Foundations of Geometry: Selected proceedings of a conference 9781487583767

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Foundations of geometry Selected proceedings of a cont erence

Foundation s of geometry Selected proceedings of a cont erence

Edited by P. Scherk

University of Toronto Press Toronto and Buffalo

©

University of Toronto Press 1976 Toronto and Buffalo Printed in Canada Reprinted in 2018

LIBRARY OF CONGRESS CATALOGING IN PUBLICATION DATA Main entry under title: Foundations of geometry Conference held at the University of Toronto, July 17-Aug. 18, 1974. 1. Geometry - Foundations - Congresses. I. Scherk, Peter. II. Toronto. University. QA681.F74 516 75-42127 ISBN 0-8020-2216-2 ISBN 978-1-4875-8240-1 (paper)

CONTENTS

Preface - vii Contributors - ix Combinatorics of finite planes and other finite geometric structures - A. Barlotti - 3 Configurational propositions in projective spaces J. Cofman - 16 Unitary planes - Martin Gotzky - 54 The Lenz-Barlotti type III - Christoph Hering - 88 Some recent results on incidence groups - H. Karzel - 114 Topological Hjelmslev planes - J.W. Lorimer - 144 On finite affine planes of rank 3 - Heinz Liineburg - 147 Every group is the collineation group of some projective plane - E. Mendelsohn - 175 Recent advances in finite translation planes - T . G. Ostrom - 183 Recent results on pre-Hjelmslev groups - Edzard Salow - 206 Finite geometric configurations - J.J. Seidel - 215 Distributive quasigroups - K. Strambach - 251 Planes with given groups - Jill Yaqub - 277

V

PREFACE

In 1974 the University of Toronto sponsored a unique experiment. For more than four weeks, from July 17 to August 18, a group of outstanding workers in the foundations of geometry lived together on the campus of the university, gave and attended lectures, and met informally. Three longer series of sixteen lectures each were given by F. Bachmann (Kiel University, West Germany)

on Hjelmslev

groups, by R. Lingenberg (Karlsruhe, West Germany) on

s-

groups, and by J. Tits (College de France, Paris) on generalized polygons.

Two of these are being prepared for publi-

cation as separate volumes. In addition, fifteen mathematicians gave not the usual brief accounts of their work but detailed reports which required up to seven hours of lecturing. reports appear in this volume.

Thirteen of these

The other two reports, by

E. Sperner (Hamburg University, West Germany) on affine structures and by J. Tits on quadratic forms and the geometry on quadrics, unfortunately could not be included. The editor wishes to thank the University of Toronto Press and the Blythe Foundation for making publication of the volume possible.

The papers represent an invaluable basis

for further progress in the field, as has already been demonstrated by the appearance of one important paper which received its stimulus from the conference.

vii

CONTRIBUTORS

BARLOTTI, ADRIANO Universita di Bologna, Istituto Matematico, Bologna, Italy COFMAN, JUDITA Fachbereich Mathematik der Johannes Gutenberg Universitat, 6500 Mainz, Postfach 3980, West Germany GOTZKY, MARTIN Mathematisches Seminar der Christian Albrechts Universitat, 2300 Kiel, Neue Universitat, Angerbau Al , West Germany HERING, CHRISTOPH Mathematisches Institut der Universitat Tubingen, 7400 Tubingen 1, Auf der Morgenstelle 10, West Germany KARZEL, HELMUT Institut fur Geometrie der Technischen Universitat, 8000 Munich 2, Arcisstrasse 21, West Germany LORIMER, J . W. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S !Al, Canada LONEBURG, HEINZ Fachbereich Mathematik der Universitat Kaiserslautern, 675 Kaiserslautern, Pfaffenbergstrasse, West Germany MENDELSOHN, ERIC Department of Mathematics, University of Toronto, Toronto, Ontario, M5S !Al OSTROM, T.G . Department of Mathematics, Washington State University, Pullman, Wash . 99163, U. S.A.

ix

SALOW, EDZARD 28 Bremen, Katrepelerstrasse 7, West Germany SEIDEL, J.J. Department of Mathematics, Technological University Eindhoven, PO Box 513, Eindhoven, The Netherlands STRAMBACH, KARL Mathematisches Institut der Universitat Erlangen-Niirnberg, 852 Erlangen, Bismarckstrasse l½, West Germany YAQUB, JILL C.D.S. Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210, USA

X

FOUNDATIONS OF GEOMETRY Selected proceedings of a conference

COMBINATORICS OF FINITE PLANES AND OTHER FINITE GEOMETRIC STRUCTURES

A. Barlotti

In the following pages, we survey recent results on some of the topics considered in section 3.2 of P. Dembowski's book Finite Geometries (1968; for other references on this subject, see A. Barlotti, 1974). 1.

SYSTEMS AXIOMATIZING FINITE PLANES AND OTHER FINITE

STRUCTURES Many authors have contributed to the study of systems of axioms for finite projective and affine planes (see Dembowski, 1968, pp. 138-139).

Using the notation of Dembow-

ski, we state the following results. THEOREM A (A. Basile, 1970).

Let n > 1 be an integer.

Then the following is a complete list of minimal axiom systems, each one of which will ensure that a non-degenerate incidence structure is a finite projective plane of order n: (a)

(l'), (2"), (4"), (5'),

(a')

3

(1"), (2'), (3"), (6');

(b)

(l'),(2 11 ),(4 11 ),(6'),

(b')

(1 11

(c)

(l'),(2 11 ),(3'),(6 11 ) ,

(c')

(1"), (2'), (4'), (5");

(d)

(l'),(2 11 ),(5'),(6 11

(d')

(l 11)

(e)

(1'),(2 11 ),(6),

(e')

(l 1 1 ) , (2'), (5);

(f)

(2'),(3 11 ),(4 11 ),(5'),

( f')

(1

(g)

(1'),(3 11 ),(4 11 ),(5'),

(g')

(2'), (3 11

),

(4"), (6');

(h)

(1 11 ),(2'),(3"),(4'),(6 11

(h' )

( 1' ) , ( 2 11

)

(

),

),

1 )

t

(2'), (3"), (5');

(2 t)

,(3 11

)

t

(5 11

,(4 11

)

t

),

(6

I);

(6');

),

,

3' ) , ( 4 11 )

,

(i)

(2 11

),

(i')

(1"),(3'),(4),(5 11

(j)

(l"), (3), (4'), (6"),

(j')

(2 11 ),(3'),(4),(5");

(k)

(2'), (3"), (5'), (6"),

(k')

(1'),(4"),(5"),(6');

(,Q,)

( l ' ) , (3"), (5'), (6"),

(,Q,')

(2'),(4"),(5"),(6');

(m)

( 3 11

(mt )

(3'), (4"), (5"), (6');

(n)

(2'),(3"),(6)

(n')

(1'),(4"),(5);

(o)

(1'),(3"),(6),

(o')

(2 I)

(p)

(3"),(4'),(6),

(p')

(3'),(4"),(5);

(q)

(l"), (3'), (4'), (5"), (6 11

),

),

) t

(r)

(2

(s)

(1 1 )

(3), (4'), (6 11

(

4

I ) t

(

5

I ) t

(

6 H)

t

t

(4")

t

);

(2"), (3'), (4'), (5"), (6");

(6 11

)

,

(r')

(1'), (5"), (6);

(5)

t

(6 II)

t

(s')

(2'),(5"),(6);

(t)

(4'),(5),(6"),

(t')

(3 I)

(u)

(3'), (5), (6"),

(u')

(4'), (5"), (6).

t

(5")

t

THEOREM B (P. Brutti, 1969; G. Pabst, 1974). be an integer .

(6) ;

Let n > 2

Then the following is a complete list of

minimal axiom systems, each one of which will ensure that

4

;

(q') t

t

5 11 )

(5);

(5)

I) t

(

a non-degenerate incidence structure is a finite affine plane of order n: (a)

(3)

(b')

(5), (7), (9");

(c')

(5),(8),(9);

(d')

(3'), (5"), (7"), (9);

(e)

(3")

(f)

(3'), (5"), (8"), (9);

(g')

(3")

I

(5 I)

I

(8 I)

I

(9") i

(k)

(5")

I

(7")

I

(8 I)

I

(9 I);

(Q,)

(5'),(7

(m)

(3'), (5"), (7'), (8"), (9");

(n)

(3"), (5'), (7"), (8').

(5)

I

I

I

(9) ;

(5 I)

1 ),

I

(7 I)

I

(9") i

(8"), (9");

Clearly additional hypotheses will modify the above systems.

C. Bernasconi (to appear, a) studied the problem

with the additional assumption that the structure is connected, and M. Crismale (to appear) changed (3'), (4'),

(4") to axioms which require that in (3) and (4)

"=" be replaced by "-~ 11 or

11

~

bols in Dembowski (1968], v 1 b

1

~

(3"),

n + 1, b

1

~

in average (with the sym-

11

~

n + 1, v 1

~

n + 1,

n + 1).

Similar systems of axioms have been studied for other finite structures:

for inversive planes (R. Bumcrot,

private communication) and for projective designs (C. Bernasconi, to appear, b).

5

In connection with the above questions, see P . Biscarini (to appear), N.G. de Bruijn and P. Erdos (1968), R. Magari (1964), and I. Reiman (1963, 1968).

In particular,

we wish to point out that D.A. Drake (1974) has proved that in a finite Hjelmslev plane the parallel axioms cannot be replaced by cardinality assumptions. 2.

REPRESENTATION OF NETS AND PLANES

We can use the representation of nets and planes by sets of mutually orthogonal latin squares to study the properties of the plane (see R. C. Bose and K. R. Nair, 1941; G. Pickert, 1955). For recent results in this field, see A. Barlotti (to appear), J . Denes and A.O. Keedwell (1974), and P. Hohler (1970, 1972). 3.

GALOIS GEOMETRIES

Galois geometries are the study of the sets of points in finite spaces.

In the following, PG(r, q) will denote an

r-dimensional projective space of order q .

If r

= 2,

the symbol PG(2, g) will be used only for a desarguesian plane whereas n(q) will denote any projective plane of order q. A (k; n)-arc of n(q) is a set of k points of n(q) such that n is the largest number of them which are collinear.

In a given plane, a

there does not exist a

(k; n)-arc is complete if

(k'; n)-arc which contains it with

6

k' > k. (k; 2)-arcs are simply called k-arcs; i.e. a k-arc of n(q) is a set of k points of n(q) such that no three of them are collinear. 3.1.

Some properties of k-arcs

In the last twenty years, k-arcs have been studied extensively, especially in PG(2, q).

We shall recall here some

of their properties. A line of n(q) is called a secant, a tangent, or an external line to a k-arc Kif it meets Kin two, one, or zero points.

It is very easy to prove that in n(q) the

number of tangents of a k-arc passing through a point not on the arc has the same parity ask. A question which in general is still open is determining the maximum value of k for which, in a given n(q), a k-arc will exist (the packing problem fork-arcs). In the general case, we have the following result: THEOREM 3.1.1 (R.C. Bose, 1947). in n(q), then k

$

If there exists a k-arc

q + 1 when q is odd and k

$

q + 2 when

q is even. The following theorems give the existence of maximal arcs in the desarguesian planes. THEOREM 3.1.2. In PG(2, q), an irreducible conic is a (q

+ 1)-~. 7

THEOREM 3.1.3 (B. Quist, 1952). of a

In TT(q), all the tangents

(q + 1)-arc are concurrent when q is even. This implies that in TT(q), if q is even, then every

(q + 1)-arc is incomplete and can be uniquely completed to form a

(q + 2)-arc.

The point at which all the tangents

of a (q + 1)-arc meet is called the nucleus of that {q + 1)-arc. The question of the existence of {q + 1)-arcs in every projective plane TT(q) is still open. THEOREM 3.1.4 (G. Korchmaros, to appear) .

Each Hall plane

of odd order contains a {q + 1)-~. A graphic characterization for conics in PG(2, q), q odd, is given by the following theorem. THEOREM 3.1.5 {B. Segre, 1955).

In PG(2, q), if q is odd,

then every (q + 1)-arc is an irreducible conic. In PG(2, 2h), we can obtain a

(q + 2)-arc by adjoin-

ing the nucleus to a conic (see Theorem 3.1.3).

By delet-

ing a point other than the nucleus from this (q + 2)-arc, we obtain a {q + 1)-arc which is not a conic provided q > 5.

For if this arc is a conic, then by the theorem

of Bezout, it cannot have more than four points in common with the original conic.

The question arises as to

whether or not there exist (q + 2)-arcs which do not

8

contain a conic.

The following theorem provides the

answer. THEOREM 3.1.6

(B. Segre, 1957).

h

In PG(2, 2 ), there

exist (q + 2)-arcs which do not contain q + 1 points forming a conic when h = 4 or h = 5 or h

~

7.

It is not known if the same property holds for h When h

= 6.

= 1, 2, or 3, every (q + 2)-arc can be obtained

by adding the nucleus to the q + 1 points of a conic. We state now a theorem which shows how a geometrical property of an arc can determine the nature of the plane

in which the arc can be embedded: THEOREM 3.1.7

(F. Buekenhout, 1966).

If in n(q) there

is a (q + 1 ) - ~ K such that every hexagon whose vertices are points of K is pascalian, then n(q) is pappian and K

is a conic. We wish to point out that the introduction of "segments" in the geometry over a field by B. Segre (1973) leads to results of deep interest in the study of conics. For more details on the study of k-arcs, the reader is referred to A. Barlotti (1965) and B. Segre (1961, 1965, 1967). 3.2.

Some properties of (k; n)-arcs

The study of (k; n)-arcs, for n > 2, is not as developed

9

as the study of k-arcs.

The "packing problem" has not

been solved, not even for the case of PG(2, q).

A. Basile

and P. Brutti (1971, 1973) and H.-R. Halder (to appear) have obtained interesting results on this problem with the aim of proving the Lunelli-Sce conjecture (see L. Lunelli and M. See, 1964). In the following, we present a brief list of topics with which (k; n)-arcs are connected or in which they are of interest. (a) The use of (k; n)-arcs in the construction of geometric structures. class of (p

3m

+ 1, p

m

R.C. Bose (1958) used a special + 1)-arcs to derive BIB-designs.

T.G. Ostrom (1962) used arcs to construct Bolyai-Lobachevsky planes and J.A. Thas (1973, 1974) used (k; n)-arcs to construct partial geometries.

The connection between

arcs and certain (finite) Laguerre-m-structures is also of interest (see H.-R. Halder and W. Heise, to appear). Structures in high-dimensional finite spaces, corresponding to the k-arcs of the projective plane over the total matrix algebra of then x n matrices with elements in GF(q), were used by J.A. Thas (1973) to construct special classes of 4-gonal configurations.

See also the

constructions due to J. Tits in P. Dembowski (1968). (b) The relationship between {k; n)-arcs and other subsets of points of n{q).

A. Bruen and J.C. Fisher

{1973) found an interesting connection between complete

10

k-arcs and "blocking sets."

(A blocking set Sin n(q) is

a subset of the points of n(q) such that every line of n(q) contains at least one point in Sand at least one point not in S [see, for example, A. Bruen, 1971].) (c) To obtain information on planes, whose existence is not yet known, by studying (k; n)-arcs in the planes (with the hypothesis that the planes exist).

As examples:

R. H. F. Denniston (1969) proved that no 6-arc is complete in n(l0) and A. Barlotti (1971) obtained some (k; n)-arcs in the plane n(l2) with a subplane of order 3. 3.3.

k-caps and the "packing problem"

A set of k distinct points of PG(r, q)

(r

of which are collinear, is called a k-cap.

~

3), no three We shall denote

by m(r, q) the maximum number of points in PG(r, q) that belong to a k-cap.

The value of m(r, q) is very important

for some applications in coding theory and in statistics. However, it is a very difficult problem to determine m(r, q)

("packing problem") in the general case.

The following are the known values of m(r, q)

(r

~

m(3, q) = q 2 + 1, m(r, 2) = 2r, m(4, 3)

20

m(S, 3) = 56

(G. Pellegrino, 1970), (R. Hill, to appear).

A list of the known bounds for m(r, q) is given in a paper by B. Segre (1967, p.166).

11

3):

For a wide illustration of further problems on caps, see the papers due to appear by G. Tallini and M. Tallini Scafati. REFERENCES Barlotti, A. 1965. Some topics in finite geometrical structures. Lecture Notes, Chapel Hill, N.C. 1971. Alcuni procedimenti per la costruzione di piani grafici non desarguesiani. Conferenze Sem. Mat. Bari 127: 1-17. 1973. Some classical and modern topics in finite geometrical structures. In: A Survey of Combinatorial Theory, edited by J.N. Srivastava et al., pp. 1-14. North Holland Co . 1974. Combinatorics of finite geometries. In: Combinatorics, edited by M. Hall Jr. and J.H. van Lint, Part I, pp. 55-59. Mathematical Centre Tracts 55, Math. Centrum, Amsterdam. (to appear) Alcune questioni combinatorie nello studio delle strutture geometriche. In: Atti Convengno Teorie Combinatorie, Acc. Lincei, Rome 1973. Basile, A. 1970. Sugli insiemi di proprieta che definiscono un piano grafico finito. Le Matematiche 25: 84-95. Basile, A. and P. Brutti. 1971. Alcuni risultati sui {q(n - 1) + l; n}-archi di un piano proiettivo finito . Ren. Sem. Mat. Univ. Padova 46: 107-125. 1973. On the completeness of regular {q(n - 1) + m; n}-arcs in a finite projective plane. Geom. Dedicata 1: 340-343. Bernasconi, C. (to appear) (a) Strutture di incidenza connesse e definizione assiomatica di piani grafici e affini. In: Ann. Univ. di Ferrara. (to appear) (b) Sistemi di assiomi che caratterizzano i disegni proiettivi. Biscarini, P. (to appear) Sets of axioms for finite inversive planes. Bose, R.C. 1947 . Mathematical theory of the symmetrical factorial design. Sankhya 8: 107-166 . 1958. On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements. The Golden Jubilee Commemoration Volume, Calcutta Math. Soc. 1973. On a representation of Hughes planes. In: Proc. Internat. Conf. on Projective Planes, edited by M.J. Kallaher and T.G. Ostrom, pp. 27-57. Wash. State Univ. Press . 12

Bose, R.C . and K.R. Nair. 1941. On complete sets of Latin squares. Sankhya 5: 361-382. Bruen, A. 1971. Blocking sets in finite projective planes. SIAM J. Appl. Math. 21: 380-392. Bruen, A. and J.C. Fisher. 1972. Arcs and ovals in derivable planes. Math. z. 125: 122-128. 1973. Blocking sets, k-arcs and nets of order ten . Advances in Math. 10: 317-320. Brutti, P. 1969. Sistemi di assiomi che definiscono un piano affine di ordine n. Ann. dell'Univ. Ferrara, ser. VII, 14: 109-118. Buekenhout, F. 1966. Plans projectifs a ovoides pascaliens. Arch. Math. 17: 89-93. Crismale, M. (to appear) Sui sistemi minimi di assiomi atti a definire un piano proiettivo finito. de Bruijn, N.G. and Erdos. 1968. On a combinatorial problem. Indag. Math. 10: 421-423. Dembowski, P. 1968. Finite Geometries. Springer-Verlag, Berlin-Heidelberg-New York. Denes, J. and A.D. Keedwell. 1974. Latin Squares and Their Applications. Academic Press, New York and London. Denniston, R.H.F. 1969. Nonexistence of a certain projective plane . J. Austral. Math. Soc. 10: 214-218. Drake, D.A. 1974. Near affine Hjelmslev planes. J. Comb . Theory 16: 34-50 ... Halder, H.-R. (to appear) Uber syrnrnetrische (k; n)Kurven in endlichen Ebenen . Halder, H. -R. and W. Heise. (to appear) On the existence of finite Laguerre-m-structures and k-arcs in finite projective spaces. Hall, M. Jr. 1959. The Theory of Groups. Macmillan, New York. Hill, R. (to appear) On the largest size of cap in s 5 , 3 . Rend. Acc. Naz. Lincei. (to appear) Caps and groups. Hohler, P. 1970. Verallgemeinerung van orthogonalen lateinischen Quadraten auf hohere Dimensionen. Diss. 4522, Eidg. Technischen Hochschule Zurich. 1972. Eigenschaften van vollstandigen Systemen orthogonaler Lateinischer Quadrate, die bestirnrnte affine Ebenen reprasentieren. J. Geom. 2: 161-174. Korchmaros, G. (to appear) Ovali nei piani di Hall di ordine dispari. Lunelli, L. and M. See. 1964. Considerazioni aritmetiche e risultati sperimentali sui {k; n}-archi. Rend. Ist. Lombardo (A) 98: 3-52. Magari, R. 1964. Sui sistemi di assiomi "minimali" per una data teoria. Boll. UMI 19: 423-435. Menichetti, G. (to appear) q-archi completi nei piani di Hall di ordine q = 2k.

13

Ostrom, T.G. 1962 . Ovals and finite Bolyai-Lobachevsky planes. Amer. Math. Monthly 69: 899-901. Pabst, G. 1974. Private communication. (The result is in the author's master thesis.) See also "Quantitative axioms for affine planes" in Abstract of Communications ICM, Vancouver, 1974, p . 109. Pellegrino, G. 1970. Sul massimo ordine delle calotte in s 4 . Le Matematiche 25: 149-157. ,q 1972. Procedimenti geometrici per la costruzione di alcune classi di calotte complete in S 3 . Boll . UMI (4) 5 : 109-115. r, Pickert, G. 1955. Projektive Ebenen. Springer-Verlag, Berlin-Gottingen-Heidelberg. Quist, B. 1952 . Some remarks concerning curves of the second degree in a finite plane . Ann. Acad. Sci. Fenn. 134 : 1-27. Reiman, I. 1963 . Su una proprieta dei piani grafici finiti. Rend. Acc. Naz. Lincei 35: 279-281. 1968. Su una proprieta dei due disegni. Rend . Mat. e Appl. 76-81. Segre, B. 1955. Ovals in a finite projective plane. Canad . J . Math . 7: 414-416. 1957. Sui k-archi nei piani finiti di caratteristica due. Rev . Math . Pures Appl . 2: 289-300 1961. Lectures on Modern Geometry. Cremonese, Rome . 1965 . Istituzioni di geometria superiore, Vol. I, II, III, Lecture Notes . Istituto Matematico, Universita di Roma. 1967 . Introduction to Galois geometries. Memorie Lincei (8) 7: 133-236. 1973. Proprieta elementari relative ai segmenti ed alle coniche sopra un campo qualisiasi ed una congettura di Seppa Ilkka peril caso dei campi di Galois. Ann . Mat . Pura e Appl . ser. IV, XCVI: 289337 . Tallini, G. (to appear) Graphic characterization of algebraic varieties in a Galois space . In: Atti Convegno Teorie Combinatorie, Acc. Lincei, Rome 1973. Tallini Scafati, M. (to appear) The k-sets of type (m, n) in Galois spaces S (r ~ 2). In: Atti Convegno r,q Teorie Combinatorie, Acc. Lincei, Rome 1973 . Thas, J.A. 1970. Connection between then-dimensional affine space A and the curve C, with equation n,q y = xq, of the affine plane A2 n· Rend. Trieste 2: 146-151. ,q 1973 . 4-gonal configurations. In: Finite Geometric Structures and Their Applications, C.I.M.E. II ciclo 1972, pp . 249-263 . Cremonese, Rome.

14

1973. On 4-gonal configurations. Geom. Dedicata 2: 317-326. 1973. Construction of partial geometries. Simon Stevin 46: 95-98. 1974. Construction of maximal arcs and partial geometries. Geom. Dedicata 3: 61-64. (to appear) Some results concerning {(q + 1) (n - l); n}-arcs and {(q + 1) (n - 1) + l; n}-arcs in finite projective planes of order q. (to appear) On 4-gonal configurations with parameters r = q 2 + 1 and k = q + 1.

15

CONFIGURATIONAL PROPOSITIONS IN PROJECTIVE SPACES

Judita Cofman

A configurational proposition about points and lines of a projective space Eis any statement about points and lines in E of the following kind: (C)

For any choice of points P 1 choice of lines £ 1 equalities P. i

i

,

£2 ,

P., t J

s

,

••• ,

i

P2

,

••• ,

Pn of E and any

tm of E from certain in-

it, certain incidences

P. I t , and negations of some incidences P. J t , i s i s where i, j E {l, 2, •.• , n'}, with n'

~

n, m'

~

s, t

E

{l, 2, •.. , m'}

m, i i j, s i t , certain incidences

of the same type follow. The points P 1

,

••• ,

Pn and the lines £ 1

,

••• ,

tm are called

the variables of (C); the points Pn+l' ••• , Pn' and the lines tm+l'

••• , tm' are the fixed elements of (C).

If

(C) has no fixed elements, we say that (C) is universally valid. Well-known examples of configurational propositions are the proposition of Desargues and the proposition of Pappus.

Their importance in the theory of projective 16

spaces justifies the study of configurational propositions in general.

It would be desirable to establish hierarchies

between propositions of given classes of projective planes (see, for example, Amitsur [1]); however even special cases of this problem turn out to be extremely difficult.

On

the other hand there are other possibilities for the study of propositions by relating them to different properties of the corresponding spaces.

For instance, R. Baer [3]

has pointed out that (P, g)-transitivity of a projective plane n is equivalent to the property that n is (P, g)desarguesian.

This result has a large number of important

consequences for projective planes (see, for instance, problems related to the classification of Lenz-Barlotti in [7], p. 126).

Or questions of the following kind can

be investigated:

given a configurational proposition (C)

in a restricted form in a projective space E, is it possible to remove some of the restrictions imposed on (C); in particular under which circumstances is (C) universally valid in E? Here I would like to consider some recent results on configurational propositions. be considered:

The following topics will

(1) configurational propositions and cor-

relations in projective spaces; propositions in spreads;

(2) configurational pro-

(3) configurational propositions

in derivable planes.

17

I.

CONFIGURATIONAL PROPOSITIONS AND CORRELATIONS

I.l.

Projective Planes

A correlation of a projective plane the point setP of

TI

is a 1-1 mapping of

TI

onto the line set2' of

the incidence relation in

TI.

preserving

TI

A point X (line x) is called

an absolute point (line) of a correlation a of only if it is incident with its image x 0

(x 0

)

TI

if and

under a.

A correlation of order 2 is called a polarity. Although polarities of projective planes have often been investigated, not much is known about correlations of arbitrary power. In this section we shall discuss results of Herzer

(9) on correlations with a large number of absolute points. Correlations with many absolute points were studied by Baer (3) in 1942 . plane

TI

a

Call a correlation

o of

a projective

(P, g)-correlation if and only if there is a

non-incident point-line pair (P, g) in

TI

such that x 0 =

PX for any point XI g and £ 0 = £ n g for any line£ IP. It is easy to verify the following:

(a) the points of

g and the lines through Pare the only absolute elements of

o;

(b) for any point Y f P, Y

t

g and any line

h I PY n g, h f PY, g there is at most one (P, g) correlation of

TI

mapping Y onto h.

A plane

TI

is said to be

(P, g)-homogeneous if and only if for a given non-incident point-line pair (P, g), any point Y f P, Y

18

t

g and any

line t I PY n g, t f PY, g there is a of

mapping Y onto t .

TI

(P, g)-correlation

Baer has proved that a plane

is

TI

pappian if and only if there are two distinct lines m, g in M

such that

TI

#

is (M, g)-homogeneous for any point MI m,

TI

m n g.

Consider now a correlation o of a projective plane

TI

with the property that the set of absolute points of~ contains the point set [g] of a line g of subset.

c

correlation TI

as a proper

Then the following is true:

PROPOSITION 1.1.

g of

TI

If the set

of

of

of absolute points of a

contains the point set [g] of a line

TI

as a proper subset, then there exists a unique

line i of

J'f=

such that

TI

[g] u [t].

PROOF STEP 1.

A

=

96-1

t

For, the image of any line i # g through A

lute lines.

is the point Ao

n

(t n g)

follows that to

=

i

lines i

1 ,

=

g and£~= i

t

1

n

i

g; all lines through A are abso-

2

g.

n

c.

Since (t n g)o

n

i

n

g, it

There are at least two distinct

through A distinct from g . 2

=

g are distinct.

The points£~ Hence

At

g.

All

lines through A are absolute . STEP 2.

g 0 = B # A.

Otherwise

c

would be an (A, g)-

correlation, contradicting the existence of a point in

/t-

[g].

Clearly, B

STEP 3.

t

g.

No point of AB distinct from P

19

=

AB n g is

an absolute point of o. such that Y concurrent.

f

points A,

x,

t

Hence Y0

=

g, P 0

£

r/t -

[g), x 0 I A.

g,

/t" -

ferent from g.

Xe

I

x,

and (AX n g)o I AX n g are

Thus

Xe

= AX.

[g) = [£) for a line£ through P, dif-

Since for any point X

£

,It- [g) the line

x 0 is incident with A i t follows that Ao £

jt - [ g) .

Take any point z of £

It is easy to see that Zo point.

The

AX n g are three distinct points, hence

three concurrent lines.

X

AB, and Y0 are

Y.

For any point X

their images Ao

STEP 5.

=

The images A0

P.

STEP 4.

For, suppose Y is a point of AB

=

AZ.

=

Ao

-1

I X for every

-1 I

z

t

£ n g.

Thus z is an absolute

Since every point of£ is absolute and since

there are no absolute points on AB except P, it follows that g, £, and AB are concurrent. DEFINITION.

A correlation o is said to be a

correlation (i

=

(A . , g . )1.

l.

1, 2) if and only if the setv~ of abso-

lute points of o consists of the points of gi and Ai 0-1 = gi for i = 1, 2. Our aim is to investigate projective planes admitting (Ai' gi)-correlations. following questions:

(A., g.)-correlations? l.

l.

We shall consider the

Which projective planes can admit How is it possible to characterize

pappian projective planes in terms of (A., g . )-correla1.

tions?

20

l.

The following definitions are needed: DEFINITION.

An (A . , g . )-configuration consists of two 1

1

distinct lines g 1 , g 2 intersecting in a point P and of two distinct points A 1 , A 2 incident with a line h through P

i P i A2 •

where A 1

DEFINITION.

For a given (A . , g.)-configuration consider 1

t

any line 2

1

P intersecting gi in Ri' i = 1, 2, and any

point SI 2, S i t n h, R 1

i Si R 2 •

If for all possible

choices of 2 and S the points SA 1 n g

THEOREM 1.1.

A projective plane

SA 2 n g 2

1 ,

admits a

TI

,

and

(A., g . )1

1

correlation if and only if it is (A., g . )-pappian. 1

1

PROOF (a) Assume that set of points in lines in mapping

TI

is (A., g.)-pappian. 1

o of

ft into ~:

cr is a bijection of

J

'JS

onto

£

J

and the set of

let Y0 be the line

•

It is easy to verify that

i-

Moreover, three collinear

are mapped onto concurrent lines of$·

if three points A, B, Care on a line t

-

the lines A0 (where Ri

Denote the

Consider the following

for any Y

joining YA 1 n g 1 to YA 2 n g 2

points of

JJ

which are not on h by

TI

not through P by~

TI

1

=

-

,

B0

,

t

P then each of

c 0 is incident with the point A 1 R 2

2 n gi,

i

=

1, 2) since

TI

Namely,

n A2 R1

is (Ai' gi)-pappian.

If A, B, Care on a line t IP, consider the intersection

21

-

-

Q of A0 and B0

The point Q must belong to h; otherwise

•

the lines through Q would represent the images of the

cr,

points ofJ} on a line not through P under possible.

which is im-

But if QI h, this implies that Q

and similarly Q

a is

Thus

tained from

-

= A0

-

c0

n

since Q

,

= A0

-

= A0

-

B0

n

h.

n

an isomorphism of the affine plane c1f,

ob-

by deleting h together with its points onto

TT

the dual of the structure obtained from together with the lines through P.

by deleting P

TT

Obviously,

cr

can be

uniquely extended to an isomorphism o of the corresponding projective planes.

Thus o is a correlation of

If

TT.

T £ g. for i l.

= TAi+l

o is a

Thus TI T0

(i + 1 taken mod 2).

in other words

,

(A., g.)-correlation. l.

l.

(b) Leto be a

(A., g.)-correlation of l.

l.

From the

TT.

investigations in the proof of Proposition 1.1 it follows

1

that A.

l.

g. and that all lines through A. are absolute l.

l.

for i = 1, 2. set of P.

TT

Denote by h the line A 1 A 2

,

byJ the point

outside h, and by$ the line set of

TT

not through

For any point X £ f i t s image x 0 is the line through

XAi n g i, i

= 1, 2.

The image t O of any line t

point R 1 A 2 n R 2 A 1 where Ri

= t

n gi.

£

$

is the

The correlation o

is uniquely determined on the elements of

?

u

'ii.

As shown

in (a) this implies that o is uniquely determined on

TT.

The points of gi are mapped onto the lines through Ai+i· Thus gf

= Ai+l

for i

=

1, 2,

22

(i +1 taken mod 2).

Hence

=

(g 1 n g 1 ) 0

o

A 1 A 2 and (A 1 A 2 ) 0

interchanges hand P.

g1 n g2•

The points A 1

they form together with g Let i be any line of

=

and g 2 a

1

l.

that

TI

,

Pare collinear;

l.

not through P and let S be any point

TI

,Q,

l.

o maps

A2

,

(A . , g . )-configuration.

of i distinct from i n hand from R . Since

In other words,

i = 1, 2.

n gi,

R 1 , S, R 2 onto concurrent lines, it follows

is (A., g.)-pappian. l.

l.

This proves Theorem 1.1. REMARK 1.1 . 02 since A .

Ai+l

REMARK 1 . 2.

A (A., g . )-correlation is uniquely determined

l.

A (A., g.)-correlation l.

=

(i

l.

TI

the plane

1, 2, i + 1 taken mod 2).

l.

by Ai' gi, i = 1, 2 . tion in

cannot be a polarity,

l.

Thus for a given (Ai' gi)-configuraTI

can admit at most one (A., g . )l.

l.

correlation. DEFINITION.

A projective plane

TI

is said to be (Al, gl,

g 2 )-pappian if it is (Ai' gi)-pappian for fixed elements A1

,

g

1 ,

g 2 and any point A 2 on h = A 1 P distinct from A 1

and P. We are now able to prove the following characterization of pappian planes: THEOREM 1 . 2 .

PROOF.

A projective plane is pappian if and only

We have only to show that (A 1

23

,

g1

,

g 2 )-pappian

implies that

is pappian.

TI

This will be done in several

steps. STEP 1.

TI

is (A 1 , g 2 )-transitive.

For, take two

distinct points R, Son h, different from P and A 1 • Denote by 6R and 6s the (A 1 , R, g 1 g 2 )-correlations respectively. collineation of

TI

,

g 2 ) - and (A 1 , S, g 1 , -1

Obviously, 6R6S

mapping R onto s.

-1

Moreover 6R6S

every line through A 1 and every point on g 2 is a

is a

(A 1 , g 2 )-homology mapping R onto S.

•

fixes -1

Thus 6R6S

By fixing Rand

varying S over the set of all points on h distinct from A 1 and P we obtain that STEP 2.

TI

is (A 1

,

TI

is (A 1

,

g 2 )-transitive.

h)-transitive.

Take a point R on

h distinct from P and Al and a line s I P, s -I gl, g 2, h. -1

Denote by s the image of s under the inverse 6R (A 1 , R, g 1 , g 2 )-correlation that 6 R

-1

6R •

us 6 R is the (A 1 , R, g 1 J:

,

of the

It is easy to verify s)-correlation of

TI.

By fixings and varying Rover the points of h distinct from A 1 and P, we get (A 1 , X, g 1 , s)-correlations for all points XI h, X-/ A 1 , P. plane

TI

Thus, in view of Step 1, the

is (A 1 , s)-transitive.

The (A 1 , s)-transitivity

and the (A 1 , g 1 )-transitivity together imply that

is

TI

(A 1 , h)-transitive. STEP 3.

TI

is a translation plane with respect to h.

Namely, the collineation 6 2 interchanges A 1 and A 2 implies that

TI

is also (A 2

transitivity and (A 2

,

,

h)-transitive.

(A 1 ,

h)-transitivity imply that

24

This

•

h)TI

is a

translation plane with respect to h. STEP 4.

is the dual of a translation plane with

TI

respect to P.

This follows from the fact that

o

inter-

changes hand P. STEP 5.

In the ternary ring~ coordinatizing

with

TI

respect to the quadrangle O = (o, o), E = (1, 1), V = {00 ) ,

U = (o) by the method of Hall [8], with V = P, VO= g 1 , VE= g 2 , U =A, and A I

-

2

= (a) for any element a £ ~ -

{o}

the following relation is satisfied: (*)

(a - ax)z = az -

for all x, z £f/.

(az)x

For, consider the incident point-line pair Q =

(x, y), g = [az, o]

Then Q 0 = ( [ (x, y),

with y = (az)x.

(o) J n OV)

( [ (x, y),

(a) 1 n EV)

= [a - ax, y] and

g 0 = [((az, o]

n OV),

(a)]

[(az, o] n EV,

n

(o)]

= (z, az) (using the rules given in the proof of Theorem 1.2 (b)) . Since 0 6 I g 0 , it follows that az = (a - ax)z + y = (a - ax)z + (az)x, or (*)

(a - ax) z = az STEP 6.

Jl,

(az)x

for all x, z £

is a commutative field.

TI

Jr.. is a

translation plane with respect to hand the dual of a translation plane with respect to V; all a distributive quasifield. is (A, g )-transitive and since I

2

25

thus~ is first of

Moreover, the plane TI

TI

is a translation plane

with respect to PA 1 , it follows that transitive.

is also (A 1 , g 1 )-

TI

This implies that multiplication is associa-

tive in

J1,

any a£

Jl- {o}.

In view of Step 5, relation (*) is valid for Take a= l; this implies that xz = zx

for all x, z in.,l •

The ternary ring

Ji

is cornrnutati ve.

In other words, we have proved that REMARK 1.3.

TI

is pappian.

As we have shown in Step 1, if

g 2 )-pappian then

TI

TI

is (A 1 , g 2 )-desarguesian.

is (A 1 , g 1 , Using the

same arguments as in Step 1, one can prove that if pappian, then

TI

TI

is

is (A, g)-desarguesian for all non-inci-

dent point-line pairs in TI.

This implies that TI is (X, g)-

desarguesian for all point-line pairs in TI, i.e. TI is desarguesian.

Thus we have a new proof of the well-known

statement: a pappian plane is desarguesian. Our next aim is to give a sufficient condition for the existence of a

(A . , g.)-correlation in a projective 1

1

plane: THEOREM 1.3. plane V. (*)

TI

Let~ be the ternary rinq 1 of a projective

with respect to the points of reference: 0, E, U,

If TI is (V, UV)-transitive and if the relation (a - ax)z = az -

(az)x

is satisfied for all x, z

£Ji,

correlation with A 1 = (o),

A2

then

--

TI

admits a

(A . , g. )1

1. Throughout the whole section the coordinatization method of Hall (8) is used. 26

1

= (a), g 1 = OV, g 2 = EV for

:,f - {o}.

a fixed element a£ PROOF . and by

Denote by

i:

'J

the set of the points (x, y) in

the set of all lines [m, kl in

mapping 6 on the set

Ju

ij

TI.

TI

Define the

as follows:

( x, y) 6 = [ a - ax, y l and

[az, kJ 6 = (z, az + k).

It remains as an exercise to verify that 6 is an isomorphism of the affine plane ~obtained from its points onto the affine plane

'11

TI

by deleting UV with

which is the dual of

the structure obtained from

TI

by deleting P together with

all lines through P.

6

can be extended uniquely to

a correlation

o of

TI.

Hence

It is easy to check that the points

of VO and VE are absolute points of (a) play the role of A 1 and A 2 • COROLLARY.

o;

the points A 1 and

This proves Theorem 1.3.

There exist non-desarguesian projective planes

admitting (A., g.)-correlations. -

PROOF.

1

1

There are known examples of distributive quasi-

fields with commutative multiplication which are not fields {see, for example, Dembowski [7), 5.3).

All of

these satisfy relation {*) with a= 1 and their corresponding projective planes are (V, UV)-transitive .

Thus any

projective plane coordinatized by a distributive quasifield with commutative multiplication admits a correlation.

27

(A., g . )1

1

REMARK 1.4.

It would be interesting to know which classes

of the Lenz classification for projective planes can admit (A., g.)-correlations. i

i

(For the definition of the Lenz

classification see Dembowski [7].)

I shall mention with-

out proof the following result of Herzer ([9], Satz 8): THEOREM 1.4.

If a projective plane

TI

is (X, y)-transitive

for at least one incidence point-line pair (X, y) and if TI

admits an (A., g.)-correlation, then i

i

------

one of the following classes:

--

TI

must belong to

Lenz class II, Lenz class

III, or Lenz class V. REMARK 1.5.

It is not known whether there exist planes

of Lenz class III admitting (A., g.)-correlations. i

i

Planes

of Lenz class II admitting (A., g . )-correlations have been i

i

constructed by Herzer ([9], p. 251).

Distributive quasi-

fields with commutative multiplication provide examples for planes of class V with (A., g.)-correlations. i

I.2.

i

Desarguesian projective spaces

For an integer n

~

3 and a not necessarily commutative

field K denote then-dimensional left vector space over by V(n,K).

Let the projective space isomorphic to the

lattice~ of the subspaces of V(n,K) be denoted by E. A correlation of Eis an anti-automorphism of$.

A sub-

space A of Eis called totally isotropic, isotropic, or non-isotropic with respect to a correlation

28

of E if and

only if /1. n /1. 0 ively.

=

/1., or /1. n /1. 0 -:f .

Obviously,.,f-parallelism is an equi-

valence relation in T;t·

Moreover, for any

Q

E

T.,f and any

point X s.;t there is exactly one line through X which is ,:,f--parallel to a... We are now able to define two configurational propositions (D) and (P) for regular spreads. DEFINITION.

Let /f.

regular spread

!J7

9, ':c be

three distinct elements of a

and let A., B . , C., i = 1, 2, be points l.

l.

l.

of the projective space E such that Ai s.;f, Bi C.

l.

E

ff.

and

We say that A . , B . , C. satisfy the configurational l.

l.

l.

proposition (D) if and only if A 1 B 1

DEFINITION.

A 2B 2 and A 1 C 1

1I

.,t

A 2C 2

g:, if and only if (D) is true for any

three distinct elements ,,f, choice of Ai

Ed,

DEFINITION.

Let,A,

ff

1I

vt

The configurational proposition (D) is uni-

versally valid in

spread

sjJ,

Bi

E

1J,

$,

g'.

E

Take

Let O and P be

of points of L not contained

nX

X ➔ ,Q,P

where ,Q,P is the

JI-

parallel to OX through the point P and X the element of ff through

x.

It is easy to verify that

Jonte itself.

Our aim is to show that

extended to a collineation the following:

6OP

cOP

of L.

is a 1-1 map of

6OP

can be uniquely

For this we observe

(1) Points of jJ on a transversal of any

J)E 9'

are mapped onto points of a transversal in view of

(D) .

(2) Points of 'fi on a line h

mapped onto collinear points.

.S

E

.i E

fP are

This follows from the fact

that h can be embedded into a plane any element of !f'; in view of (1) 34

for any

TT

not contained in

C TT

OP is a plane.

(3)

Co-

planar points of I can be embedded into three-dimensional spaces not contained in any element of9'; hence coplanar points of I are mapped onto coplanar points.

Thus COP

can be extended to a map c OP of the point setP of I into itself in such a way that c Op is a collineation of L cOP fixes every element of J' .

Clearly,

Denote by b. the

set of all collineations cOX for a fixed point O £~,t and all points X

£;f.

The set b. is a group acting as a

transitive permutation group on the point set of every

jj £!/'. Conversely, if b. is a collineation group of I with the required properties, then for any :iJ -transversal g of ~and any

d£

b.,

the points ofJ . LEMMA 2. 2.

Let

gc

11

,l

g.

The group b. is transitive on

Hence (D) is universally valid in J'.

fl, J be two distinct elements of the

regular spread~ and let g be a line intersecting both

A

Define the map og on the point set of .,f;-and ,I as C og .R, follows: X = X,g n j for any X £vi- and X g = .R, X,g nv"f;-

and,B.

for any X through X.

£

2.

Here .R.X ,g denotes the If-parallel tog

Proposition (P) is universally valid in~ if

and only if for any two distinct elements v4 , 13 of 9' and any three lines a , b , c intersecting A and J3 the following relation is true:

(*) PROOF.

Suppose that (P) holds. 35

If any two of the lines

a, b, c coincide, then (*) is satisfied, since 6 2 = 6 2 = a b 0 C2 = 1.

Otherwise take any point X

£

rJt

and consider the

6aobocoa oaob configuration Al = X, A 2 = X , A3 = X , Bl 0 o obo c o a ob oaoboc X , B 2 = X a, and B3 = X a Because of (P) 6 (6a6b6c) 2 B C = X i t follows that X Thus Al 3 (oaoboc)

= 1.

2

Conversely, by assuming that (*) holds for the above configuration A 1 , A 2 A 1 B 3 I IA 3B 1 , i.e.

,

B2

,

B 3 , we deduce that

(P) is satisfied.

universally valid.

Let.,t,

g a fixed line meeting

Ji}

,

Let y> be a regular spread in which (P) is

LEMMA 2 . 3.

and

A 3 , B1

/t

JJ

and

be two distinct elements of[?,

1J

and h any line of L meeting

Then the set 6 g = {o g • oh induces on the projective space

for all h meeting

tit

A

an abelian col-

lineation group transitive on the points of,4. PROOF.

Leth and h' be two arbitrary lines intersecting

d and iJ • ~

= (oh,)

-1

According to Lemma 2. 2, oh 6g oh, = ( oh 6g 6h, ) -1

-1

og oh= 6h,6goh.

Hence (ogoh) (ogoh,) =

6g(6h6goh,) = og(oh,ogoh) = (ogoh,) (ogoh). 6

g

-1

Therefore

induces a commutative permutation group on the point

set of#. Let X and Y be two arbitrary points of.It. The 0 points X g and Y determine a unique line h such that Y o oh = X g hence 6g is transitive on the points ofc/t. From the definition of og and oh it follows that 6g induces a collineation group onvf-, which proves the lemma. 36

II.3. Main results We are now able to prove the following theorems: THEOREM 2.1.

A regular spread 9' is desarguesian if and

only i f (D) is universally valid in ~. THEOREM 2. 2.

y

A spread g, is pappian if and only i f

is

regular and (P) is universally valid in!J'. THEOREM 2.3. Then

J'

Let

J' be a spread over a field K with I.Kl> 2.

is a Moufanq spread which is not desarguesian if and

only if .!I' is regular and (D) is not universal in g,. For the proofs of these results some well-known properties of spreads are needed (see Andre (2) and Bruck and Bose (4),

[SJ):

PROPERTY 1 (Bruck-Bose (5), p. 155).

E.

Let j?be a spread of

Denote by V the vector space over){ from which Eis

obtained .

For any element

ponding subspace of V.

X

£

ff' denote by

X:

the corres-

Take an ordered triple;!,

:B,

~ of

distinct elements of!/; there is a unique linear trans-+-

-+-

-

formation a+ a' of ,A, into formation ; + ; + ; ' maps element

X of

1

-

;;t,

such that the linear trans-

into

1/.

The vectors of any

the space corresponding to

expressed in the form ;

Y'£ +

a'

where ;

;r £ Y' - .Jt £

can be

,;,f, ; ' is the

corresponding vector of 1j defined above, and

$Pf is a

unique linear transformation of ~ into itself determined

37

f'.

by Ci:

In particular~-= 0 and

:B

of the linear transformations

the properties:

b £ cit and

9'-

X

E

Ci:

;

'f _ -

r

'f _

JI

r-g, = Cf_ X

I.

The collection

for all X

E

Ef' -

vt has

'J:

i f a,

-

is non-singular i f X-:/

+

-:/ ; , then there exists exactly one element

ar_X

+

such that

+b.

PROPERTY 2 (Bruck-Bose (5], p. 158) .

Let E' be a pro-

jective space containing E as a hyperplane and let TT be the translation plane obtained from E', E, and Y as described in II.l.

Then there is a ternary ring of TT iso-

morphic to the algebraic structure~(+, ·) defined as follows:

( 1) the elements of _$( +, •) are the vectors of +

(2) "+" is the vector addition; non-zero vector of +

,j;

(3) let e be a fixed

then for any two vectors:,

+

the product a · bis the vector +

a 7+ b +

transformation of Ci: mapping e onto b. the role of the identity.

where

f+

b

b Ed-

is the +

Clearly e plays

,I(+, ·) is a quasifield.

PROPERTY 3 (Bruck-Bose [5], p. 159).

If Ci: is closed under

multiplication, then (€, •) is a group with zero isomorphic to

/t( · ) .

PROPERTY 4 (Herzer (11]). ~+, •).

Xis contained in the kernel of

Moreover, J< is contained in the centre of.if(+,·)

if and only if ff is a regular spread.

38

PROPERTY 5 {Andre [2)).

Let

Q

zing a translation plane

TT;

then the multiplicative group

of the kernel of homologies of

TT

Q

be a quasifield coordinati-

is isomorphic to the group of all

with a given affine centre and the im-

proper line as axis. We shall now indicate the proofs of Theorem 2.1, 2.2, and 2.3. PROOF OF THEOREM 2.1.

Suppose that {D) is universally

valid in a regular spread.9'.

Then, according to Lemma 2.1

the space I admits the collineation group 6.

I

+

+

~

+ + = < {x, y)>c.

6* imply the existence

+

+

➔

+

of a homology mapping onto . Hence

➔

Let

{Vis represented in the form ;it$;/-).

TT

is {R, t 00 )-transitive, i.e. 39

TT

is desarguesian

(see Pickert [14)). Conversely, if n is desarguesian, then n is (R, i 00 ) Hence I admits a collineation group 6 fixing

transitive.

all elements of !.f and acting transitively on the points of each element in

J'

(because of 6 "X*IJof a

42

derivable projective plane

TI

is an affine hyperplane of a

three-dimensional affine space 'T contained in TI 0 TI

is desarguesian.

fixing

TI 0

Hence

TI.

Moreover any Baer collineation of This

pointwise induces a collineation ofr.

gives the following information about the group of all Baer collineations of

TI

fixing

TI 0

IT(TI 0 )

pointwise:

IT(TI 0 )

is a subgroup of the group of all perspectivities of

with axis

TI 0

contained in

'T

and centres on an improper line of .7'not TI 0 •

Our investigations are carried out on the dual planes of derivable planes; the Baer collineations and Baer subplanes belonging to a "dual derivation set" in these planes have the above-mentioned properties of III.2.

IT(TI 0 ) .

Main results

Throughout this section plane,

and

TI 0

r:J

TI

denotes a derivable projective

a derivation set in

TI,

and

TI

0

a Baer subplane

belonging toot>. Let us denote by

TI

the plane dual to

of/) i n ; is a set.;c) of lines in

TI

TI.

The dual

through a point L

with the property that for any two distinct lines a, b of

A

TI

not through L for which the intersection of a and b A

belongs to a line ofoiJ there is a Baer subplane ing a, b, and L such that the lines of A

exactly the lines ofdl>. mined by a, b, andJ>.

Obviously

;o

A

TI 0

A

TI 0

contain-

through Lare

is uniquely deter-

We say that a Baer subplane of; be-

43

longs to

A

,;I)

if and only if it contains the lines of

us denote by~ the class of all Baer subplanes of

A

al). TI

Let

belong-

A

ing to~.

Take a line g

tobtained from

TI

and consider the affine plane

E ~

by deleting g together with its points.

Since any Baer subplane

A

TI 0

of

A

TI

belonging to ,;3 contains

g, the affine points and lines of tJ!. contained in an affine subplane d{ 0 of t7l obtained from g together with its points in

TI 0 •

A

TI O

A

TI 0

form

by deleting

Denote by~ the class

of all affine Baer subplanes of t:l;obtained in the above way from the subplanes of.$ • The affine points of any plane of $'are points on the A

lines of - {g}.

Let D be an affine point

of Cl on a line k of :f> - {g}, different from h. by Pt, 1 the plane of~ through A, B, D.

Denote

Take an arbitrary

point X on g which is not an improper point of (l 1 • ~ 1

is a Baer subplane of rJ{,, the correspondence

9':

Since Y

XY n h where Y runs over all affine points of (l 1 is a

46

+

one-one mapping of the affine points of t( 1 onto the affine points of h.

In order to verify that is an affine

plane whose lines are blocks of .7'it remains to show that the blocks of Yon the line hare the images of the affine lines of

a

1

points of h.

r•

under

Let M, N be two arbitrary distinct

Denote by Uthe unique point of XM in

by V the unique point of XN in ot,1 •

a

1 ,

and

If U =Mand V = N

then clearly MN is the image of the affine line UV of cl1 under

f·

Suppose U

~

M.

Then U, V, Mis a triangle, and

according to (1) there is a unique plane t1[ 2 of it.

Let

A

7T 2

u,

V the plane

A

7T

2

contains the points X

and L, thus it contains the point N the line UV of ;

r•

tl1

containing

be the plane of jj from which tJ[ 2 was obtained.

Together with M,

7T 2 •

~

XV n ML.

Moreover

is projected from X onto the line MN of

2

Thus the block MN is the image of the block UV under Conversely, by the same argument every affine line of is mapped under

f

onto a block of ✓ on h.

This finishes

the proof of (3).

:r contains

two types of planes:

planes of type 1B

which are the Baer subplanes o f f and planes of type L which are the lines structure

J

J> - {g}.

According to Lenz [ 12] the

is an affine space if the following properties

are satisfied:

(i) between the blocks an equivalence

relation called parallelism is defined;

(ii)

for any

point-block pair (P, b) there exists a unique block b'

47

incident with P and parallel to b;

(iii) for any four

distinct points, A, B, C, D such that AB is parallel to CD and for any point P € AC, either P s CD or AB and PD have a point in common.

We shall define parallelism as

follows: Two distinct blocks of :rare parallel if and only if

(IP)

they are disjoint sets of points belonging to a common plane of 7'.

Each block is parallel to itself.

In view of (3) all planes of ✓ are affine planes; properties (ii) and (iii)are satisfied .

hence

It remains to prove

that parallelism (~) is an equivalence relation.

Clearly,

(~) is reflexive and symmetric; transitivity has to verified.

Thus we have to show that: If b

(4)

1 ,

b2

,

and b 3 are three blocks of

.r

such that b 1

is parallel to b 2 and b 2 is parallel to b 3 , then b 1 is parallel to b 3 • For the proof of (4) the following cases can be distinguished : CASE 1 . of

Y.

b1, b2

,

b 3 are contained in a common plane

Since the planes of :T'are affine planes,

(4) follows

immediately . CASE 2.

m,

b 1 and b 2 are contained in a plane of type

and b 2 and b 3 are contained in a plane of type L.

note the plane through b

1

and b 2 by

48

Cl1 and the plane

De-

through b 2 and b points of ~

2 •

/12 •

by

3

Let 2 be the line carrying the

Take two arbitrary points X, Y of b 1

and an arbitrary point Z of b 3 •

In view of (1) the points

X, Y, Z determine a unique subplane ~ 3 off.

(l 3

The plane

intersects 2 in a block b; through Z having no point In the affine plane 9;2 there is exact-

in common with b 2 •

ly one block through Z containing no point of b 2 the block b 3 •

Thus b

3

,

namely

and b; coincide, that is b 1 and

b 3 are parallel lines of

d/ 3 •

Hence b 1 and b 3 are parallel

in./. CASE 3.

b 1 and b 2 are contained in a plane tt' 1 of

type Band b 2 and

b

3

also belong to a plane carrying b 1 respectively. subplane from

,

b2

,

b 3 by 2 1

,

22

,

23

Since any line of~ contains lines of any

r,

there is a line b, of

Pll on

of (1) the blocks b 3 and b 4 are parallel in b, is parallel to b 1 •

23.

In view

J; obviously,

From the investigations in Case 2

it follows that b 1 and b 3 are parallel blocks of.J': This proves (4). Thus (~) is an equivalence relation, which implies: (5)

:r

is an affine space; the affine lines of :J-- are the

blocks of j'; the affine planes of .J"' are Baer subplanes of

i

and the affine line of 1ft, belonging to£.

Consider an affine plane

£-

{g}.

,Z1 of j represented by a line of

Take a point P of .7'in

49

tt 1

and two distinct blocks

b1

,

b 2 in {l{_ 1 through P 1 •

For any other affine plane,~\

of ✓ represented by a line of ~ - { g} and for any point P 2 of~ there exist unique blocks b 3 and b 4 through P 2 parallel to b 1 and b 2 respectively. and ,!£ 2 are parallel in J.

Hence the planes

£

1

On the other hand any affine

plane of :T represented by a Baer subplane of ~: in a block, that is in an affine line of

g, intersects

r.

This

implies that (6)

J'"'is a three-dimensional affine space. We have completed the proof of the following theorem:

THEOREM 3.1.

Let

jective plane

TI

.J) in

TI.

g

TI

be the dual plane of a derivable pro-

and let cl)" be the dual of a derivation set

O[ is the affine plane obtained from

A

TI

by delet-

ing a line g s ~ together with its points, then the affine points of~ on the lines of sional affine spacer.

io -

{g} form a three-dimen-

The affine lines of Jare the

affine lines of those Baer subplanes C(i of dl, which are obtained from the Baer subplanes of deleting g with its points. affine plane of

A

TI

A

belonging to ;lJ after

Any such subplane fZ. is an l

---

::r'.

As a corollary we obtain the following statement: THEOREM 3.2.

If

jective plane

TI

TI

is the dual plane of a derivable pro-

and if

ftJ

is the dual of a derivation set

50

J) of 71 in 71, then any Baer subplane of 71 belonging to ,:J:; A

is desarguesian. Our next aim is to consider Baer collineations of 71 . A Baer collineation of a projective plane is a collineation fixing pointwise a Baer subplane of the plane.

Clearly,

the Baer collineations fixing pointwise a given Baer subplane form a group . Let ; , 3.1.

£,

g,

ct,

J"', and O[ . be defined as in Theorem l.

Denote by IT(Q . ) the group of the Baer collineations l

A

fixing pointwise a Baer subplane ~ - belonging to ;!J. l.

Any

element a of IT (i . ) fixes the lines of (Jl . ; hence a fixes l

A

all lines ofo!J. in

J.

~

It follows that a induces a collineation

Since {l . is a hyperplane of l.

:T the collineation

a

induced by a in ✓ is a perspectivity of ✓with axis ~i . A

The collineation a fixes all lines of~; of

a must

hence the centre

lie on the improper line of ✓ common to all A

affine planes&'. of 7represented by the lines ofci)- {g}. l.

In other words we have the following result: THEOREM 3.3. Let 71 be the dual plane of a derivable proA

jective plane 71 and let 71 0 be a Baer subplane of 71 belonging to the dual of a derivation set of 71 contained in 71. If IT(71 0 ) denotes the group of all Baer collineations of 71 fixing

;o

pointwise, then IT(; 0 ) is isomorphic to a sub-

group of the group rx,$'('of all perspectivities in a threedimensional projective space ':f with a common axis

51

-~

isomorphic to contained in

TI 0

and with centres on a line i of 9' not

£.

The investigations carried out for the duals derivable planes

TI

can be dualized step by step.

TI

of

If G

denotes an arbitrary but fixed point in a derivation set d)of

TI,

of

through the points of ;JJ - { G} which are different

TI

then it is not difficult to show that the lines

from the line containing ol)are the points of a threedimensional affine space

:r. A

By completing j t o the corresponding projective spacer it follows that the Baer subplanes of to ,:,l) are hyperplanes of r.

TI

belonging

Thus the statements of Theo-

rems 3 . 2 and 3 . 3 can be formulated for derivable planes: THEOREM 3.4.

Any Baer subplane belonging to a derivation

set of a derivable projective plane is desarguesian. THEOREM 3.5.

Let IT(TI 0 ) be the group of the Baer col-

lineations of a derivable projective plane

TI

fixing point-

wise a Baer subplane TI 0 which belongs to a derivation set of TI. group

Then IT(TI 0 ) is isomorphic to a subgroup of the

rx,~

of the perspectivities in a three-dimensional

projective space L with a common axis:J( isomorphic to

TI 0

and with centres X on a line i of L not contained in~.

52

REFERENCES

1. 2.

3. 4. 5. 6.

7. 8.

9. 10.

11. 12. 13. 14. 15. 16.

s.A. Amitsur . Rational identities and applications to algebra and geometry. J. Algebra 3 (1966): 304359. .. . J. Andre. Uber nicht-desarguessche Ebenen rnit transitiver Translationsgruppe. Math. Z. 60 (1954): 156-186. R. Baer. Homogeneity of projective planes. Amer. J. Math. 64 (1942): 137-152. R.H. Bruck and R.C. Bose. The construction of translation planes from projective spaces. J. Algebra 1 (1964): 85-102. - - Linear representations of projective planes in projective spaces. J. Algebra 4 (1966): 117-172. J. Cofrnan. Baer subplanes of affine and projective planes. Math. Z. 126 (1972): 339-344. P. Dembowski. Finite Geometries. New York: Springer (1968) . M. Hall. Projective planes. Trans. Arner. Math. Soc. 54 (1943): 229-277. A. Herzer. Dualitaten rnit zwei Geraden aus absoluten Punkten in projektiven Ebenen. Math. Z. 129 (1972): 235-257. Ableitung Zweier selbstdualer, zurn Satz von Pappas aquivalenter SchlieSungssatze aus der Konstruktion von Dualitaten. Geom. Dedicata 2 (1973): 283-310. Charakterisierung verschiedener regularer Faserungen durch Schliesungssatze. To appear in Archiv der Mathematik. H. Lenz. Zur Begrilndung der analytischen Geornetrie. Sitz-Ber. Bayer. Akad. Wiss. (1954): 17-72. Vorlesungen ilber projektive Geornetrie. Akad. Verl. Ges. Geest und Portig, Leipzig (1965). T.G. Ostrom. Semi-translation planes. Trans. Amer. Math . Soc. 111 (1964): 1-18. G. Pickert. Projektive Ebenen. Berlin-GottingenHeidelberg: Springer (1955). o. Prohaska. Endliche ableitbare affine Ebenen. Geom. Dedicata 1 (1972): 6-17.

53

UNITARY PLANES Martin Gotzky

In the following the intention is to develop unitary geometry of the plane, with the help of quasisymmetries. At the heart of the considerations is the theorem of the third quasireflection (see Gotzky, 1973).

Part I

deals with unitary-euclidean geometry, and part II with unitary-minkowskian geometry (see Gotzky, 1970). I.

GENERALIZED EUCLIDEAN GEOMETRY

Let P be a projective plane and Sa set of perspective collineations

i 1.

If a is a straight line of P, let

S(a) be the set of elements of S with axis a .

Assume

the following property: (*l)

{l} u S(a) is a group for every straight line a.

Obviously {S(a); a is a line of P} is a partition of s. Let G

= be the group generated bys.

We are

interested in generated groups (G, S) with (*l), for which the following statement holds:

54

THEOREM OF THE THIRD QUASIREFLECTION.

Let a, b, c be

straight lines through a given point O with a f b. a€ S(a), S € S(b), and caS f c.

Let

Then there exists a

y € S(c) and a straight lined passing through Osuch that aSy € S(d). Fixing a straight line g of P we may consider the affine plane

A= Pg instead of P.

If S 0 is the set of

line-reflections of A and A a pappian plane with Fano axiom, then Veblen (Veblen and Young, 1918, vol. 2, §52) has already stated this theorem for (, S 0 ) .

Only

(*1) does not hold. There are many examples of generated groups (G, S) with (*l) for which the theorem of the third quasireflection holds.

For instance let S 2 be the set of all axial

collineations of with S 1 the set of all shears and line-reflections of an affine Moufang plane Fano axiom.

A=

P with g

Then (G, S . ) for i = 1, 2 is a generated 1

group with (*1) holding.

The theorem of the third

quasireflection holds for

(G,

arguesian and for

S) exactly

i f A is des-

(G, s 1) exactly i f A is pappian

(Gotzky, 1972). For the case (G, s 2) '

let a € S (a),

s

€ S (b), and

y € S(c) with a, b, c and O, P, Q as shown in figure 1. Further let Q

= QaBy.

Then P

PaSy for every point P

on the connecting line P of O and Q if and only if aSy € S.

55

FIGURE 1

This shows that quasireflection is closely related to Desargues' theorem. Now let V = Vn{K, f, 1) be an n-dimensional left vector space where K is a skew field of char. f 2, 1 is an antiautomorphism of K with 1 2

= 1, and f is an 1-hermitian

form, that is a function mapping each pair a, b of vectors onto an element of Kand satisfying f{sa, b)

=

s f{a, b)

f{a' + a." , b) = f

(a

It

if

s EK,

b) + f {a", b),

f(a, b) 1 = f{b, a).

(If 1

1, then K is commutative and f a bilinear form,

hence V is a metric (orthogonal) vector space.) The set of all linear mappings = f(a,

7T

satisfying f{a

b) is a group called the unitary group U{V).

7T

I

b n)

The

subset of all linear mappings n of U(V) fixing each element of rad V := {x EV; f{x, V) = O} is also a group called the restricted unitary group U*{V).

56

Let a I rad V ands EK\ {O}, then ( 1)

xcr ( s, a)

f ( x, a) • s • a

:= x -

defines a linear map cr(s, a) fixing the hyperplane {y;

f(y, a)

= O} vectorwise.

Also, cr(s, a) E U*(V) if

and only if

(2)

S

-I

l

+

S

-I

Let S(Ka) S (V)

=

f (a, a). := {cr(s, a); s- 1 1 + s- 1 = f(a, a)} and S

u

a/rad V

(Ka) .

Obviously S(Ka) u {l} is a group. THEOREM.

Consider the generated restricted unitary group

(U*(V), S(V)).

Then, for the one-dimensional subspaces

of V not contained in rad V, we have:

if Ka~ Kb and

Kc~ Ka+ Kb and also a E S(Ka), BE S(Kb), and (Kc)aS Kc, then there exist a y E S(Kc) and a subspace Kd

+ Kb with a Sy E S (Kd)

~

~

Ka

(Gotzky, 1964, 1965) .

Let n = 3 and P = PV be the desarguesian projective plane with one-dimensional subspaces of Vas straight lines, two-dimensional subspaces as points, and incidence defined by inclusion.

Then, for P,

(PU*(V), PS(V)) is

a generated group satisfying (*1) and by the theorem also satisfying the theorem of the third reflection. REMARK.

If S contains only reflections, the theorem of

the third quasireflection for (, S) becomes the theorem of the three reflections (see Bachmann, 1959 or 1973).

57

P = PV is called a unitary plane. f(x, y)

=

L

Ky if

0, an orthogonality relation for these planes

is defined. plane.

By Kx

If dim rad V = 1, Prad Vis a unitary affine

In this case we say

Prad Vis unitary-euclidean if index f =

o,

Prad Vis unitary-minkowskian if index f = 1. REMARK.

If index f = 0 for (PU*(V), PS(V)), the theorem

of the third quasireflection holds even if we delete the assumption ca8

1

c; we say the stronger theorem of the

third reflection holds.

Also the theorem of the third

quasireflection keeps holding if ca8; c is replaced by

ct

c.

In this case we say that the varied theorem of

the third qusireflection holds. Next we concentrate on affine translation planes

A= P, with a point involution jong which defines an g affine orthogonality relation by a Lb if the intersecting points of a with g and b with g correspond under j (see figure 2).

FIGURE 2

58

To avoid distinction between cases we restrict ourselves to cases with j free of fixed points.

Thus there

are no self-orthogonal straight lines. Assume a system S with (*1) such that: (*2)

S(a) contains an involution for all straight lines a. Obviously then there is a

unique reflection on every

straight line a of Pg and Char Pg~ 2; that is, the Fano axiom holds. We assume that the stronger theorem of the third quasireflection holds for

(G, S) with G = and assume

furthermore that (*3)

S contains all perspective collineations of G.

(*4)

G preserves the orthogonality induced by j. The question is whether A is unitary or not (if

unitary it is unitary-euclidean).

For a better under-

standing of this question, I shall prove some little statements.

(For the following discussions if

of the same group, we define

sY

:=

S, y are elements

y- 1 Sy.)

LEMMA 1.

The point reflections are in G.

LEMMA 2.

Suppose a

E S(a), SE S(b), and a II b.

Then

aS

is a translation if and only if there is a point reflection C such that

aS C = 1.

59

PROOF.

a

- 1

a

e = e a e is a translation.

Let aS be a transThen

lation and ea point reflection with be= a. aSe· (S- 1 )e = aS where (S lation. LEMMA 3.

-1

e

) Sand aS are translations, so aS

e e Moreover, as E S(a), so as =

e

is a trans-

1.

Let a E S(a), s E S(b) with ola ~ s1O.

Then

there is a o E S(d) for all d such that oaS is a glidequasireflection. REMARK.

If a ES u {l} then a is called a quasireflection.

If Tis a translation and a ES

u

{l}, then Ta is called

a glide-quasireflection. PROOF.

Let e be a point reflection such that de IO. C

There is a o E S(d) such that y := o aS Es.

If y E S(c)

then c IO (theorem of the third quasireflection).

Now

oaS = o(o- 1 )cy is a glide-quasireflection. DEFINITION 4. points XY, then

If TIE G and MXY is the midpoint of the HTI:

X ➔

MXY defines a point map called

the midpoint map of TI. If a midpoint map H preserves collinearity, TI then it is a collineation or else there is a line passing

LEMMA 5.

through all the points of image Hn. PROOF.

See Gotzky (1972).

60

LEMMA 6.

Ha preserves collinearity for all a Es.

From now on, we shall not distinguish between a point and its point reflection. PROOF OF LEMMA 6. C

-f a.

Let a E S (a), 0 I a, and O I

C

I p but

We show that, regardless of how P was chosen,

there is a line m passing through 0 such that P 8 a is on m. Consider the identity O•POPa = (PPaO) 0 = PaoP.

Let

b La and b IO such that O is the product of the line reflections in a and b. and SE S(b).

Then aO

=

a'S for some a' E S(a)

By the theorem of the third quasireflection,

there is a y E S(c) such that y := ya'S ES. 3.)

Obviously Pao= P 0 , so O•POPa

= P 0 P.

(See figure

Leto E S(d).

Then dis a line through O perpendicular to the connecting line of O and POPa.

Since d does not depend on the

choice of Pon c, the connecting line also does not.

FIGURE 3

61

If

this connecting line is called m, the lemma is established. If w E G and Tis a translation, then H WT

LEMMA 7. PROOF. p

=

H H . WT

Using -1

H

MPPWT

p T

MppwH T

The uniqueness of the midpoint yields H ) T = (M M PPWT PPw 11

p

11

HwT

H H (P W) T

H H p WT

This proves the lemma. LEMMA 8.

If Tis a translation and if w ES, then HTw

and H preserve collinearity. WT PROOF .

Since S contains all the axial transformations by

(*3), it follows that TWT H

TWT

-1

•T

-1

ES.

Furthermore, we have

Hence we only have to prove that HwT

preserves collinearity.

But H WT

preserve collinearity. LEMMA 9. PROOF.

If a, SES then HaS preserves collinearity. Let a E S (a) and S E S (b).

If a

11

b then

aS ES u {T; T translation}, hence aS preserves collinearity.

Thus we may assume that a f band a, b IO for some point 0.

62

There is a Y

Let c be a line.

is a glide-reflection (Lemma 3).

H H P aS = P yaB for all P I c.

€

S(c) such that yaS

Now Pas= PyaS and hence

Since HyaB preserves collinearity, then HaB also has to preserve collinearity. LEMMA 10.

Let O, P, Q be non-collinear points.

exists an axial collineation

o

o0

such that

=

o

There and P 0 = Q.

Let O, P, Q be non-collinear and a, b, c, R be as

PROOF.

shown in figure 4.

FIGURE 4 (1)

Assume ls(x)

Choose a'

E

I

> 1 for all lines x.

S(a), B"

E

S(b), and y

E

S(c) such that

y is an involution but a' and B' are not. y -1 Since a' := y a•y is a non-involution, Lemmas 5 and 6 together yield that H Y is a collineation with axis a' aY. It is easy to show that Lemmas 5 and 9 together yield that H,

a Y

is a collination (use the fact that

63

at

C).

a'y Now X all XI c.

= Xa'Y

Ha•y for all XI c, so X

Therefore y•

:= H, H- 1 Y fixes every point of

a ya' c; that is, y' is axial with axis c

s (c)).

=

(it might not be in

Similarly, y" := Ha" H- 1 y is an axial collineation µ y B"

= HY with axis C. Because H a' a'Y we obtain

= y,

p(y'y)

Ha'

•y

P I a HS"

R = Q

y

=

Q

y II y H B"y

Q

Q I

(y"y)

Ha" µ

I

·y

b

Using RI c, we are led to

= Since (y'y)

Ha,

Q

(y"y)

Ha" µ

and (y"y)

Ha" µ

are obviously axial and

both have the same direction PQ, the product o := (y'y)Ha'[(y"y/B"rl is also axial.

This

o

satisfies o 0

=

O and p 0

Q, as

required. (2)

Assume IS (all

a~ b ~ c ~ a.

=

1 for one a I

o.

Let b, c I O,

Suppose, if possible, that IS(c) I > 1 and

choose a e S(a), Be S(b), and distinct y, y" e S(c). The theorem of the third quasireflection yields that o

=

yBa and o'

=

y'Ba are both in S.

to the sequence of inferences

o'o- 1

But this would lead =

o•o- 1

e S(c) u {l};

y, y', o, o' e S(c); hence Bae S(c), which is impossible.

64

So the assumption ls(c) I > 1 was incorrect . that IS(c) I= 1 for all lines c.

This proves

But in this case the

theorem of the third quasireflection yields the theorem Hence A is pappian (see Bach-

of the three reflections.

mann, 1959 or 1973) and Lemma 10 holds. THEOREM. PROOF.

Pis a Moufang plane. It is sufficient to show that for every pair

a.++ b in A = P there is a line reflection with axis a in g the direction of b (Pickert, 1955).

Since there are line

reflections in A it is therefore sufficient to show that

Au~(A) is transitive on the non-parallel pairs of lines of A. Thus we want to map a

-tf-

b onto c -}t d.

Since it is

a translation plane, we may assume that a, b, c, d IO for a point 0.

Using Lemma 10 we furthermore may assume

that a= c.

Fix e, P,

Q as in figure 5

there is an m and aµ

E

(c

II

d).

By Lemma 10,

S(m) such that Oµ = a and Pµ = Q;

hence aµ= a= c and bµ = d.

FIGURE 5

65

This proves the theorem.

Whether the plane A is unitary under the given conditions now depends on the existence of both a nondesarguesian affine Moufang plane

A=

P and a given pointg

involution jong that is free of fixed points, which together admit a generated collineation group (G, S) satisfying (*1) to (*4) and the theorem of the third quasireflection.

It is still unknown whether such planes

exist. In the next section, unitary-minkowskian planes

A=

P are investigated. g

Using an additional condition

denoted there as A.6, a full characterization of those planes is given. II. II.l.

UNITARY-MINKOWSKIAN PLANES Axioms

The purpose of this section is to introduce the group plane of a given generated group (G, S) with S the set of generators of G, and further to present a set of axioms for the elements of s which ensures that the group plane has certain desired properties. BASIC ASSUMPTION. l

JS

(1)

Let (G, S) be a generated group with

such that the following properties hold: There is a partition SG = {s

(x);

x

E

G} of S which

is invariant under inner automorphisms.

(2)

S(x) u {l} is a group for all x

66

E

G.

(3)

For all

X

E G such that S(x)

u { l}

is non-abelian,

the set S(x) contains an involution.

(4)

If a 'I band B E S (b), then S(a)B

=

S(b) yields

S(b) £: CG(S(a) u { l}) • -1 (Remember that S(a) 8 : = B s (a) B.)

DEFINITION 1.1 .

P := {a$; a, 8 ES and a, B, aB involu-

tory}. Suppose S(a) and S(b) are elements of the partition

P; and consider the following properties :

SG and A

E

(a)

A

CG(S(b) u {l}).

(b)

A /

(c)

S(a) u {l} s CG(S(b) u {l}).

(d)

If S(a) or S(b) contains an involution, then a 'I b.

E

S (b).

DEFINITION 1.2. a

I

A

Ib

precisely when (a) and (b) hold.

b precisely when (c) and (d) hold. The triplet (P, G, 1) with Pa point set, Ga line

set and

I

as both an incidence relation and an orthogonal-

ity relation is now an incidence structure with orthogonality relation between lines .

This incidence structure is

called the group plane E(G, SG) . From now on let (G, S) always satisfy the basic assumption .

We are interested in the following set of

axioms:

67

P, there is a c E G such that c I A, B.

A.l.

I f A, B E

A.2.

I f a, b E G, either there is a c E G such that

cl a, b or there is a C E P such that C A.3.

b.

I f A, B E Panda, b E G and A, B I a, b, then A

or a A.4.

I a,

=

I a,

B

b.

Let a, b,

o

=

C

b, c.

E G and O E P such that a :/ b and Further let a E S(a) and 13 E S (b).

If

S(c) contains an involution, then there exist a y E S(c) and ad IO such that A.5.

o

:= al3y E S(d).

Let a E S(a), 13 E S(b), and a, b la where a, b, dEG. Then one and only one of the following conditions holds

A.6.

(a)

al3ES.

(b)

There is a CE P such that al3c

= 1.

Let a, b, c E G and DE P such that a, b, c ID. If neither S(a) nor S(c) contains an involution and if further a :f b :/ c, then there is a 13 E S(b) u {l} such that S(a) 13

Ex. A. A

S(c).

There exist A E P and b E G such that b I b but

I b.

(A+ b means that A I b does not hold.

a+ b means that

a jb does not hold.) THEOREM 1 . 3.

The group plane E = E(G, SG) = (P, G,

j)

is a Fano translation-plane (that is, a translation plane satisfying the Fano axiom) with an orthogonality relation definable by a point involution j on the line of infinity.

68

The involution j has at least four fixed points. For all w longing tow.

G, let

E

wbe

the inner automorphism be-

Then (G, S) is a generated group, withs

a set of axial collineations of E, and the following properties are satisfied: (*l)

S(a)

(*3)

Each axial collineation of Gisin s.

(*4)

G preserves the orthogonality of E.

(*5)

If a

u

E

{l} is a group for every a

G satisfies a

group of shears with axis a.

I a,

If a

E

G.

then S(a) u {l} is a E

G satisfies a .}a,

then S(a) u {l} is a group of homologies with axis a containing an involution. Theorem 1.3 holds for generated groups (G, S) satisfying the basic assumption, A.l to A.6, and Ex. A (see Gotzky, 1970).

The proof needs a large number of element-

ary properties of the partition SG, which follow from the assumptions.

It includes a proof of the existence and

uniqueness of the perpendicular, and also a proof of the fact that a quadrilateral with three right angles is a rectangle. From now on, let (G, S) always satisfy the basic assumption along with A.l to A.6 and Ex. A so that, in particular, Theorem 1.3 holds. REMARKS.

Axiom A.4 yields the varied theorem of the

third quasireflection for the plane E.

69

By A.6, the group

S(a) u {l} acts transitively on the set of self-orthogonal lines different from a which pass through a given point 0 on a. Reduction theorem

II.2.

Each element of G is a product of elements of S.

We are

interested in representations of a given element w

E

G

as a product of elements of S with a minimum of factors. Suppose that a, bare non-parallel lines of the group plane E and c is a further line satisfying c a

E

S(a),

.f c.

S(b) and let O be the point with O

SE

Finally let dllc with d

IO.

and a line m lo such that aB

By A.4, there are a 6

=

µ6 for someµ

E

Let a, b. E

S(d)

S(m).

Using this result, one easily proves: LEMMA 2.1. fying a

II

Let w b, c

choice of a

E

II

E

G.

There are lines a, b, c, d satis-

d, and d,rd, such that for a certain

s (a), B

E

S (b), y

E

S (c), and 6

E

S (d) the

following holds:

w = aBy6. Using A.5, one deduces from Lemma 2.1 the following: LEMMA 2.2.

Let w

E

G.

Then either there are non-parallel

lines a, b such that, for certain a

E

S(a) and

BE

S(b),

we have w = aB; or there are two points A, Band a line c such that, for a certain y

70

E

S(c), we have w

= ABy.

(If

a,

Sare axial collineations such that their axes

have a point in common, then aS is called a quasirotation. The elements if

y is

ABy

EPPS are called glide-quasireflections

a quasireflection, or glide-shears if

y is

a

shear.) Lemma 2.2 tells us that the elements w of Gare either quasirotations or glide-quasireflections, or glidequasishears.

Because w is also the inner automorphism

of G belonging tow, Lemma 2.2 yields: THEOREM 2.3.

The centre of G is trivial and therefore

G ~ G/Z(G) = G. Lemma 2.2 also yields REDUCTION THEOREM 2.4.

II.3.

If OE Pis any point of E, then

The theorem of the antiorthological i-quadrilaterals

The theorem referred to in the title of this section is a special case of the theorem of the antiorthological quadrilaterals, due to Schlitte (1955), which characterizes the affine orthogonality relations induced by hermitian forms. DEFINITION 3.1.

Let i = l, 2, 3, 4 and j = 1, 2 .

Consider

4 lines ai and 2 self-orthogonal lines sj such that there A.

1

71

with a 0 := a s

s

1 ,

and sj IAj, Aj+ 2 •

4 ,

s

2 ,

a

a4

3 ,

;

The a. are the sides and the s . the J

i

diagonals of the quadrilateral. 1 ,

a

,

which may be abbreviated as (ai; sj), is called

2 ),

an i-quadrilateral.

A Is

Then (a 1

If there is a point

then (ai' sj) is called almost complete and A

2

is referred to as the diagonal point of the i-quadrilateral (see figure 6).

A,, /

,,,

/

I I

/

/

/

/

/

I

/

/~

FIGURE 6 The following two lemmas are the core of the theory to be considered here: LEMMA 3.2

(first lemma of the i-quadrilaterals).

Let

(a.; s.) be a non-degenerate, almost complete i-quadrii

J

lateral with vertices Ai and diagonal point A. 2, 3 let bi be the line satisfying b:!. a line through A.

I

For i = 1,

A, a .• i

Then the following two statements are

equivalent:

72

(1)

=

(2)

A,

I

I

I

I

I

I

s 2 for i

1, 2,

I

A FIGURE 7

PROOF .

There exists exactly one line satisfying (1).

Assume if possible that there are two elements B•, BI4 such that (2) holds.

s. S l

=

s2 implies b 4 -/

Then Sl

-1

s~s.

€

8 (S l)

and, since

-/ b~, it follows that

So there is at most one 8 4

€

€

s'

Sl

4

s·s- = 1

4

4

s

1.

S satisfying (2), which means

that it is sufficient to show that (1) implies (2). Assume 8 1

,

8 2, 8 3 ,

8 4 satisfy (1) anu the assumptions

of (2). the relations: and

A.

l.

It follows that:

73

(a)

s.

A i+1' A.1

and that

1

s-:-1 A. 1

(b}

1

1 ai'

I

a., s 1

l

S·1

A.

Now (a) yields Ai+l

s.

for i = 1, 3

52

1

for i = 2, 4 and A 5 = A 1 . for i = 1, 3 and (b) yields

-1

= A. 1

for i = 2, 4.

1

Reviewing the situation we know that Aw= Ai A 1 = A~ and s 1

I A,

A1 .

Thus Lemma 2.2 yields w

S(s 1 ).

E

This

proves the lemma. LEMMA 3.3 (second lemma of the i-quadrilaterals).

Let

(ai' sj) be a non-degenerate, almost complete i-quadrilateral with vertices Ai and diagonal point A. 4, 3 let bi be the line satisfying ai

be any line passing through A.

II

bi

I

A.

For i = 1, Let b 2

Then the following state-

ments are equivalent: (1)

b2 11 a2.

s.1

(2)

If Si E S (b . ) satisfies 51 1 -1 - 1 3, 4, then 8184 8382 E s (s 1 ).

PROOF.

= 52 for i

1, 2,

As in the proof of 3.2 it is sufficient to show

that (1) implies (2). Assume 8 1 , 8 2 , B3 , 13 4 satisfy (1) and the assumptions of (2).

Choose ai -1

-1

v = a 1 a 4 a 3a 2

a.

E

S (ai) such that s 1 1 -1

and w = 13 1 13 4 8 3 8 2 .

statements: (i)

s

1

-1

II

s 2•

We prove a sequence of

-

is a fixed line of both v and w.

74

Define

The proof is straightforward. (ii)

There exist points X, YI s

For i = 1, 2, 3, 4, parallel to s

1

1 B.a7 l l

1

such that w = vXY.

fixes the pencil of lines

and has as its centre the pencil of perl So -Biai is a translation.

pendiculars to bi and ai.

I

I

FIGURE 8 Since

vw-

1

1 B.a7 l l

is a translation for i

must be a translation.

=

1, 2, 3, 4, also

So (ii) holds because of (i).

(iii)

w is a quasirotation with centre A.

(iv)

vis a quasirotation or a translation.

whas

the fixed point A and v the fixed line s

1•

Therefore the statements (iii) and (iv) are consequences of Lemma 2.2 (remember that s

75

1

Is

1

and that the elements

of Sin particular are quasirotations). (v)

vis either a shear with axis s

or a transla-

1

tion. Assume that vis not a translation. quasirotation fixing s

v

fixed point Q of

Then it is a

[(i) and (iv)] and there is a

1

(the centre of the quasirotation).

Since every fixed line of a quasirotation goes through its centre, we conclude that either v

possibilities:

s

Now there are two

1•

S(s 1 ) or

E

fixed point, which is Q.

QI

v has

Therefore Q

s2.

because A1

=

C(l Sl

II

C(

1

s2,

I s 2, C( 1 Sl

v

Assume, if possible, that

C(l

has one and only one fixed point Q. the only fixed point of

exactly one

V

C( 1 I

hence

=

s2.

In this case Q

is

which also has the fixed line

oa i I

C(l Sl

I

s2 and hence,

The last equation forces

A, which is impossible for non-degenerate i-quadri-

laterals (ai' sj).

The conclusion is that v

E

S(s 1 ).

So (v) holds. (vi)

w

E

S(s 1 ) .

If vis a translation, then by (ii) so is wand, by (iii), it follows that w

=

If v

S (s 1 )

then, by (ii),

the collineation ~ has to be a glide-shear.

But, by (iii),

w is a quasirotation.

l.

E

The conclusion is that w must be a

shear, which proves (vi) and the lemma 3.3. THEOREM 3.4 (theorem of the antiorthological i-quadrilaterals).

Let (ai; sj) and (bi' tj) be two non-degenerate

76

i-guadrilaterals with s i = l, 2, 3 .

\

1

Then also

It2 and a4 Ib4•

s

2

I

t

1

and a 1.

I b 1.

for

\

\

\ \ \S1 \

\

\

B, \

\

FIGURE 9

\

THEOREM 3 . 5 (theorem of the parallogical i-guadrilaterals). Let (ai, sj) and (bi, tj) be non-de9:enerate i-g:uadrilaterals with s. 11 t. and a. l. J J 2, 3. Then also a 4 _ 11 b 4 .

II

b. for j = l, 2 and i = l, l.

--

Theorems 3.4 and 3.5 are both consequences of Lemmas 3.2 and 3 . 3 of the i-quadrilaterals.

This is obvious if

the i-guadrilaterals in question each have a diagonal point.

The case in which the diagonals are parallel can

be reduced to the general case with diagonal point (see Gotzky, 1970) •

77

II.4.

Coordin ates 1

In this section the results of section II.3 are used to prove that the group plane E minkow skian plane.

=

(G, SG) is a unitary -

The method is to constru ct both the

skew field which coordin atizes the plane and the hermiti an form which represe nts the orthogo nality relatio n of E.

FIGURE 10 Let a E S(s) withs Is and EE S(e) with Further let t

=

sE and,

such that Ole, s, t. Ff,

crE

E

S(t).

E2

=

1.

O may be a point

Let F I s and E = F E , such that

0 f, E.

1. Compare the introdu ction of coordin ates in Wolff (1967). 78

IO

From now on, for all the lines a, b, •. • satisfy s t a t t, s t b

t

that

t, ... , let a, S, .. • represent

elements of S(a), S(b), • .. respectively, such that sa sS = t, ... B := F

(remember Axiom A.6).

t,

Further let A:= Fa,

B, ••••

Next let K be the set of all points incident with t, and define: A + B := AOB A

0

B := Fm: B

A

0

0

:= 0 =: 0

+

Then K

0

A

for all A, B

E

K,

for all A, B

E

K\ {O},

for all A

E

K.

(standing for K with respect to the operation

+) is an abelian group with unit 0, in which A has the

inverse -A= OAO .

Further K0

(representing K\ {O} with

respect to the operation o) is a not necessarily abelian group with unit E = F£, in which A has the inverse A £a-1£ £ (remember that £ 2 = 1 and hence t = s). = F Concerning the proof of these statements there is only one problem worth mentioning , and that is the associativity of K0 which we now prove. Ao B = A£B; therefore (Ao B) o C = (A A£B£Y.

O

B) £y =

Let D = B ° C, hence D

a'= 8£yo

-1

has the fixed points O and F.

the reduction theorem yields a'

E

S(s).

A£B£y = A£cr'o = A£o =Ao D; hence (A o B)

o

C

Ao D =Ao (Bo C).

79

Since O

Thus

t

F

Since (A+ B)

O

C = (A+ B)Ey = (AOB)EY = AEYosEY

AoC + B°C, K is a skew field if and only if (i)

Ao

(B + C) = AoB + AoC

also holds.

Therefore K is a skew field if there is an

antiautomorphism

of K.

TI

CONSTRUCTION OF AN ANTIAUTOMORPHISM

OF K.

TI

be lines incident with O and let a*, b*,

Let a, b, ... be the per-

pendicular lines from O onto a, b, ... respectively. mapping

j:

TI:

K with 0 F

. 1 utory.2 is 1nvo

a*O

The

0

= A*O

A

for

We shall see that

TI

E

Ko

is an antiautomorphism

of K with respect to the operations+, o. First, let D =A

a'= a£- 1 So- 1

E

Considering figure 6 yields

a* o*

-1

S* £*

-1

E

s (s) (Lemmas 3. 2 and

Thus S*E*

2.

B.

S(s) and the second lemma of the i-quadri-

laterals yields a" 3.3).

O

-1

a*o*

-1

E

S(s

a*~*- 1 u

)

= S(s).

Since E(G, SG) is a translation plane, the theorem

of the parallogical i-quadrilaterals (Theorem 3.S)implies that the line reflection a with axis sin the direction of O a- 1 a*O O·Oa- 1 a*O t exists. Therefore (A*) = A = A = (AO)oa

-1

a•aa*a holds for all A

E

K0

•

If now d

IO

such

that Ao= F 0 , then o = oao and o* = aa*a: hence (A*)o = 0 o- 1 o* 0 (A) = (A )*.

So *O = O*, and therefore

80

TI 2

= * 2 -0 2 = 1.

E and E* -1 = E*, hence F $*Ea* = F o*O , hence

But E*O

DTI = Fo*O = FB*Ea = (FB*O)Ea*o. If y

such that F

Es

a*Oy- 1

E

S(s).

a*O

= F

y

and c Io for y

E

S(c), then

Hence

On = (FB*O)Ea*O = (FB*O)EY = Bn

O

An.

So TI is at least an antiautomorphism of K0 Second, let D =A+ B.

•

Consider then figure 11.

s

FIGURE 11

The theorem of the antiorthological i-quadrilaterals (Theorem 3.4) applied to the i-quadrilaterals with vertices F, B, FOB, D and F, B*, FOB*, D* yields (FOB)D Since FA 11 (FOB) D and FA* FA

I

I

I

(FOB*)D*.

FA we get

(FOB*)D* II FA*.

So F(FOB*) = A*D*, hence A*D* = OB*, hence D* = A*OB*. The last equation yields

81

=

Dn

A1TOB1T

which proves that n is an automorphism of K+ Son is an antiautomorphism of K, and the following theorem is proved: THEOREM 4.1.

K

(K, +, o) is a skew field with anti-

automorphism n. Next we introduce coordinates in E

=

E(G, SG) using

K as skew field of coordinates.

Since Eis a translation plane, there is for each point Pi of E exactly on&pair Xi' Yi (ii)

e

K such that

P.l. = x.oY:. l. l.

Conversely every pair xi, Yi by (ii).

e

K determines a point Pi

We have to show that every line considered as

a set of points can be described by a linear equation. First, we describe lines parallel to s o r t . Obviously (1)

lines parallel to s are described by equations X = C,

(2)

lines parallel t o t are described by equations Y = C. Second, we describe lines incident with O.

and O -:/ P 1

,

I a.

P2

Consider figure 12 .

Let a IO

By the theorem

of the parallogical i-quadrilaterals (Theorem 3.5)

a perpendicular

Hence there is a line

both the lines X 1 Yi and X 2 Y; and passing through O. ~-1

Y~

l.

x~ l.

for i

=

1, 2, so x .

l.

82

Y.

l.

0

A.

to Then

a

a

-------t

FIGURE 12

Thus (3)

lines a which are incident with O but different from sand tare described by equations X =Yo A.

Finally, let s, t ~ a IO and b I/ a. There is a point OZE zE Z EK such that b = a = a (remember that Eis a translation plane).

If now P

= XOYE is a point of b,

the above result yields: E:

x = yz O

o

A

(ozEyzEo)

E:

A=

o

E:

= (YOOZ O) o A= (Y + OZ O) o ~ zEo ~ ~ Let B := 0 o A; then X = YoA + B.

83

(YzEozo:o)

A= yoA

o

A E:

+ OZ O

o

A.

This proves (4)

lines b

parallel neither to s nor t o t are described ~ by equations X = Y0 A + B. ~

Since K is a skew field a general line equation can be derived from (1) to (4):

(5)

X o A+ Yo B + C If now V

O.

= V 3 (K) is the 3-dimensional right vector

space over the skew field K, it is easy to realize that the lines of our plane can be represented by the 1-dimensional subspaces of V which are different from (0, 0, E} o K. E

= PV the planes P(O,O,E)oK and

Moreover, with P

= E(G, SG) are isomorphic .

(Compare the introduction

of coordinates in Artin, 1957.) Next we describe the orthogonality of E THEOREM 4 . 2.

Let a, b be lines of E

by the vectors (A 1 , A 2

,

= E(G, SG).

= E(G, SG) represented

A 3 } and (B 1 , B 2

B 3 ).

,

Then alb

if and only if

PROOF FOR A 2

= O.

Since a, bare lines of E, it is easy

to see that the inequalities A 1 o A 2 i o i

B 1 o B 2 hold.

Now A 2 = 0 means that a II s or, since s Is, that a So a

Ib

if and only if B 2

0.

s.

But assuming A 2 = O, the

matrix equation holds if and only if B 2

84

J

= o.

PROOF FOR

A2

1 o.

Some considerations similar to those

just used show that we may assume B 2 generality.

101

Let therefore A 2

1 B2

0 without loss of •

The matrix equation holds if and only if AnoB + I 2 A:oB 1 = O, hence if and only if (A 1 oA~)n + B 1 oB 2 = 0. Using the symbols out of (4), this last equation becomes transformed into -(A-)n -

B-

= O, or An=

aIb

The last equation is equivalent to

a0 ,

or A*= B.

(see figure 12

and remember the meaning of* used for the construction n). It remains to show that

a I b if

and only if a

Let P 1 := X 1 0Y~ 1 o 1 Q 2 := x 2 0Y! with P 1 Q2

I b.

aIb

To visualize this, refer to figure 12.

i f and only i f XIY~

I X2Y!.

I

b.

a and We have

Hence, applying the

theorem of the antiorthological i-quadrilaterals to the quadrilaterals with vertices O, P 1

x2

respectively, we find that

,

aI b

E

Y1

,

X 1 and O,

Q2

if and only if a

,

E

Y2

,

I b.

This completes the proof of 4.2. MAIN THEOREM 4.3. Let f be the TI-hermitian form (right linear, since V 3 (K) is a right vector space) defined by the matrix

0

E

0

E

O

0

0

0

0

Further, let V = V 3 (K, f, n).

Then E(G, SG) ~ PRad V for P

= PV and G = G = U!(K, f, n).

Hence, in particular, E(G, SG) is a unitary-minkowskian plane.

85

PROOF.

Taking Theorem 4.2 into consideration, we know

already that E(G, SG) holds.

~

PRad

v·

By Theorem 2.3, G

~

G

We now compare the generated groups (G, S) and

(PU*, PS(V)) Part I) .

(for the definition of the last group see

Since PS(V) is the set of all axial collinea-

tions of PRad V which preserve the orthogonality, we infer from Theorem 1.3 that identifying E(G, SG) and PRad V the inclusion

S

~ PS(V) becomes valid.

A.6 yields PS(V) ~

S,

Because now Axiom

we get (G, S) ~ (PU*, PS(V)).

finally U* := U~(K, f, TI)

~

Since

PU* the main theorem must hold.

It remains only to remark that if U*

=

u;(K , f, TI)

is a restricted unitary group of the minkowskian type, then (PU*, PS(V)) satisfies the basic assumption and the axioms of section II . l . REFERENCES Artin, E. 1957. Geometric Algebra . Interscience Tracts, No. 3, Interscience Publishers, New York. Bachmann, F. 1959. Aufbau der Geometrie aus dem Spiegelungsbegriff. Die Grundlehren der mathematischen Wissenschaften, Band 96. Erste Auflage 1959, zweite Auflage Berlin-Heidelberg-New York 1973 . Gotzky, M. 1964 . Eine Kennzeichnung der orthogonalen Gruppen unter den unitaren Gruppen . Arch . d. Math . 15: 161-165 . 1965. Eine Kennzeichnung der unitaren Gruppen uber einem Schiefkorper der Charakteristik # 2. Dissertation, Kiel . 1970 . Aufbau der unitar-minkowskischen Geometrie mit Hilfe von Quasispiegelungen. Habilitationsschrift, Kiel. 1972 . Mittelpunktsabbildungen in affinen Ebenen. Abh. Math. Sem. Hamburger Univ. 37 : 133-146. 1973. Der Satz von der dritten Quasispiegelung. Supplement 15 in Bachmann (1973) . Pickert, G. 1955 . Projektive Ebenen. Springer, BerlinGottingen-Heidelberg. 86

Schutte, K. 1955. Ein Schliessungssatz fiir Inzidenz und Orthogonalitat. Math. Ann . 129: 424-430. Veblen, O. and J.W . Young. 1910, 1918. Projective Geometry, vols . 1, 2. Boston. Wolff, H. 1967. Minkowskische und absolute Geometrie. Math. Ann. 171: 144-193 .

87

THE LENZ-BARLOTTI TYPE III

Christoph Hering

1.

INTRODUCTION

A finite projective plane admitting a collineation group G of Lenz type III is desarguesian.

The first published

proof of this theorem (see [SJ) uses the classification of finite groups with a split BN-pair of rank 1.

There-

fore a complete presentation of this proof necessarily must be very long.

Also, the proof is quite indirect.

It involves the characterization of a large variety of groups, namely the groups PSL(2, q), Sz(q), PSU(3, q), groups of Ree type, and sharply doubly transitive groups, although in the end, because of geometric reasons, only the groups PSL(2, q) actually occur.

Therefore it seems

worth while to look for a more direct proof which always stays within the range of the 2-dimensional linear fractional groups.

For planes of odd order such a proof

actually exists, and it is the main purpose of this paper to describe this shorter, more economic way of argumentation.

88

Instead of the classification of finite groups with a split BN-pair of rank 1 we use here a system of three theorems on group spaces.

At first glance, this system

looks somewhat artificial.

The main idea is to save for

Ga sufficient amount of geometric information without losing the possibility of applying induction.

This is

quite difficult and could only be achieved by partitioning the problem into three different cases.

One of the three

cases (see Theorem C) deals with groups in which all involutions are central.

Theorem 2.1 and Corollary 2 . 2

list a few simple properties of such groups.

Possibly

one can find stronger results of this kind, which might lead to further simplifications.

The proof presented

here does not use any group-theoretical classification theorem apart from the classification of finite groups with dihedral Sylow 2-subgroups by Gorenstein and Walter [2]

(which of course uses the theorem of Feit and Thomp-

son) .

Even this theorem is not used in its full generality

but only in a certain quite special situation (see Proposition 3.6) . Our notation is fairly standard.

a

a and b elements of G, and Then a

b

< [a, bl

= b

Ia

-1

E

ab,

[ a , b] = a

(J{, and be.$, >.

-1

Let G be a group,

as well as §&-subsets of G. b

-1

Also,

ab, and

}G

uP, [a, oll']

=

is the centre of G,

~(lthe centralizer of t'tin G, J!Gdtthe normalizer oft( in G, G' the commutator subgroup of G, G# the set consisting

89

of all non-trivial elements in G, o(G) the largest normal subgroup of odd order of G, and ated by all involutions in G. element of order 2.

Q2

(G) the subgroup gener-

An involution is always an

A dihedral group is a group with two

generators a and band the relations an= b 2 = 1, where n

:2:

bab = a- 1

2.

Suppose that in addition to the group G we have a set

and a binary operation

Q

= a

x G

Q

for all a

E

Q

+

such that

Q,

and g, h

E

G.

Then

we call the pair (Q, G) together with the given binary operation a group space.

2.

a

{w

E

Q

w

{g

E

G

I a?l.

Furthermore, we denote

= w for all a

E

('.7[}

1

Q(tfl)

and G al ... an

= a. for lsisn}, if al, l.

... ,

a

n

E

n.

GEOMETRIC AND GROUP-THEORETICAL TOOLS

In this section we present the most specific group-theoretical and geometric results which will be used in our proof of the Type III Theorem. THEOREM 2.1.

Let G be a finite group containing a sub-

group Z such that Q 2 (G)

sz

sjG,

and let Ube a subgroup of G containing

z.

Choose a sub-

group N of Z maximal with respect to the property that Q2

(U/N) s Z/N.

90

Then the following statements hold: (a)

Z/N is an elementary abelian 2-group.

(b)

If 1 -:j x E Z/N, then there exists an element

y E U/N - Z/N such that y 2 = x. (c)

The number r of conjugacy classes of involutions

in G/Z intersecting U/Z non-trivially is at least IZ/NI - 1. In particular, r ~3, unless the Sylow 2-subgroups of U/Z are cyclic or dihedral. (d)

PROOF.

I Z/N I :;;

I U/Z I •

~ru,

Denote G = G/N, u = U/N, and z = Z/N.

a subgroup of odd order of

z.

As

z

an involution in U/A, then l I contains an involution x and xA =

xA

A~ U.

z contains

y EU - Z.

If XA is

= 2IAI so that xA

E Z/A.

Z/A, and by maximality of N we obtain A= 1. that

Let A be

So S1 2 (U/A) ~ Suppose now

an element z such that y 2 -:J z for all

For each involution x in U/ the group

is an abelian 2-group containing the cyclic group as a subgroup of index 2. cyclic or =

X

Hence either is

, where Xis an involution.

In the first case there exists an element y such that y2

z and = .

that x E Z/.

As y 2 = z, we have y E Z, so

In the second case

But this implies that = So S1 (U/) 2

x E Z by assumption. ~ z and x E Z/.

:;; Z/, and by maximality of N we obtain

= 1, which proves (b).

91

Suppose now that Z contains an element a of order 4. By (b), there exists an element y E U - Z such that y 2 = a 2 .

JU,

As a E

order 8.

the group is an abelian group of It contains two cyclic groups of order 4, namely

and .

Therefore is the direct product of a

cyclic group of order 4 and a cyclic group of order 2. But n Z

= .

which do not lie in

Hence we have involutions in ,

Z,

a contradiction.

Z is

So

an ele-

mentary abelian 2-group. Let z 1 and z 2 be two different non-trivial elements in

z.

u-

By (b), there exist elements u 1 and u 2 in

such that

u:

=

zl and u~

=

z2.

Z

Clearly ulz and u2Z are

U/Z. Suppose that these G/Z. Then there exists an

involutions in

involutions are

conjugate in

element gE G

such that g

-1

-

u 1 gZ

=

u 2 Z.

But this implies that g

u 2 z for a suitable element

= (u2z) tion.

2

z E Z,

and hence z 2

= (g- 1 ulg) 2 = g- 1 u:g = g- 1 zlg = zl

I

=

-1

u 22

u 1g

=

=

u 22 ;

2

a contradic-

So the number of conjugacy classes of involutions

in G/z intersecting

U/Z

non-trivially is at least as large

as the number of non-trivial elements in is at least three, unless

1-1 ,z =

2 and

z.

This number

u contains

exactly

one involution, in which case the Sylow 2-subgroups of U/Z are cyclic or dihedral.

Thus the proof of (c) is

complete, and as an immediate consequence we obtain (d). Finally, we consider the special case U to the following corollary:

92

=

G.

This leads

COROLLARY 2.2.

Let G be a finite group containing a sub-

group Z such that n 2 (G) s Z

s3G.

If G/Z has at most two

classes of involutions, then the Sylow 2-subgroups of G/Z are cyclic or dihedral. LEMMA 2. 3 (Zassenhaus [11]). a finite group g

T

= g

-1

PROOF .

G.

for all g If

X €

If oCG, €

Let , be an involution acting on

=

1, then G is abelian and

G.

G, then (x

-1 TT X

)

=

T (x )

i . e . T inverts the element x- 1 xT . assumption implies that the map x

-1

X

= (x

-1

X

T -1 )

On the other hand, our 1+

x - ix T for all x

one-to-one and therefore a map of G onto itself . inverts each element of G. ((xy)-1)-1 = all x,y

€

,

E

G is

Thus T

This implies that xy =

((xy)T)-1 = (xTyT)-1 =

(x-1y-1)-1 = yx,

for

G, so that G is abelian.

BRAUER-WIELANDT THEOREM 2 . 4 (see Wielandt [10]). E = {l, e 1 , e

2 ,

e

3}

Let

be an elementary abelian group of

order 4 acting on a group G of finite odd order.

THEOREM 2.5 (Gorenstein and Walter [2]) .

Then

A finite group

with dihedral Sylow 2-subgroups satisfies one of the following conditions: (i)

G/O(G) is isomorphic to a subgroup of PfL(2, q)

containing PSL(2, q) for a suitable odd prime power q. (ii) G/O(G) is isomorphic to the alternating group A 7

93

•

(iii)

G/O(G) is isomorphic to a Sylow 2-subgroup of

G. LEMMA 2.6 (see (3, Hilfssatz 6]). Let G be a finite doubly transitive permutation group which contains involutions fixing 2 letters but no involutions fixing more than 2 letters.

Then the Sylow 2-subgroups of Gare dihedral

groups or quasidihedral groups. LEMMA 2.7.

In PSL(2, q), q odd, the normalizer of each

elementary abelian subgroup E of order 4 is transitive on E - {l}. PROOF.

Let q be an odd prime power and G

Then G is not 2-nilpotent .

~

PSL(2, q).

Hence by the Transfer Theorem

of Frobenius (7, p. 436, Satz 5.8] G contains a 2-group E such that3'GE/~E is not a 2-group .

As Eis a dihedral

group, it follows that Eis elementary abelian of order 4. Hence the order of 3tGE/~E is divisible by 3, so that ')'{GE

is transitive on the set of non-zero elements of E.

We finish the proof by showing that the automorphism group of G acts transitively on the set consisting of all subgroups of G which are isomorphic to E.

Let F~ G be a

second elementary abelian subgroup of order 4 .

As all

involutions are conjugate in G, we can assume without loss of generality that En F contains an involution a. be the centralizer of a in PGL(2, q).

94

Let C

Then C is a dihedral

group and E/ and F/ are involutions in (C nG)/. These involutions must be conjugate in C/, as E and F are not cyclic. LEMMA 2 . 8 (Baer [1]).

An involutory automorphism of a

projective plane of finite odd order is either a homology or a Baer involution. LEMMA 2.9 (Ostrom [8] and Liineburg). automorphisms of a projective plane

Let G be a group of

q, .i),

a

E

,:t_··, b

E .~(-

a,

.!.!_ a and Bare involutions, then aB is the unique involution in G(anb, AB). PROOF.

As the commutator [a, BJ fixes each point on a

and b, it is trivial.

Hence aB is an involution.

involution fixes a and exactly two points on a.

This A Baer

involution fixes at least three points on each of its fixed lines.

Therefore the product aB is a perspecti-

vi ty, which clearly must have a n b as centre and AB as axis.

Finally, if y is any involution in G(an b, AB),

then by the part of the theorem which we have already proved, a(aB) and ay lie in G(B, b). a(aB) (ay)

-1

On the other hand

fixes each point on AB and of course also each

point on b, which implies that a(aB) = ay.

Hence aB is

unique . 3.

THREE THEOREMS ON GROUP SPACES

Let (Q, G) be a finite group space with the property:

95

(*) For each a En the stabilizer Ga contains a normal subgroup Na which is sharply transitive on n -{a}, such that for a E n and g E G we have (Na) g = N ag In this section we derive some properties of (n, G) which will be useful for the proof of our main theorem . we introduce the following notation:

s = .

E

¢ is the representation of G on n determined by the group space (n, G).

K is the kernel of¢.

If U is a

subgroup or an element of G, then we denote u¢ =

U.

If u is a subgroup or an element of G, then n(U) is the set of fixed points of u . If U is a subgroup or an element of G and ln(U) I ~ 3, then S (U)

3, then PropoHence in this case

S = S

and Sis isomorphic to a homomorphic image of the representation group of s. Proposition 3.3.)

(Note that S

n

K s

j

S because of

By Schur [9, pp. 119-120] it follows

that S ~ SL(2, q) ors~ PSL(2, q) except, possibly, if

101

q = 9, in which case the Schur multiplier of s has order 6.

However, we also know that S n K is a p - group:

Let

Y be the Sylow p-subgroup of Sn K, and Pa Sylow p-subgroup of S

a

containing Y.

Then P = Y

N a

x

As ls:s a

q + 1, P actually is a Sylow p-subgroup of s.

I

=

Hence by a

theorem of Gaschiitz (see (7, p.121, Hauptsatz 17.4]) there A

A

exists a subgroups of S such that S = SY and Sn Y = 1. A

As Y is central, S plies Y = 1.

S, and S/S

!

~

Y.

But S' =Snow im-

So S _ SL(2, q) or PSL(2, q), whenever S' =

s. There still remains the case q = 3. ~ A4

•

Let T be a Sylow 2-subgroup of

subgroup of order 4 in S, so that T

g

s.

In this case S Then Tis the

Sand T(S n K)

9

S.

Now the normalizer of T contains T and the central subgroup (Sn K), hence the product T(S n K), so that Tis normal in T(S n K).

This implies that Tis characteristic

in T(S n K) and therefore normal even in S.

Hence TN CJ, is

a group and as Tis transitive, we have S = TNa. Because of Sylow's theorem, 3 Clearly T' s Sn K.

f Is

n Kl, so that Sn Ks T.

Also, by a lemma of Zassenhaus (see

(7, p. 350, Satz 13.4]) T/T'

~

(Sn K)/T'

x

Tl/T', where

T 1 /T 1 is an elementary abelian group of order 4, such that T 1 is normalized by Na and Na acts non-trivially on T 1 /T'. Here T 1 is transitive on II, so that S = Na T 1, T 1 = T, and T' =Sn K.

Thus Tis generated by two elements a and b.

We now determine the order of the commutator subgroup T'.

102

Let x, y ET and consider the commutator [x, y).

x ES n K, then [x, y) = 1 . in T -

(S n K).

If

So we can assume that x lies

As~ is transitive on T/S n K - {l}, a

there exists an element u EN such that xu = a•z, where a

z ES n K.

Furthermore,yu has the form arbsz,, where

r, s E {l, 2} and z' E Sn K. [x, y)

=

[x, y) u

Hence

= [xu, yu) = [az, arbsz' J = [a, bs).

Therefore T' = and

IT' I~ 2, as [a,b)

2

=

[a,b 2 )

= 1. THEOREM A.

Assume that there exists a subgroup U in SK

containing Kasa normal subgroup of index 2 such that IQ(U)

I

= q + 1 and S(U)/K(U) ~ PSL(2, q)

odd prime power q. PROOF.

for some

Then IQ! > q 2 + 1.

See [6, pp. 448-452).

THEOREM B.

Assume that (Q, G) has the following proper-

ties: (a)

If a, SEQ and a t S, then Gas contains exactly

three involutions.

One of them fixes all points in Q while

the remaining two don't fix any point different from a and

s.

All three are contained in JGaS but at least one is

not contained in

JG {a, S} .

(b)2

We consider the group space (Q(U),at'GU).

As the

three involutions in Gas are central, they are contained inoC'Gu.

Let Ebe the group generated by these involutions

and suppose that E ~J-,({U){a,s}·

In the doubly transitive

group space (Q(U),ot"'GU), there exists an element x interchanging a and 6.

As x centralizes E it follows that

G{a,S} = centralizes E, which is impossible by So E ~ }(~U)aS but E / j-(~U){a,s} •

(a).

Also u

so that u is an even permutation of Q and hence ln(U)I

=2

(mod 4).

~U).

for a suitable prime power - 1)

2

= lnJ

Therefore we can apply induction to (Q(U),

It follows that S(U) ~ SL(2, k

S,

E

q.

q)

Because

and IQ(U) I = q + 1

q+

1 = IQ(U)

I

~

+ 1, we obtain a contradiction to Theorem A.

So all involutions in Shave at most 2 fixed points. As they induce even permutations on Q and lnl they actually all have exactly 2 fixed points.

104

=2

(mod 4),

Hence by

Lemma 2.6 the Sylow 2-subgroups of Sare dihedral groups or quasidihedral groups.

Also, we can obtain information

about the involutions ins.

Clearly the kernel K contains

exactly one involution by (a).

If S contains only one in-

volution, then we are finished by Proposition 3.6. So we may assume that S contains an involution t which is not contained in K.

Then by the above t has two fixed points,

w.l.o.g. a and 8.

Let Ebe the elementary abelian sub-

group of order 4 generated by all involutions in Gas· Eis not central in G{a,B}' there exists an element x G{a,B} such that E

= .

E

Hence E $Sand therefore

Sn K ~En K, which is a group of order 2.

s

As

Furthermore,

S' by Proposition 3.5, and therefore Sn K $

JS

n S'

is isomorphic to a subgroup of the Schur multiplier of In particular, the Schur multiplier of

s has

s.

even order.

Hence by Schur (9, p. 108, Theorem V] the Sylow 2-subgroups of Sare not quasidihedral groups.

So they are

dihedral, and we are finished by Proposition 3 . 6. THEOREM C.

Assume that each involution in S acts trivially

on n and that 4 $ lnl

=2

(mod 4).

Then S ~ SL(2, q) and

lnl = q + 1 for a suitable odd prime power q. PROOF.

Let (n, G) be a counterexample of smallest possible

degree.

By our assumption lnl is even so that bec~use of

Proposition 3.5 each element ins induces an even permutation on

n.

Hence each involution in S fixes at least 2

105

points.

Suppose that each involution in S fixes exactly 2

points.

Then all involutions in Sare conjugate by Pro-

position 3.4(b), and therefore the Sylow 2-subgroups of Sare cyclic or dihedral by Corollary 2.2.

But then G is

no counterexample because of Proposition 3.6.

Hence

contains involutions fixing more than two points.

S

Concern-

ing such involutions we prove (1)

If u is an involution in S fixing more than 2 points

and U is the pre-image of in G, then S(U) and S(U)

~

ln(U) I

SL(2, q)

PSL(2, q) for a suitable odd prime power q.

Furthermore, PROOF.

~

lrl(U)

I

= q + 1 < (lnl

- 1)

!-,:

2

+ 1.

As u ES, it is an even permutation, so that

=

lnl - 2 (mod 4).

Hence we can apply induction to

the group space (S"l (U), LGU). SL(2, q) and ln(u)I

=

It follows that s (U) ~

q + 1 for a suitable oddprirrepowerq.

By our assumption the unique involution in S(U) must lie in K, so that S(U)

~

PSL(2, q) .

Because of Theorem A we

have the inequality lnl > q 2 + 1. Suppose now that S contains an involution fixing exactly two points .

By the above, S contains involutions

which fix more than two points.

Hence by Proposition

3.4(c) S contains an involution u such that lrl(u) ~

(lnl - 1)

!-,:

2

+ 1.

I

But this is a contradiction to (1).

Hence each involution in S fixes at least three points in

n. 106

(2)

If a, B, and y are pairwise different points in Q,

then SaBy does not contain any elementary abelian subgroup of order 4. PROOF.

Suppose that SaBy does contain an elementary

abelian subgroup X of order 4. such that In (a 1 l I In (a3 l I

3.

x S(T).

Let

X be a subgroup of S(T) isomorphic to the alternating group A 4

•

(A subgroup of this kind exists.

107

See (7, p.213,

Hauptsatz 8.27].)

Let F be the subgroup of order 4 and x

an element of order 3 in X, and denote E = x centralizes t and hence normalizes E.

F.

x

Now

In fact, has

three orbits in E - {l}, namely x, F - {l}, and a second orbit of length 3. We now choose an element d pre-image of in G by D. ol~.

We consider the centralizer

In this centralizer we have the normal subgroup

S(D) and the product E•S(D). A

Sand the second by L. odd prime power q.

We denote the first group by

Note that S ~ PSL(2, q) for some

s,

Let a,

differ~nt points in SI (D). L

F - {l} and denote the

E

and y be three pairwise

Then L A

A

A

;); LaSyS/S s L/S = ES/S

aSy

aSy

===

n

s A

1 and therefore

s. A

E/E n

Hence LaSy is an elementary abelian 2-group, and by (2) we obtain IL a..,y I s 2, so that O

=

LaSy

.

This implies that ILi s 2(q + l)q(q - 1) and hence IE n §I

~ 2, so that IE n

SI =

2 or 4.

We consider at first the case IE n

sl

=

4.

In this

case the normalizer'Jt8 (E n S) contains an element y which induces an automorphism of order 3 of En

{d},

(see Lemma

Here y centralizes and hence normalizes E

2. 7) •

S

(E n

X

sJ.

Also, y has three orbits in E

(E n s) it, and a third orbit of length 3.

IF n (E n

d

_>

sJ I

=

2I

E# - { t } .

d

~

F# u

-

{ l}' namely

As

(E n slit and actually

Also, t i s not invariant under y.

108

This implies that is transitive on E#, which leads to a contradiction to Theorem 2.1.

sl =

Assume now that IE n

2.

A

Then L/S is an element-

ary abelian group of order 4, and we have three subgroups H1 , H2 , and H3 in L containing Sas a subgroup of index 2. {HI, H2, H3} and assume that HaSy -I 1. Then and H = XS. So two of the groups H al3y = L al3y =

Let H

€

A

HI , H2, and H3 have a trivial stabilizer on a, 13, and y, which because of their order implies that they are sharply triply transitive on n(D). that His of this kind.

=

x

En H.

H.

Let

Q

Choose HE {H 1 , H2 , H 3 } such

Then we have IE n HI

=

4 and L

be a Sylow 2-subgroup of H containing

By Lemma 2.6, Q is a dihedral group or a quasi-

dihedral group.

As IHI

=

(q

+ l)q(q - 1), we have IOI ~ 8.

Hence there exists a subgroup Y such that (En H) and IY: (En H) I= 2.

Let z E Y -

(En H).

4

Y s Q

Then z does

not centralize E, since in a dihedral or quasidihedral group each elementary abelian group of order 4 is selfcentralizing.

On the other hand, z centralizes d and hence

normalizes E = x in E.

(En H).

So z induces a transvection

We consider the group acting on the project-

ive plane determined by E:

If has three orbits,

then these orbits must be {t}, F#, and the remaining elements in E#.

This implies that the elation z fixes t, d,

and F so that is its axis and d its centre .

But

then z fixes a line not passing through its centre, namely

109

En H, a contradiction.

So has at most two orbits

in E#, and we can again apply Theorem 2.1. 4.

PROJECTIVE PLANES OF LENZ TYPE III

THEOREM 4. 1.

r- .t

and

Let

(f, :L)

a subset of

Q

(f,Zl

Assume that

be a projective plane,

.t E ,Z,, P E

.t containing at least 3 points.

admits a finite group G of collineations

leaving invariant P and Q such that for all XE Q the group G(X, XP) is transitive on

Q

{X}.

Assume in addi-

tion that each involution in G is a homology and that IQI

=2

(rr,od 4) .

Then there exists an odd prime power q such

that ~ SL(2, q),

IQI = q + 1,

and (,,

/1:>

PROOF.

We consider the group space (Q, G).

contains a desargues ian subplane of order q. Clearly

this group space has the property (*) defined in section 3.

If each involution in the subgroup S

=

lies in G(P, .t), then the first two statements of our theorem follow from Theorem C. ists an involution t

ES - G(P, .t).

homology whose centre z lies on

Assume that there exThen t must be a

.t and whose axis a passes

through P, because obviously Gleaves invariant the line

.t.

By Proposition 3.5 each element in S induces an even

permutation of Q.

This implies that t must have fixed

points in Q, so that Zand an

llO

.t lie in

Q.

Also,

(Q, G)

is doubly transitive, and hence G contains an element x interchanging Zand an .t.

But then tx E G(a n .t, ZP).

By Lemma 2.9 the product ttx is an involutory homology in G(P, l).

Also, this lemma implies that t, tx, and ttx are

the only involutions in the stabilizer G2

, an"--0

have all properties required in Theorem B. cases ~ SL(2, q) and

suitable prime power q. that

(f, c\C)

l~I

•

Hence we

So in both = q + 1 for a

By [4, Theorem 2.8] we now obtain

contains a subplane of order q.

THEOREM 4.2.

Let

be a projective plane of finite

odd order which contains a line .t and a point P not incident with .t such that (y,:;/;1 is (X, XP)-transitive for each point XE .t . PROOF.

Then

(r

~ is desarguesian.

Let ({,~) be a minimal counterexample and denote

G

=

.

We consider the group space (.t, G), which again has the property (*) of section 3.

By Baer's Involution Lemma

2.8, each involution in G is a homology or a Baer involution.

Suppose that G contains a Baer involution t.

Then

the set/J/ of fixed points o f t generates a subplane of order q and

(,,£)

has order q 2 for some odd q.

This sub-

plane contains P and .t, and because of 3.1 it is again (X, XP) -transitive for each X tion,

Of is

desarguesian.

E

.t n

tf .

Hence, by induc-

This implies that q is a prime

power and S(t) induces a group of collineations of fwhich

111

is isomorphic to SL(2, q). 3. )

(We define S(t) as in section

Hence the group of permutations of

/j,

n

l induced by

S(t) is isomorphic to PSL(2, q), but this is impossible by Theorem A. Thus each involution in G is a homology. (mod 4), then Theorem 4 . 1 implies that guesian .

Assume that Ill

=0

(mod 4).

((,£:->

If Ill

=2

is desar-

Then each involu-

tion in G lies in G(P, l), as otherwise it induces an odd permutation of l .

But IG(P, l)

I

Ill - 2

=2

(mod 4).

This implies that the Sylow 2-subgroups of G contain only one involution.

So they are cyclic or quaternion groups,

and Proposition 3 . 6 implies that G ~ SL(2, q) for a suitable prime power q .

As in 4 . 1 this finishes our proof.

REFERENCES 1. 2. 3. 4. 5.

6. 7. 8. 9.

R. Baer. Projectivities with fixed points on every line of the plane . Bull . Amer. Math. Soc. 52 (1946): 273-286. D. Gorenstein and J . H. Walter. The characterization of finite groups with dihedral Sylow 2-subgroups. I . J. Algebra 2 (1965) : 85-151 . c. Hering. Zweifach transitive Permutationsgruppen, in denen 2 die maximale Anzahl von Fixpunkten von Involutionen ist . Math . z. 104 (1968): 150-174 . On projective planes of type VI. To appear . C. Hering and W.M. Kantor. On the Lenz-Barlotti classification of projective planes. Arch . Math . 22 (1971): 221-224. C . Hering, W.M. Kantor, and G.M. Seitz. Finite groups with a split BN-pair of rank 1. J. Algebra 20 (1972) : 435-475. B. Huppert. Endliche Gruppen I . Berlin-HeidelbergNew York: Springer Verlag (1967). T.G. Ostrom. Double transitivity in finite projective planes. Canad. J. Math. 8 (1956): 563-567. I. Schur. Untersuchungen iiber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 132 (1907): 85-137. 112

10. 11.

H. Wielandt. Beziehungen zwischen den Fixpunktzahlen von Autornorphisrnengruppen einer endlichen Gruppe. Ma th • Z• 7 3 ( 19 6 0 ) : 14 6 -15 8 • H. Zassenhaus. Kennzeichnung endlicher linearer Gruppen als Perrnutationsgruppen. Abh . Math . Sern. Univ. Hamburg 11 (1936): 17-40.

113

SOME RECENT RESULTS ON INCIDENCE GROUPS

H. Karzel

1.

INTRODUCTION

For the study and foundation of absolute planes, it is useful to consider the associated kinematic space which is also called a group space.

To each line X of an abso-

lute plane there corresponds exactly one line reflection X fixing all points of X and preserving the structure of the absolute plane (compare [12], section 17).

Let be

the set of all line reflections, r := the group generated by all line reflections, 0 2 := {XY : X, YE D}

(we de-

note the line and the line reflection by the same letter), and J := {yE r: y 2 = 1, y # l } .

If a Er, let~ := {X ED:

a.XE J}; when aE 0 2 \ {l}, the set~ is called a pencil. L be the set of all pencils.

Let

Two pencils~, SE Lare call-

ed joinable i f ~ n ~#~,and a pencil which is joinable to every other pencil is called a proper pencil . L 0 denote the set of all proper pencils.

Let

The pair r, Dis

then a reflection group; that is, r, D fulfills the following four axioms ([12], section 19):

114

(S1)

Dis a system of generators of r.

(S2)

If~ E Land X, Y, Z E

(S3)

L0

(S4)

For each XE D, the cardinality

'cl,

then XYZ ED.

°I ¢.

!{a

E Lo: XE

a}!

°I 1, 2. If r is a reflection group, one can prove that D 2 is a subgroup of rand that, for each a E D 2 \ {l}, the set ~ 2

:= {XY: X, YE

a}

forms a commutative subgroup of D 2

•

The kinematic space associated with a reflection group r, Dis defined as follows: the point set o~ the kinematic space is the subgroup G := D 2 and the set of lines is G := {a•i 2 : a, BE D 2 ,

B -I l}; this means that each line

is a subgroup or a coset of D 2

•

One proves that the pair

(G, G) is an incidence space (see section 2).

Therefore

the set G is provided at the same time with a group structure and an incidence structure.

Both structures fulfill

the following compatibility conditions: V9.,

G For each y E G, the mapping Y9.,: {G-+ E;, ➔

yf;,

is an automor-

phism of the geometric structure (G, G). V r

For each y E G, the mapping y

r

:

{G-+ G E;,

➔

E;,y

is an automor-

phism of the geometric structure (G, G). Vk

Each line X

REMARK .

E

G with 1 E X is a subgroup of G.

The kinematic space of an absolute plane is a

dimensional) projective space or a

(3-

(3-1)-slit space (see

section 2) if and only if the incidence structure of the

115

absolute plane is a projective plane or an affine plane (see [3),

[15) respectively); such an absolute plane is

called a (generalized) elliptic or euclidean plane, respectively. This situation led us to introduce the concept of an incidence group [1] : A triple (G, G, • ) is called an incidence group if (G, G) is an incidence space,

(G,

•) is a group, and the

compatibility condition vi is valid . This paper will be a continuation of the reports [4] and [11) on incidence groups.

In section 2, we give the

geometric notations used and, in section 3, the algebraic ones .

For the algebraic description of the class of punc-

tured affine incidence groups, G. Kist has introduced the concept of a near-field extension, which is a generalization of the normal near-fields used for the algebraic representation of projective incidence groups (4.1).

Many

of the known infinite proper near-fields F do not contain sub-near-fields K such that the pair (F, K) is a normal near-field, but they do contain sub-near-fields such that they are near-field extensions.

For instance, the Kal-

scheuer near-fields (section 3) are near-field extensions over lR and over~, but not normal. Examples of incidence groups are obtained from algebraic structures (section 4), for instance , from associative unitary algebras . An important class consists of the 116

kinematic spaces which are derivable from kinematic algebras.

In section 5, we state the representation theorem

for punctured affine incidence groups by near-field extensions of G. Kist. A type of generalized incidence groups are the socalled 2-incidence groups (section 6), which can be described by pairs of near-domains.

The case in which the

underlying geometric structure is an affine space or plane will be investigated. Finally a geometric characterization of projective kinematic spaces is briefly discussed in section 7. 2.

GEOMETRIC CONCEPTS

Let P be a set and Ga subset of the power set of P; the elements of P will be named points and the elements of G lines. (Il)

The pair (P, G) is called an incidence space if: For each pair of distinct points p, q E P, there exists exactly one line A E G such that p, q EA; this line will be designated by p, q.

(I2)

For each XE G, the cardinality

!xi

is at least 2.

For the following, let (P, G) be an incidence space. A subset Tc Pis called a subspace if t implies t

1 ,

t

2

c T.

1 ,

t

2

ET, t

1

# t

Let T be the set of all subspaces.

Then Tis closed with regard to intersections and therefore the closure of a subset Mc P defined by

117

M

:= n{T:

2

is a subspace.

Mc TE T}

inf{

IMI: M

For TE T we define by dim T :=

T} - 1 the dimension of a subspace.

of dimension 2 are called planes. G(p)

Subspaces

For a point p E P let

:= {X EG: p EX} and for a subspace TE T let G(T)

:={XE G: X

c

T}.

called an m-line if:

A line LEG is

Let m E lN u {O}. (1)

ILi

~

3 if m = 0, and (2) for

each XE P\L, there exist exactly m distinct lines H1 , •••I

H

m

E G(x) n G({x} u L) such that H. n L =¢for i

i E {l, 2, •.. , m} and each XE G({x} 1 for all i

u

E {l, ... , m} satisfies X

Ix

L) with n

L

~

¢.

n

Hil

Let G

m

be the set of all m-lines. REMARKS.

An incidence space (P, G) with G = G0 is a pro-

jective space.

If G = G1 we call (P, G) a pseudoaffine

space; if in addition the relation on G defined by "A

II B A = B or A n B = ¢ and dim AuB = 2" is tran-

sitive, then (P, G) is an affine space.

This last condi-

tion can be proved if there is a line A E G with IAI

~

4.

(Compare [ll), pp. 79-81). For every subset Uc P we define the trace space (U, G ) by G u

u

: = { X n U: X E G,

IX n U I

space is again an incidence space.

~

2}.

The trace

If (P, G) is a pro-

jective space and La subspace of (P, G), then the trace space (P\L, G(P\L)) is called a slit space and if dim P = n and dim L = r it is called an (n - r)-slit space. An internal characterization of slit spaces is given in

ll8

[11].

If (P, G) is a pseudoaffine (affine) space and q

a fixed point of P, then the trace space (P\{q}, G(P\{q})) is called a punctured pseudoaffine (punctured affine) space.

Each line of (P\{q}, G(P\{q})) is either a 1-line

or a 2-line.

G. Kist [13] has given the following internal

characterization

an incidence space (P, G) is a punctured

pseudoaffine space if G = G1 u G2 and if the conditions (Pl) and (P2) hold, and a punctured affine space if (P3) also holds: (Pl)

Let Ebe a plane with G 2 (E) for each point x E E,

(P2)

:= G(E) n G2 f ¢; then

IG(x) n G2 (E) I = 1.

To every line A E G1 there exists a line BE G2 such that A II B (that is, A n B = ¢ and dim

= (P3)

2).

The relation 11

1

:= II

n

(G 1 x G1 ) is transitive

or there is a line A E G with IAI 3.

AuB

~ 4.

ALGEBRAIC CONCEPTS (NEAR-DOMAINS, NEAR-FIELDS, NEAR-

FIELD EXTENSIONS) A set (F, +, ·) provided with two operations+: F x F and•: F x F

➔ Fis

➔

F

called a near-domain [5], p. 123) if

the following conditions hold: (Fl)

(F, +) is a loop (0 denotes the neutral element) such that a, b E F and a+ b = 0 implies b +a= 0.

(F2)

(F*, •) with F* := F\{0} is a group.

119

(Fl)

For all a, b, c

(F4)

For every a

(FS)

For all a, b for all

X

E

F, a• (b + c)

E

a•b + a•c.

F, 0•a = 0.

E E

F* there exists d

F, a + (b + x)

A near-domain

=

(FI

(F, +) is a group.

+,

.)

=

a,b

E

F such that

(a + b) + d

a, b·x.

is called a near-field if

A set (F, +, o, •) provided with three

binary operations is called a dicksonian near-field (or regular near-field) (2)

[2] if (1)

(F, +, o) is a near-field,

(F, +, ·) is a field, and (3) for all a

ping ~a: F

+

F; x

+

a- 1

•

(a ox)

F* the map-

E

(a- 1 denotes the inverse

with respect to·) is an automorphism of the field (F'

+' . ) .

REMARKS.

(1) We know that every finite near-domain is a

near-field but we do not know at present whether there are near-domains which are not near-fields.

(2) With the ex-

ception of seven finite near-fields, all known near-fields are dicksonian near-fields. A pair (F, K) is called a normal near-field if Fis a near-field, Ka sub-near-field~ F, and K*

~

F* [1].

As H. Wahling [17] proved, K is even a field and (F, Kl a left vector space . Normal near-fields are used for the algebraic representation of desarguesian projective incidence groups [1], [4] .

For the description of desarguesian punctured affine

incidence groups, G. Kist [13] introduces the concept of a near-field extension: 120

A quadruple ((F, +, o), of a near-field (F, +, (F, +, o), a mapping• a+~

a

0 )

(K, +,

0 ),

·,

~)

consisting

a sub-near-field (K, +,

o)

of

: F x K + F, and a mapping~

in the symmetric group of the set K is

called a near-field extension if the following conditions are valid: (El)

• (K x K) =: K•K

c

K and (K, +, •) is a field such

that the identity 1 0 of (K,

0 )

and 1. of (K, •)

coincide. (E2)

( (F, +),

(E3)

For all a, b a

o

(b•>..)

(K, +, ·), ·) is a right vector space. €

= (a

F*, and for all O

A EK

holds

b)·~a(>..).

In [13] Kist proves the properties (3.1) of nearfield extensions and shows how one can get examples of near-field extensions. (3.1)

Let (F, K, ·, ~) be a near-field extension, then: (a) For every a E F* the mapping~

a

is an automor-

phism of (K, +, ·). (bl (3.2)

(K, +, o, •) is a dicksonian near-field.

Let (F, +,

every y, z E F,

o)

be a near-field, KF := {x E F: for

(y + z)x = yx + zx} the kernel (the kernel

of a near-field is always a field!),

•:=

0

IF x KF, and~

the function mapping each a E F* on the identity of SK. Then (F, KF, ·, ~) is a near-field extension.

121

F

The same

is true if His a subfield of KF. REMARK.

With (3.2) we can get examples of near-field

extensions from the Kalscheuer near-fields : be the quaternions over the reals IR, c

f ]H l X where

E

Let (Ill,+,

·)

IR*,

+ IlI +

e -ic log(a • a)

•X•e

ic log(a • a)

denotes the involutorial antiautornorphism

of (Ill,+,·)

(a·a is an element of m::= {r

E

1R:0 < r})

and IlI

1H

X

]H

➔

~ { a ob

(a, b)

~

0

if

a-/ 0,

if

a = 0.

Then (Ill,+, o) is a Kalscheuer near-field. (IH, +,

0 )

The kernel of

is the field of complex numbers(«:,+,

·)

(in

a: := 1R + iIR the multiplications o and • coincide). Therefore ((Ili, +, o), (1R,+,

o),

o, 1/J") with

(«:, +, o), o, 1/1') and ((Ii,+, o),

1/1~ := 1/Jal«:(=ida:) and 1/1; := 1/JallR

(=idlR) are near-field extensions. (3.3)

Let ( (F, +, 0) ,

.:

F

{ (a,

X

K

;q

(K, +, 0) ) be a normal near-field,

➔

F

➔

a•\ := \oa

and

122

1/!:

l

F*

....

SK

a

....

1/!a :

Then ( (F, +, 0) ,

:

{

....

K

....

>,_a = ao>.. oa -1

(K, +, 0) ,

.

is a near-field extension

ljl)

and ((K, +, • ) is an anti-isomorphic field to (K, +, REMARK.

0 )).

(3 . 3) shows that every normal near-field can be

considered as a near-field extension.

However there are

near-field extensions which are not normal near-fields. For instance, the Kalscheuer near-fields do not contain any sub-near-field K such that ((Ill,+, o),

(K, +,

0 ))

is

a normal near-field. Let (F, +, o, •) be a dicksonian near-field and

(3.4) (K,

+, o) be a sub-near-field such that a

all a E F* and all A EK . with

®

= . IF

X

K and ljJ : -- a

-1

Then ((F, +, o),

{

K

....

K

>,_

....

a

-1

• (ao>..)

EK for

(K, +, o),

@,

· (ao>..)

is a near-field extension. 4.

INCIDENCE GROUPS

An incidence group (G, G, ·) is called 2-sided if both

v1

and Vr are valid, and linearly fibred or a kinematic space if v 1 , Vr, and Vk are valid.

If the incidence structure

(G, G) is for instance a projective, affine, or slit space we call (P, G, •) a projective, affine, or slit incidence group.

123

ljl)

Examples of incidence groups 1.

Let (V, K) be a left vector space and G :={a+ Kb:

a, b EV, b i O}; then the pair (V, G) is an affine space and the pair (V, +)where+ is the vector addition is a commutative group. at: V

➔

V; x

➔ a+

Since

for each a EV the mapping

xis bijective and maps lines on lines,

(V, G, +) is a commutative affine incidence group. 2.

Let (A, K) be an associative algebra over the field

K, such that K is in the centre of A and let Ube the set of all units of A.

Now we can apply an affine and a pro-

jective derivation: (a) By the affine derivation, let G :={a+ bK: a, b EA, b i O} and (U, G) be the trace space (section u 2) belonging to the set of units.

Then (U, Gu' ·) is a

2-sided incidence group, since U is a group with respect to the multiplication given in the algebra A.

In the

special case that (A, K) is a division algebra, we have U = A* := A\{O} and (A*, GA, ·) is a punctured affine incidence group (as they will be discussed in section 5). (b) For the projective derivation, if A*/K* := {K*·x: x EA*} where A* := A\{O} and K* := K\{O}, let l

2

be the

set of all 2-dimensional vector subspaces of (A, K) and X :

{

A*

X ➔

➔

A*/K*

the canonical map.

K*x

Then (A*/K*, G) with G := {x(L*): LE l

124

2 }

is the projective

space corresponding to the vector space (A, K) and since K* ~Uthe set x(U)

= U/K* is a group.

If we again take

the trace space as incidence space, then (U/K*,G U/K*' ·) is a 2-sided incidence group. a division algebra, group.

In the case that (A, K) is

(A*/K*, G, •) is a projective incidence

We obtain examples of kinematic spaces from kine-

matic algebras [7]:

(A, K) is a kinematic algebra if for

each x EA the element x 2 belongs to the set K + Kx.

The

quaternions Ill over the reals :rn. constitute a kinematic division algebra and the corresponding kinematic space is the same, which is associated with the classical elliptic plane.

The kinematic space of the classical euclidean

plane can be derived from the four-dimensional algebra (A, m) with basis {l, i,

£,

id and i

2

=

-1, e: 2

=

O,

ie: = -e:i. The projective incidence groups are the most intensely studied ones [4].

The main theorems on projective inci-

dence groups are the following: (a)

(4 .1)

Let (F, K)

be a normal near-field (section 3),

G := F*/K* and G := {x(L*): LE L 2 } , then IT(F, K) (G, G,

:=

·) is a desarguesian projective incidence group

[ 1, 4 l •

(b)

Let (G, G, ·) be a desarguesian projective in-

cidence group, then there is exactly one normal nearfield (F, K) such that (G, G, •) and IT(F, K) are isomorphic [ 1, 4 l •

125

(4.2)

Let (G, G, •) be a desarguesian projective incidence

group and (F, K) the corresponding normal near-field; then (a)

(G, G, ·) is 2-sided if and only if F is a field

(b)

(G, G, •) is a commutative incidence group if

(16].

and only if Fis a commutative field [4]. (c)

(G, G, •) is a kinematic space if and only if

(F, K) is a kinematic division algebra.

If G is not com-

mutative, then (F, K) is a quaternion division algebra (this means that K is the centre of F and [F: Kl= 4).

If

G is commutative, then K is a commutative field of characteristic 2 and Fa pure inseparable extension of K (6]. Similar results on slit incidence groups can be found

in [11]. 5.

PUNCTURED AFFINE INCIDENCE GROUPS

The punctured affine incidence groups (G, G, •) - here (G, G) is a punctured affine space (section 2) - were recently studied by G. Kist in his dissertation [13].

He

got the following main results: (5.1)

(a) Let ( (F, +, o),

(K, +, o), o,

field extension (section 3) with K {a+ b•K: a, b

E

~

~2

F*, a J b·K} u {bK*: b

be a near-

ljl)

F

,

E

~

K, and G :=

F*}; then

(F*, G, o) is a desarguesian punctured affine incidence group with G1 ={a+ b•K: a, b

126

E

F*, a J b·K} and G2 =

{b•K*: b (b)

E

F*}. Let (G, G,

o)

be a desarguesian punctured affine

incidence group; then there exists exactly one near-field extension (F, K,

~) such that (G, G,

o)

and the inci-

dence group derived from the near-field extension are isomorphic and the groups (F*, REMARK.

o)

and (G,

o)

coincide.

In section 3 we have seen that every normal near-

field can be considered as a near-field extension and that there are near-field extensions which do not come from normal near-fields.

Those punctured affine incidence

groups which can be represented by normal near-fields or even by field extensions are characterized by G. Kist (13). 6.

2-INCIDENCE GROUPS

In this section we shall consider the following generalization of incidence groups, which are closely related to the punctured affine incidence groups: A tripel (P, G, f)

is called a 2-incidence group if

(1)

(P, G) is an incidence space.

(2)

(P, f)

is a sharply 2-transitive group (this

means that, for each two pairs (a 1 , a with a y(a

1

1)

(3)

'I a 2 , b 1

-:/

b2

,

bl' y(a 2

)

=

b

Each

y

E

r

2 ),

(b 1 , b 2 )

there is exactly one

y

E

E

P x P

r such that

2 •

is an automorphism of the incidence

space (P, G).

127

In this section, we say that two lines X, Y €Gare parallel, denoted by X II Y, if X = Y or if dim and X n Y =

¢.

XuY = 2

An automorphism a of (P, G) will be called

a dilatation if, for all

X

e G, a

(X)

II

X

and a translation

if a= 1 or if a is a dilatation without fixed points. To every 2-incidence group, there corresponds a usual incidence group: (6.1)

Let (P, G, r) be a 2-incidence group, a, e € P with

(:= the trace space {X a -'I e, p a := P\{a}, G : = GP a a r := {y e r: y(a) Xe G, Ix n P I ~ 2 ), and a}. a

P:

n

a

Then

a

ra operates regularly on Pa: hence, for each x € Pa there is exactly one tion

x era

such that x(e) = x.

The multiplica-

defined by {Pax Pa (x, y)

➔

p

a

➔ x(y>

makes P

a group, such that (P, G, •) is an incidence a -------- - - - - a a

(6.2)

Each desarguesian punctured affine incidence group

~-

(G, G, ·) can be extended to a 2-incidence group (P, G, r) such that for any a, e e P, a -,f e, the incidence group (Pa' Ga,

·) and (G, G, •) are isomorphic.

Let ((F, +, o},

-~K, +, o), ·, ~) be the near-field extension corresponding to (G, G, ·), G :={a+ boK: a, b € F, b -'IO} and r

:=

{[a, b]: a, be F, b -'IO} the sharply 2-transitive group consisting of all mappings

128

[a, bl + box,

G,

then (F,

= (F*,

G,

f) is a 2-incidence group extending (G, G, ·)

o).

For the proof, it is sufficient to remark that condi-

(E3)

tion

of a near-field extension implies that boK =

b•K. Let (P, G, f) be a 2-incidence group, KEG and

(6.3)

rK := {y Er: y(K) = K}.

Then (K, rK) is a sharply 2-tran-

sitive group. PROOF.

Let (a 1

,

a 2 ),

(b 1 , b 2 ) E K

x

K with a

b 2 , and y Er uniquely determined by y(a 1 ) b2 .

Then K = a

1,

I a 2, b 1 I

1

= b1,

y(a 2 )

=

a2 = bl, b2 and y(K) = y (al), Y (a2) =

bl, b2 = K, soy E r K

.

For the further discussion of 2-incidence groups, we need the following theorems

-

from the theory of sharply

2-transitive groups stated for instance in section 11 of

[ 51 : (6.4)

Let (F, +, •) be a near-domain; for each pair (a, b)

E F x F*, let [a, bl be the mapping [a, bl: F a+ b·x and r := {[a, bl: a, b E F, b IO}.

+

F; x

+

Then (F, f)

is a sharply 2-transitive group. (6.5)

Let (P, f) be a sharply 2-transitive group, J :=

129

{y

f:

E

y

1, y ;

2

l} and 0, 1

E

P two distinct points,

then: (a) a

E

For all p, q

=

J such that a(p) Let a, 8

(b)

E

E

P with p ; q, there is a unique

q and a(q)

J, a

i 8;

=

p.

then aS is free of fixed

points. (c)

If one a

J has a fixed point, then each

E

has a fixed point and to each point x ponds exactly one x

+

J

E

P, there corres-

E

J fixing x.

There exist two operations+: P

(d)

•: P x P

E

1

x

P

+

P and

P such that (P, +, •) is a near-domain (section

3); 0 (resp. 1) is the neutral element with regard to+ (resp.

•);

r

consists of all mappings [a, bl: P

a+ b•x with a, b each x

E

E

P and b

t 0

and

r0

~

+

P; x

(P*, ·).

For

+

P* := P \ {0}, let x+ be the involution interchang-

ing O and x; then+ is given by +A a

+ b :=

C

O (b)

if a

t

0

if a = O

or

i

if

a

if

a= 0

0

according as the involutions of J have fixed points or not. (e)

The near-domain (P, +, .) has the characteristic

2 (that means 1 + 1

=

0) if and only if the elements of J

have no fixed points.

130

By Theorem (6.5) the sharply 2-transitive groups (P, r) decompose into two classes depending on whether each involution

1

E J

has a fixed point or not.

A 2-in-

cidence group (P, G, r) is said to be of type 1 if each 1

E

J has a fixed point; otherwise it is of type 2.

From (6.3) and (6.5) we obtain: (6.6)

Let (P, G, r) be a 2-incidence group, 0, 1

distinct points, K : = ~ . a n d (P, +,

P two

E

•) the near-domain

associated with (P, r) whose neutral elements are O and 1. Then

r

consists of all mappings [a, bl

with a, b

E

P, b

:P

➔

P; x

➔

a+

b•X

I 0, K is a sub-near-domain, and each

line XE G can be represented in the form X =a+ b·K with a, b

E

P and b

I 0.

The last statement follows from the fact that

r

is

2-transitive on P and, hence, it is transitive on the set

G of all lines. Conversely we have: Let (F, +, ·) be a near-domain,

(6.7)

sub-near-domain of (F, +,

•),

(K, +,

•) a proper

(F, r) the corresponding

sharply 2-transitive group (see (6.4)) and G :={a+ b•K: a, b

E

F, b i O}.

PROOF.

= u + •v•b·K

Then (F, G, r) is a 2-incidence group.

Let X := a + b·K v(a + b·K) E

E

G

and [u, v)

E

r,

then [u, v) (X)

u + (v•a + v•b·K) = (u + v•a) + d

G (see section 3); this shows that

131

r

u,v•a

preserves

the structure (F, G).

Let a, b E F be two distinct points.

By section 3 we obtain a+ (-a+ b)•l =a+ (-a+ b) = 0 + d

a,-a

·b = b.

joining a and b.

Therefore X :=a+ (-a+ b) • K is a line By the sharp 2-transitivity of r, it is

enough to show that K is the unique line joining the points 0 and 1.

Let 0, 1 Ea+ b•K; then there is A EK such

that O =a+ bA.

Thus a= -b•A = b(-A) and a+ b•K = b(-A)

+ b·K = b• (-A+ K) = b•K.

Since 1 Ea+ b•K = b · K implies

the existence ofµ EK with 1 = b•µ, we see that b = µEK and therefore a+ b•K = b·K = µ- 1 •K = K.

1

Each line

a+ b•K E G contains (at least) the points a and a+ b. Therefore (F, G) is an incidence space and (F, G, r) a 2incidence group. By the Theorems (6.6) and (6.7) we see that there is bijective correspondence between all 2-incidence groups (P, G, f) and all pairs ( (F, +, • ) ,

(K, +, ·) ) of near-

domains, where K is a sub-near-domain of F.

We shall now

study this correspondence for the class of 2-incidence groups, where (P, G) is an affine space. First we need the following lemma: (6. 8)

Let (P, G, f) be a 2-incidence 9:roue of t:r:ee 1.

Then: (a) For all x E p and all

X

(b) For all x E p and all y E G with =

cp.

132

A

E G with x E x, x(X) A

X

J Y, x(Y)

x. n y

(c) For all x

E

P, the involution xis a dilatation.

PROOF.

(a) From (6.3),

rx> fore Jx

is a sharply 2-transitive group of type 1.

(X,

= {y: y

tions of (b)

rx

€

X}

(6.5), and (6.6) it follows that There-

(see (6.Sc)) is the set of involu-

and x(X) = X as required.

Suppose, if possible, that z

ix, ~(x, z) = x;-"z by (a), and x;-"z A

{z} so x(z) = z.

€

n

y n ~ (Y) ; then z

Y

= x, z n ~(Y)

=

Since x thus has the two distinct fixed

points x and z, the sharp 2-transitivity implies that~=

x

1, contradicting

E

J.

(c) The statement (c) is a consequence of (a) and (b), since by (a) any plane E containing xis fixed by (6. 9)

x.

Let (P' G, r) be an affine 2-incidence groue with 3 for x

IXI

~

(K'

+, 0)) are an associated eair of near-domains and

€

T := J.J =

G and dim P

{xy:

x, y

€

~

2.

sueeose ( (P = F, +, 0) '

p}

or T := J u { l}

according as (P, G, f) is of tyee 1 or of tyee 2.

Then:

(a) Tis a groue consisting of translations which oeerates regularly on the set of eoints and Tis isomorehic to (F, +), and

if (t', G, f) is either of type 1 or of type 2

dim P = 2. (b)

(F, +,

near-field, if

0 )

T

is a near-field and (K, +, o) a subis a group.

133

(c)

is a translation plane

If dim P = 2, then (P, G)

and [F: K] = 2 (that is, F = K + aK for a E F\K). (d)

If (P, G) is desarguesian and Ta group, then

there exist two mappings •: F x K that ((F, +,

0

),(K, +,

0

),

➔

F and~: F*

SK such

• , ~ ) i s a near-field extension.

We prove first the statements (a), incidence groups

➔

(c) for 2-

(b),

(P, G, f) of type 1.

Results (6.8c) and (6.Sc) imply that Tis a set conLet a, b E Pandy E J be the

sisting of translations.

involution interchanging a and b, so ya ET and ya(a) = b. Since by assumption (P, G) is an affine space of dim P

~

2,

this implies that Tis a commutative group and (P, G) a From (6.Sd), we obtain T

translation space. and therefore (F, +,

0

)

is a near-field.

Now let (P, G, f) be of type 2 and l

and X E G with

EJ

would imply

dim P = 2

because Hence

l

l (X)

{1(a)} = 1(X)

assumption, that

l

(P = F, +)

~

" X .

Then {a}= X

X = {a}

n

l

(X) II X

n

Let

l(X)

contradicting the

has no fixed points. we have

dim P =2.

Therefore and

for all

X E G .

is a translation.

Since any two distinct points can be interchanged by an involution,

(P, G) is a translation plane and T =Ju {l}

is the group of all translations. T

~

From (6.Sd) we derive

(F, +) and (F, +, o) is a near-field. It remains to prove statement (d) for both types.

Theorems (6.1) and (6.5), the incidence group (F*, G 0

134

,

By 0 ),

with F* = F 0 = F \ {O} and G0 : = {X \ {O}: XE G} corresponding to the points 0, 1

F, is a desarguesian punctured

E

affine incidence group and (F*,

o)

=r

0 •

Applying the

theorem (5.1) of Kist there is a near-field extension

((F,

$,

that I!>= Problem:

), o.

(K, +, ),

•,

1/J)

representing (F*, G 0

Statement (a) finally implies that$=+ .

(P, G, f)

Let

of type 2 with

dim P

be an affine 2-incidence group and

3

be a projective double space

Then there is a quaternion divisionr• algebra (F, K) such that (P, G, 11 i, 11 r) is derivable from 11

the kinematic space associated with (F, K)

(in the sense

of section 4, Example 2(b) or Theorem (4.1)).

140

(7.6)

Let (P, G, Iii, llr) be a projective double space

with

Iii=

llr and dim (P, G) = 3.

Then there is a com-

mutative field K of characteristic 2 and a pure inseparable extension field F of rank [F : Kl = 4 such that (P, G, Iii, llr) is derivable from the kinematic space associated with (F, K) •

REMARK.

Every pure inseparable extension field F of a

commutative field K of characteristic 2 produces examples of projective double spaces with

Iii=

llr•

In the meantime H.-J . Kroll has proved in [14a] that there are no other examples of projective double spaces II R. = II r

with

Are there any (projective) incidence groups

Problem: (G,

G,

•

.)

with

(that is, where

vi , Vk

which are not kinematic spaces Or in the algebraic

is not valid)?

V r

language for the projective case:

besides the kinematic

division algebras, are there normal near-fields such that element

[F, K]

x2

~

3

and, for each

is contained in the set

x

E

F\K,

(F, K) the

K + Kx ?

H. wahling has recently solved this problem for projective incidence-groups: implies

He proved that

vr •

141

Vi

and Vk

REFERENCES 1.

E. Ellers and H. Karzel. Kennzeichnung elliptischer Gruppenraume. Abh. Math. Sem. Univ. Hamburg 26 (1963): 55-77.

2.

Endliche Inzidenzgruppen. Hamburg 27 (1964): 250-264.

3.

H. Karzel. Verallgemeinerte elliptische Geometrien und ihre Gruppenraume. Abh. Math. Sem. Univ. Hamburg 24 (1960): 167-188.

4.

- - Bericht uber projektive Inzidenzgruppen. Deutsch. Math. Verein. 67 (1964): 58-92.

5.

Inzidenzgruppen I. Vorlesungsausarbeitung von I. Pieper und K. Sorensen, Hamburg (1965).

ti.

Zweiseitige Inzidenzgruppen . Univ. Hamburg 29 (1965): 118-136 .

7.

Kinematic spaces. Istituto Nazionale di Alta Matematica, Symposia Matematica 7 (1973): 413-439.

8.

H. Karzel and H.-J. Kroll. Eine inzidenzgeometrische Kennzeichnung projektiver kinematischer Raume. Arch. Math. 26 (1975): 107-112 .

9.

H. Karzel, H.-J. Kroll, and K. Sorensen. Invariante Gruppenpartitionen und Doppelraume. J. Reine Angew . Math. 262/263 (1973): 153-157.

Abh. Math. Sem. Univ.

Jber.

Abh . Math. Sem .

10.

Projektive Doppelraume. 206-209.

11.

H. Karzel and I. Pieper. Bericht iiber geschlitzte Inzidenzgruppen. Jber. Deutsch. Math.-Verein. 72 (1970): 70-114. H. Karzel, K. Sorensen, and D. Windelberg. Einfilhrung in die Geometrie. UTB 184, Gottingen (1973).

12. 13.

Arch. Math. 25 (1974):

G. Kist. Punktiert-affine Inzidenzgruppen und Fastkorpererweiterungen. Abh. Math. Sem. Univ. Hamburg 44 (1975).

Zur Struktur geschlitzter Doppelraume. H.-J. Kroll. J. Geometry 5 (1974): 27-38. 14a. H.-J. Kroll. Bestimmung aller projektiven Doppelraume. Appearing in Abh. Math. Sem. Univ. Hamburg. 15. H. Meissner. Geschlitzte Gruppenraume. Abh. Math. Sem. Univ. Hamburg 32 (1968): 160-185. 14.

142

16.

H. Wahling. Darstellung zweiseitiger Inzidenzgruppen durch Divisionsalgebren. Abh. Math . Sern . Univ. Hamburg 30 (1967): 220-240.

17.

Invariante und vertauschbare Teilfastkorper. Abh. Math. Sern. Univ. Hamburg 33 (1969): 197-202.

143

TOPOLOGICAL HJELMSLEV PLANES

J.W. Lorimer

Hjelmslev planes are generalizations of customary projective and affine planes where two distinct points may be joined by more than one line and two distinct lines may meet in more than one point.

Two points P and Qare neighbours

(P ~lP Q) when they possess more than one joining line. Two lines land mare neighbours (projectively) if they intersect in more than one point and are neighbours (affinely)

if each point incident with either l or m possesses

a neighbour point incident with the other of l or m.

In

these cases we write l ~L m. An incidence structure H

=




is said to be

a topological Hjelmslev plane if His a Hjelmslev plane (cf.

[2],

[4]) with the following properties:

(i)

lP and Lare topological spaces.

(ii)

lP and

~L are closed sets of IP x lP and

L x L

respectively. (iii) The joining of two non-neighbouring points is a continuous function; and the meeting of two lines with

144

a unique intersection point is a continuous function. Moreover if the plane is affine we also assume the additional property: (iv)

L(P, g), the unique line through P parallel to

g, is a continuous function of P and g. Examples of such planes can be constructed over certain topological rings.

A local ring with its unique

maximal ideal equal to its set of two-sided zero divisors is an H-ring if for any two elements a and bat least one is a multiple (left and right) of the other (cf.

(2)).

Such a ring is a topological H-ring if it is a topological ring whose units form a multiplicative topological group. The subring Dn of the (n x n) matrices over a topological division ring F consisting of matrices of the form (dij) where dij

=

O if i

H-ring (cf.

>

j and di,i+k-l

=

dl,k is a topological

[ 1)) •

In fact from the constructions in (2),

(3), and (4)

we have the following result. THEOREM 1.

Every desarguesian Hjelmslev plane can be co-

ordinatized by a topological H-ring and conversely. If His a Hjelmslev plane, then the "quotient geometry" H/~

=




-1

=

l.

f: (X, Ri)

+

(Y, Si}

(i EI) is a mor-

Y is a function and for all i (f(x), f(x'))

EI

E R .• l.

we shall refer to R(2i)iE{l} as R(2) and call it the category of graphs.

The elements of R . are called i1.

(coloured)-arrows.

We shall say "f preserves R." if f: l.

x + y is a morphism from (X, Ri) to (Y, Si) in R(2).

A

loop of R. is a point x such that (x, x) l.

E Ri' an isolated

point of R . is a point such that y(x, y) l.

I Rl.. and (y, x}

I Rl.. ; a two-cycle in Rl.. is a pair (x, y) such that (x, y) and (y, x) E R. Aut(X, R.) . I = {fl f and f-1 E Hom (X, Ri)iEI' l. l.E (X, R.). I)}; l. l. E

(X, Ri)iEI is rigid if Hom (X, R . ) . 1 (X, R.). I l. 1.E l. l.E

176

Stab (

'.: : 1 • X

=

X

X,y

{f I f Aut(X, R. ) . If(x)

) ( X, R. ) . I i

iE

i

iE

f(y) = y}.

THEOREM 5 (Cayley-Frucht).

Let G be a group and (X, R) g gE: G

be a graph with IGI colours defined by X = IGI and (g 1

,

g2)

LEMMA 1.

Let G be a group.

Then there exists a multi-

coloured graph (X, R.) . I such that i

(i)

iE:

Aut(X, R.). I " G. i iE:

(ii) For N < G (including N = 1 and N = G) E:

(x, y)

Ri(N) such that Stab(x,y)((X, Ri)iEI) "N .

CONSTRUCTION.

Let I= {a, b, c}U{G x N} where N

(N = 1 and N = G included)}.

{N I NLIG

Let

G G X ={NI N < G} u• {Hg I Hg€ H' H < G},

Ra= {a rigid graph on{~ IN< G}}, G Ng) Rb= {(N,

Rc

I

G Ng E: N},

{(Ng, Kh) INg ~ Kh and N ~ K},

R(N,g) = {(Nh, Nh') LEMMA 2 .

I Nh

=

Nh'g}.

If G is a multicoloured graph, then there exists

a graph G1 with the same automorphism group, and the effect of stabilizing an arrow of G can be achieved by stabilizing an arrow of G 1 [4]. THEOREM 6 .

If G is a group, then there exists a graph

(X, R) such that A(X, R) "G, and if a~ ~o is a cardinal IXI = a!GI

(31,

(41.

177

THEOREM 7.

If G., i

-

l.

€

I is a well-ordered sequence of

groups, then there exist graphs (Xi, Ri) with X. l. R.

C

l.

+

,

X., J

and A(X . , Ri) " G. [ 8] • Rj, i < j , l. l.

THEOREM 8. (X 1

C

If G 1 , G 2 are groups then there exist graphs

R 1 ) and (X 2

,

R 2 ), and an arrow-onto map~:

(X 1 , R 1 )

(X2, R2) such that A(XI, RI) "Gl and A(X2, R2) "G2 (5). Furthermore by a technique found in [4] and [8] one

can assume that all graphs which appear in the above theorems have no loops, isolated points, or two-cycles, and that each point is related to at least two others.

These

assumptions are for technical reasons which will reveal themselves in the particular construction and proofs we shall use. RESULTS NEEDED FROM PROJECTIVE GEOMETRY We shall need the following well-known results from projective geometry: THEOREM 9.

The collineation group of Pi=oo is isomorphic

to the subgroup of the collineation group stabilizing i THEOREM 10.

[7].

Given a partial plane p there exists a free

completion of this plane to a projective plane [2]. THEOREM lL

If a partial plane is confined (i . e. contains

at least three points on each line and at least three lines

178

through every point) then the free completion has the same collineation group as the partial plane and furthermore the isomorphism is given by restriction [2]. The configuration V The configuration Vis given

by the following table where

the columns are lines and the entries points: A

C

E

A

B

D

D

C

E

A

B

B

N

0

F

K

N

0

K

M

K

N

C

L

L

G

L

K

M

G

G

0

0

D

H

F

H

M

F

H

E H

FIGURE 1 LEMMA 3 .

V has no collineations other than the identity

and is confined. The pasting of Von a graph v,

The process we shall use is analogous to the sip process [8].

Let G

=

(X, R) be a graph without loops, isolated

points, or two-cycles and with each point connected to at least two others.

We define the configuration (G) in the

179

following way: The points of V(G): P(G) = {[{PD - {K, O}} The lines of

X

R] u

x}.

V(G): L(G)

The incidence of V(G): IG by: (i)

(p, r) IG (.Q,, s) r

=

s and p ID

s) s

=

(x, y), K ID

(iii) y IG (.Q,, s) s

=

(y I

(ii)

X

IG

( _Q,

I

z) ,

.Q,;

.Q,;

0 ID .Q,.

Intuitively, we place a copy of Vin place of every arrow of R pasting the points Kand/or O together whenever they correspond.

If we represent the configuration Vas in

figure 2 and paste it on the graph of figure 3 we get the configuration of figure 4.

D

0

K

FIGURE 3

FIGURE 2

THEOREM 12. If G = (X, R) is a graph such that G has no loops or isolated points, and every point is connected to at least two others, then: 1.

V(G) is a confined configuration.

2.

There is an isomorphism between the collineation 180

D

FIGURE 4 group of V(G) and the automorphism group of the graph G. 3.

There is a one-one correspondence between onto

maps from G to G* (where G* also has the properties given in the hypothesis) and onto homomorphism from V(G) to V(G*). 4.

Stabilizing an arrow of G corresponds to stabiliz-

ing a line of V(G). Theorems 1-4 now follow from the analogous theorems for graphs and considering the plane F(V(G)). REFERENCES l. 2. 3. 4.

R. Frucht. Herstellung von Graphen mit vorgegeben abstrakter Gruppe. Composite Math. 6 (1938): 239-250. M. Hall. Projective planes. Trans. Amer. Math. Soc. 54 (1943): 229-277. z. Hedrlin, A. Pultr, and P. Vopenka. A rigid relation exists on any set. Comment. Math. V. Carolinae 6 (1965): 149-155. Z. Hedrlin and J. Lambek. How comprehensive is the category of semigroups? J. Algebra 11, No. 2 (1969): 195-212.

181

5. 6. 7. 8.

z. Hedrlin. On endomorphisms of graphs and their homomorphic images . In: Proof Techniques in Graph Theory, edited by F . Harary. Academic Press (1969). D.R. Hughes. On homomorphism of projective planes. Proc. Symp. Appl. Math. 10 (1960): 45-52. R. Lingenberg. Grundlagen der Geometry 1. Bibliographische Institute, Zurich (1969). E. Mendelsohn. On a technique for representing semigroups as endomorphism semigroups of graphs with given properties. Semigroup Forum 4 (1972): 283-294.

182

RECENT ADVANCES IN FINITE TRANSLATION PLANES

T.G. Ostrom

I.

GROUPS GENERATED BY GIVEN TYPES OF COLLINEATIONS

The work on rank-three affine planes, reported on by Luneburg at this same conference, is certainly a phase of the "recent advances" but I shall make no further mention of it here.

Most of what I have to say was reported on at a

Conference on Projective Planes held at Washington State University.

Copies of the Proceedings may be obtained

from the Washington State University Press.

My report in

those Proceedings contains what I hope is a complete listing of known finite translation planes and some of the properties of the collineation groups.

Since then I be-

lieve that Rao has some more examples of flag transitive planes.

Prohaska and Walker have an example, not yet pub-

lished, of case (e) in the theorem I shall soon be stating. I believe that the list is complete with these additions. Some of what I shall talk about is in my "Finite Translation Planes."

183

If

TI

is a translation plane of dimension rover its

kernel K, then the (affine) points of

TI

can be identified

with the elements of a 2r-dimensional vector space over the field K; the lines through the zero vector Oare rdimensional vector subspaces and the scalar transformations are the homologies with centre O and axis t 00 •

The system

of lines through the origin is called a spread. Although some of what we have to say does not require finiteness, we shall assume that everything is finite . "Point" always means "affine point" unless we say "point at infinity." K

=

GF(q) .

The letter q will denote the order of K:

The stabilizer of O in the collineation group

is a group of semilinear transformations - a subgroup of fL(2r, q).

We call it the translation complement and its

intersection with GL(2r, q) the linear translation complement.

The characteristic of K will be denoted by p. The first result, I am told, is implied by some un-

published work of John Thompson.

In the following form

it is due in part to Hering and in part to myself. THEOREM.

Suppose that the translation complement contains

(affine) elations.

Let S be the subgroup of the transla-

tion complement which is generated by all of the elations. Then one of the following holds: (a)

Sis an elementary abelian p-group and (if Sis

non-trivial) only one line through O is the axis of a nontrivial elation .

184

(b)

S

~

s

SL(2, p) for some s.

The net whose lines

through the origin are the axes of elations can be embedded

r

in a desarguesian affine plane of order q .

= 2 and S

s

(c)

p

(d)

S contains a normal subgroup N of odd order and

~

Sz(2) for some s.

index two in S. (e)

p = 3 ands~ SL(2, 5).

We have a desarguesian

net as in case (b) . Details of the proof are given in [7],

[13], and [18] .

The first part is concerned with the group generated by two elations with different axes.

It depends upon the

fact that the group generated by two matrices

lo a)1

r1

over a field of characteristic p

~

[i ~)

and

s

3 is SL(2, p ),

where GF(ps) is the smallest subfield containing a .

A re-

presentation can be used so that two elations with different axis can be represented by p 1

=

lrrr 1o)

and P 2

__ (ro

A 1 ),

where I is an r by r identity matrix and A is an r by r submatrix with an eigenvalue a. Let K' be an extension of the field K which contains the eigenvalue a of A and let We dimension 2r over K'.

w be

a vector space of

If we think of the elements of Was

ordered r-tuples of elements of K' then the vector space Ve V obtained by restricting the r-tuples to elements of K can be identified with the one in which the plane represented.

Furthermore