cu(v2) nX. Since Cu(v2) is a 2-group, we get Cx(T) =Tis a self-centralizing subgroup of X. Set P = O(X) and P = P/(c2). If y E Cp(i), i E T#, then [y,i] is both inverted and centralized by i, so it is l. Since 25 f ICa(i)I, 5 f IPI. Lemma 10.2.7 now forces P to be a 3-group. So P = (c 2) x P0 , where P0 = 0 3 (X). Since 25 I IXI, there is an element of order 5 in X acting on P0 . But 5 f I GLi(3)1, i = 1, 2. On the other hand 25 f ICa(3A)I. Thus P0 = 1, O(X) = (c 2), and O(X) = l. Let M be a minimal normal subgroup of X. Since 5 I XI, X is not a 2-group, by Glauberman's Z*-Theorem 4.7.3 there exists an x EX such that v cf. vx ET. If T were normal in X, then IXll12, a contradiction. Thus Mis a simple group and TE Syl2(M). By Theorem 15.2.1 in [47], its p. 421, M c::c PSL 2(q), q = ±3 mod 8. Moreover, IMll2 2 ·3 2 ·5 3 , thus q = 5 and M c::c As. Since IX: Ml is odd, we get Cx(M) = 1 and X = M. Therefore X is an extension of Zs by As. Since the Schur multiplier of As is Z2, we obtain Ca(c2) c::c (c2) x As. Since Nu(c2)/Cu(c2) c::c Z4, we obtain Na(c2) c::c ( (c2) x As) : Z4. MAGMA shows that Na(c2) c::c (Zs : Z4) x As. Clearly, the involution v E As. The existing additional generator w of As can be chosen to be of order 5 satisfying the given relations with v.

10.2 Fusion

471

(d) In particular, R E Syl 5 (G) is not abelian, c E Z(R), and Nc(R) :