Geometry of Sporadic Groups: Volume 1, Petersen and Tilde Geometries (Encyclopedia of Mathematics and its Applications) (v. 1)

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

EDITED BY G.-C ROTA Editorial Board R. S. Doran, M. Ismail, T.-Y. Lam, E. Lutwak, R. Spigler Volume 76

Geometry of Sporadic Groups I Petersen and tilde geometries

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 4 6 11 12 18 19 21 22 23 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 70

W. Miller, Jr. Symmetry and separation of variables H. Mine Permanents W. B. Jones and W. J. Thron Continued fractions N. F. G. Martin and J. W. England Mathematical theory of entropy H. O. Fattorini The Cauchy problem G. G. Lorentz, K. Jetter and S. D. Riemenschneider Birkhoff interpolation W. T. Tutte Graph theory J. R. Bastida Field extensions and Galois theory J. R. Cannon The one-dimensional heat equation A. Salomaa Computation and automata N. White (ed.) Theory ofmatroids N. H. Bingham, C. M. Goldie and J. L. Teugels Regular variation P. P. Petrushev and V. A. Popov Rational approximation of real functions N. White (ed.) Combinatorial geometries M. Pohst and H. Zassenhaus Algorithmic algebraic number theory J. Aczel and J. Dhombres Functional equations containing several variables M. Kuczma, B. Chozewski and R. Ger Iterative functional equations R. V. Ambartzumian Factorization calculus and geometric probability G. Gripenberg, S.-O. Londen and O. Staffans Volterra integral and functional equations G. Gasper and M. Rahman Basic hypergeometric series E. Torgersen Comparison of statistical experiments A. Neumaier Intervals methods for systems of equations N. Korneichuk Exact constants in approximation theory R. A. Brualdi and H. J. Ryser Combinatorial matrix theory N. White (ed.) Matroid applications S. Sakai Operator algebras in dynamical systems W. Hodges Model theory H. Stahl and V. Totik General orthogonal polynomials R. Schneider Convex bodies G. Da Prato and J. Zabczyk Stochastic equations in infinite dimensions A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler Oriented matroids E. A. Edgar and L. Sucheston Stopping times and directed processes C. Sims Computation with finitely presented groups T. Palmer Banach algebras and the general theory of *-algebras F. Borceux Handbook of categorical algebra I F. Borceux Handbook of categorical algebra II F. Borceux Handbook of categorical algebra III A. Katok and B. Hassleblatt Introduction to the modern theory of dynamical systems V. N. Sachkov Combinatorial methods in discrete mathematics V. N. Sachkov Probabilistic methods in discrete mathematics P. M. Cohn Skew Fields Richard J. Gardner Geometric tomography George A. Baker, Jr. and Peter Graves-Morris Pade approximants Jan Krajicek Bounded arithmetic, propositional logic, and complex theory H. Gromer Geometric applications of Fourier series and spherical harmonics H. O. Fattorini Infinite dimensional optimization and control theory A. C. Thompson Minkowski geometry R. B. Bapat and T. E. S. Raghavan Nonnegative matrices and applications K. Engel Sperner theory D. Cvetkovic, P. Rowlinson and S. Simic Eigenspaces of graphs F. Bergeron, G. Labelle and P. Leroux Combinatorial species and tree-like structures R. Goodman and N. Wallach Representations of the classical groups A. Pietsch and J. Wenzel Orthonormal systems and Banach space geomery

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Geometry of Sporadic Groups I Petersen and tilde geometries

A. A. IVANOV Imperial College, London

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge, CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain © Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 A catalogue record of this book is available from the British Library Library of Congress Cataloguing in Publication data Ivanov, A. A. Geometry of sporadic groups 1, Petersen and tilde geometries / A.A. Ivanov. p. cm. Includes bibliographical references and index. Contents: v. 1. Petersen and tilde geometries ISBN 0 521 41362 1 (v. 1 : hb) 1. Sporadic groups (Mathematics). I. Title. QA177.I93 1999 512'.2-dc21 98-45455 CIP ISBN 0 521 41362 1 hardback Transferred to digital printing 2002

Contents

Preface

page ix

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14

Introduction Basic definitions Morphisms of geometries Amalgams Geometrical amalgams Universal completions and covers Tits geometries ^4/£7-geometry Symplectic geometries over GF(2) From classical to sporadic geometries The main results Representations of geometries The stages of classification Consequences and development Terminology and notation

1 2 5 7 9 10 11 16 17 19 21 23 26 33 42

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Mathieu groups The Golay code Constructing a Golay code The Steiner system 5(5,8,24) Linear groups The quad of order (2,2) The rank 2 T-geometry The projective plane of order 4 Uniqueness of 5(5,8,24) Large Mathieu groups Some further subgroups of Mat24

49 50 51 53 56 59 62 64 71 74 76

vi

Contents Little Mathieu groups Fixed points of a 3-element Some odd order subgroups in Mat24 Involutions in Mat^ Golay code and Todd modules The quad of order (3,9)

81 85 87 90 95 97

Geometry of Mathieu groups Extensions of planes Maximal parabolic geometry of MatiA Minimal parabolic geometry of Mat24 Petersen geometries of the Mathieu groups The universal cover of ^(Mat2i) &(Mat23) is 2-simply connected Diagrams for Jf (Mat24) More on Golay code and Todd modules Diagrams for Jf(Mat22) 3.10 Actions on the sextets

100 101 102 106 112 117 122 124 130 132 138

2.11 2.12 2.13 2.14 2.15 2.16 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

4.10 4.11 4.12 4.13 4.14

Conway groups Lattices and codes Some automorphisms of lattices The uniqueness of the Leech lattice Coordinates for Leech vectors Cou C02 and C03 The action of Co\ on A4 The Leech graph The centralizer of an involution Geometries of Co\ and C02 The affine Leech graph The diagram of A The simple connectedness of ^{Co2) and ^{Co\) McL geometry Geometries of 3 • 1/4(3)

141 141 147 150 153 158 160 163 169 173 178 189 193 198 203

5 5.1 5.2 5.3 5.4 5.5 5.6

The Monster Basic properties The tilde geometry of the Monster The maximal parabolic geometry Towards the Baby Monster 2 £6(2)-subgeometry Towards the Fischer group M(24)

210 211 216 218 222 224 227

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Contents 5.7 5.8 5.9

5.10 5.11 5.12 5.13 5.14 5.15

Identifying M(24) Fischer groups and their properties Geometry of the Held group The Baby Monster graph The simple connectedness of &(BM) The second Monster graph Uniqueness of the Monster amalgam On existence and uniqueness of the Monster The simple connectedness of &(M)

From Cn- to Tn-geometries On induced modules A characterization of 0(3 • Sp4(2)) Dual polar graphs Embedding the symplectic amalgam Constructing T -geometries The rank 3 case Identification of J(n) A special class of subgroups in J(n) The f(ri) are 2-simply connected 6.10 A characterization of f(ri) 6.11 No tilde analogues of the ^/tygeometry

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

vii 231 236 242 244 256 259 265 268 271 272 273 276 280 285 288 290 293 295 297 301 303

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7

2-Covers of P -geometries On P -geometries A sufficient condition Non-split extensions ^(3 23 • Co2) The rank 5 case: bounding the kernel ^(3 4371 • BM) Some further s-coverings

307 307 313 315 318 321 327 330

8 8.1 8.2 8.3 8.4 8.5 8.6

7-groups Some history The 26-node theorem From Y-groups to 7-graphs Some orthogonal groups Fischer groups as 7-groups The monsters

332 333 335 337 340 345 351

9 9.1

Locally projective graphs Groups acting on graphs

358 359

viii 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

Contents Classical examples Locally projective lines Main types Geometrical subgraphs Further properties of geometrical subgraphs The structure of P Complete families of geometrical subgraphs Graphs of small girth Projective geometries Petersen geometries

Bibliography Index

362 367 370 374 379 383 386 389 392 394 398 406

Preface

Sporadic simple groups are the most fascinating objects in modern algebra. The discovery of these groups and especially of the Monster is considered to be one of the most important contributions of the classification of finite simple groups to mathematics. Some of the sporadic simple groups were originally realized as automorphism groups of certain combinatorial-geometrical structures like Steiner systems, distanceregular graphs, Fischer spaces etc., but it was the epoch-making paper [Bue79] by F. Buekenhout which brought an axiomatic foundation for these and related structures under the name "diagram geometries". Buildings offinitegroups of Lie type form a special class of diagram geometries known as Tits geometries. This gives a hope that diagram geometries might serve as a background for a uniform treatment of all finite simple groups. If G is a finite group of Lie type in characteristic p, then its Tits geometry ^(G) can be constructed as the coset geometry with respect to the maximal parabolic subgroups which are maximal overgroups of the normalizer in G of a Sylow p-subgroup (this normalizer is known as the Borel subgroup). Thus ^(G) can be defined in abstract group-theoretical terms. Similar abstract construction applied to sporadic simple groups led to maximal [RSm80] and minimal [RSt84] parabolic geometries, most naturally associated with the sporadic simple groups. Notice that besides the parabolic geometries there are a number of other nice diagram geometries associated with sporadic groups. Tits geometries are characterized by the property that all their rank 2 residues are generalized polygons. Geometries of sporadic groups besides the generalized polygons involve c-geometries (which are geometries of vertices and edges of complete graphs), the geometry of the Petersen IX

x

Preface

graph, tilde geometry (a triple cover of the generalized quadrangle of order (2,2)) and a few other rank 2 residues. In the mid 80's the classification project of finite Tits geometries attracted a lot of interest, motivated particularly by the revision program of the classification of finite simple groups (see [Tim84]). It was natural to extend this project to geometries of sporadic groups and to try to characterize such geometries by their diagrams. For two classes of diagrams, namely

and

the complete classification under the flag-transitivity assumption was achieved by S.V. Shpectorov and the author of the present volume [ISh94b]. Geometries with the above diagrams are called, respectively, Petersen and tilde geometries. A complete self-contained exposition of the classification of flag-transitive Petersen and tilde geometries is the main goal of the two volume monograph of which the present is the first volume. To provide the reader with an idea what sporadic group geometries look like we present the axioms for the smallest case. A Petersen geometry of rank 3 is a 3-partite graph ^ with the partition

which possesses the following properties. For a vertex x e ^ of the element set of Jf7 into the element set of ^ which maps incident pairs of elements onto incident pairs and preserves the type function. A bijective morphism is called an isomorphism. A surjective morphism cp : J f -» ^ is said to be a covering of ^ if for every non-empty flag Q> of J f the restriction of cp to the residue res^ ( ^ is a covering and §? is simply connected, then xp is the universal covering and 9 is the universal cover of 9 of arbitrary incidence systems is called an s-covering if it is an isomorphism when restricted to every residue of rank at least s. This means that if O is a flag whose cotype is less than or equal to s, then the restriction of cp to res^(<E) is an isomorphism. An incidence system, every s-cover of which is an isomorphism, is said to be s-simply connected. The universal s-cover of a geometry exists in the class

6

Introduction

of incidence systems and it might or might not be a geometry. In the present work we will mainly use the notion of s-covers either to deal with concrete morphisms of geometries or to establish s-simple connectedness. For these purposes we can stay within the class of geometries. It must be clear that in the case s = n — 1 "s-covering" and "covering" mean the same thing. An isomorphism of a geometry onto itself is called an automorphism. By the definition an isomorphism preserves the types. Sometimes we will need a more general type of automorphisms which permute types. We will refer to them as diagram automorphisms. The set of all automorphisms of a geometry ^ obviously forms a group called the automorphism group of ^ and denoted by A u t ^ . An automorphism group G of ^ (that is a subgroup of Aut ^ ) is said to be flag-transitive if any two flags Q>i and Q>2 in ^ of the same type (that is with t(Q>\) = t(Q>2)) are in the same G-orbit. Clearly an automorphism group is flag-transitive if and only if it acts transitively on the set of maximal flags in ^ is a 2-covering of geometries then y and ^ have the same diagram. It turns out that many properties of ^(G) can be deduced from its diagram and in many cases the diagram of ^(G) (including the indices) specifies ^(G) up to isomorphism. Without going into details, this important and beautiful topic can be summarized as follows. Existence in G of the Weyl group W as a section imposes on ^ an additional structure known as a building. The buildings of spherical type (i.e. with underlying geometries having spherical diagrams) were classified in [Ti74] by showing that they are exactly the parabolic geometries of finite groups of Lie type. Later in [Ti82] it was shown that under certain additional conditions the structure of a building can be deduced directly from the condition that all rank 2 residues are generalized polygons. That is the following result was established.

Theorem 1.6.3 Let & be a Tits geometry of rank n>2. Then & is covered by a building if and only if every rank 3 residue in & having diagram

is covered by a building.

•

1.6 Tits geometries

15

We should emphasize again that in view of the main result of [Ti74] the buildings of spherical type are exactly the parabolic geometries of finite groups of Lie type. We formulate another important result from [Ti82]. Theorem 1.6.4 Every building of rank at least 3 is 2-simply connected.

•

If G is a finite group of Lie type then its parabolic geometry is a building and hence it is 2-simply connected by the above theorem. In view of (1.5.3) this means that G is the universal completion of the amalgam of rank 2 parabolic subgroups associated with the action of G on ^(G). This reflects the fact that G is defined by its Steinberg presentation. In fact every Steinberg generator is contained in one of the minimal parabolic subgroups associated with a given maximal flag and for every Steinberg relation the generators involved in the relation are contained in a parabolic subgroup of rank at most 2. Thus the Steinberg presentation is in fact a presentation for the universal completion of the amalgam of rank 2 parabolics. The last important topic we are going to discuss in this section is the flag-transitive automorphism groups of parabolic geometries of groups of Lie type. Let G be a Lie type group in characteristic p, & = &(G) be the parabolic geometry of G, B be the Borel subgroup and U — OP(B). An automorphism group H of ^ is said to be classical if it contains the normal closure UG of U in G. In this case if G is non-abelian, then H contains the commutator subgroup of G. The following fundamental result [Sei73] (see Section 9.4.5 in [Pasi94] for the corrected version) shows that up to a few exceptions the flag-transitive automorphism groups of classical geometries are classical. Theorem 1.6.5 Let & be the parabolic geometry of a finite group of Lie type of rank at least 2 and H be a flag-transitive automorphism group of &. Then either H is classical or one of the following holds: (i) (ii) (iii) (iv)

& is the projective plane over GF(2) and H = Frob]; & is the projective plane over GF(S) and H = Frob913); ^ is the ^-dimensional projective GF(2)-space and H = Alt-]; & is the generalized quadrangle of order (2,2) associated with Sp4(2) and H^Alt6; (v) & is the generalized quadrangle of order (3,3) associated with and H is one of24 : Alt5, 2 4 : Sym5 and 2 4 : Frob45;

16

Introduction

(vi) & is the generalized quadrangle of order (3,9) associated with 1/4(3) and H is one o/L 3 (4).2 2 ; L4(3).23 and L 3 (4).2 2 ; (vii) ^ is £/ie generalized hexagon of order (2,2) associated with G2(2) and H S G2(2)' ^ t/ 3 (3); (viii) ^ is £/ze generalized octagon of order (2,4) associated with 2 F 4 (2) and H ^ 2 F 4 (2)' (the Tits group). D 1.7

Let us discuss the exceptional rank 3 residues from (1.6.3). By (1.6.2) there do not exist any thick generalized 5-gons, so as long as we are interested in thick locally finite Tits geometries we should not worry about the H3 -residues. On the other hand there exists a thick flagtransitive C3-geometry which is not covered by a building. This geometry was discovered and published independently in [A84] and [Neu84] and can be described as follows. Let Q be a set of size 7 and G = Alt-? be the alternating group of Q. Let n be a projective plane of order 2 having Q as set of points. This means that n is a collection of seven 3-element subsets of Q such that any two of the subsets have exactly one element in common. Let G\ = Alte be the stabilizer in G of an element a e Q. Let G2 be the stabilizer in G of a line of n containing a, so that G2 = (Sym3 x Sym^f where the superscript indicates that we take the index 2 subgroup of even permutations. Finally let G3 be the stabilizer of n in G, so that G3 = L3(2) is the automorphism group of n. Let s/ = {Gi,G2,G3} and 9 = ^ ( G , J / ) . Then ^ is a Tits geometry with the following diagram: C3(2) : o 31

'

2

o 2

2

If {xi,x2jx3} is a maximal flag in ^ where x,- is of type i then res#(x3) is canonically isomorphic to n9 res^(x2) is the complete bipartite graph K^3 and res^(xi) is the (unique) generalized quadrangle of order (2,2) associated with Sp4(2) on which G\ = Alts acts flag-transitively (1.6.5 (iv)). Notice that G, n Gj = Sym4 for 1 < i < j < 3 and B ^ D8. The C3-geometry ^(Alt-j) was characterized in [A84] by the following result (see also [Tim84], p. 237). Theorem 1.7.1 Let l, over GF{2) and let *F be a non-singular symplectic form on V. If {v\,...,v\,v\,...,vfy is a basis of V then up to equivalence *F can be chosen to be

(here and elsewhere ij) is a generalized digon. If i = j — 1 and j < n then ^f ,• and ffl) correspond to all 1- and 2dimensional subspaces in the 3-dimensional GF(2)-space F)+i/K/_2 and

18

Introduction

is the projective plane of order 2. Finally, if i = n — 1 and j = n then Jf t and Jtfj correspond to 1- and 2-dimensional subspaces in the 4-dimensional space V^_2/Vn-2 that are totally singular with respect to the symplectic form on this space induced by *F. In this case res^( 2; (WP2) for 1 < i < j < n the intersection B := P{ n P, is a 2-group, which is independent of the particular choice of i and j ; (WP3) Pi/O2(Pi) ^ Syms for 1 < 1 < n; (WP4) if P o = (P,-,P;-> for 1 < i < j < n and Qtj = O2(Pij) then B is a Sylow 2-subgroup of Ptj and Sym3 x 5^m3 if j — i> 1, L3(2) ifi = j-landj 4 in which the residual rank 3 EDPS's are all isomorphic to £(U4(2)\ so that 5 = t = 2. Let A be the graph on the set of elements of type 2 in $ in which two elements are adjacent if they are incident to a common element of type 3 but not to a common element of type 1. Then every connected component of A is the derived graph of a Petersen geometry of rank n — 1. In this way the EDPS's *(Af(22)), /(3-M(22)) and *(Af(23)) are related to P-geometries 2(M(24)). It is worth mentioning that there is a geometry j f (3 • M(24)) with diagram o 2

o 2

-o 2

o 2

possessing a morphism onto J4?(M(24)) whose simple connectedness was also established in [Iv95]. (C) The EDPS S(M) of the Monster group was constructed in [BF83] and [RSt84]. The simple connectedness proof for S(M) in [IMe97] reduces the problem to the simple connectedness of the T5-geometry of the Monster. That is, we show that the universal completion U of the amalgam of maximal parabolic subgroups corresponding to the action of M on S(M) contains subgroups C, N and L as in (1.10.4) which implies that U = M. It was noted in [Iv96] that 3-local and 2-local parabolics in the Monster are related via a subgeometry Jf (224 • Co\) in S(M) with diagram

acted on flag-transitively by C/Z(C). In [IMe97] it was shown that the universal cover of S(M) also contains Jf (224 • Co\) as a subgeometry. At this stage we applied the main result of [ISt96] which proves the simple connectedness of the EDPS ^(M(24)) which is also a subgeometry

42

Introduction

in S(M). It is worth mentioning that the geometry ^f(224 • Co\) is not simply connected since it possesses a double cover [ISh98] (which might or might not be universal). 1.14 Terminology and notation

In this section we fix our terminology and notation concerning groups and their actions on graphs and geometries. Let G be a group. Then 1 is the identity element, G# = G \ {1}, Z(G) is the centre, G — [G, G] is the commutator subgroup and Aut G is the automorphism group of G. Let g be an element in G (written g € G) and if be a subgroup of H (written H < G). Then (g) is the subgroup of G generated by g, CG(g) and CG(H) are the centralizers in G of g and if, NG(H) is the normalizer of H in G. If NG(H)/CG(H) s Aut If then H is said to be fully normalized in G. By G/H we denote the set of right cosets of H in G. In the case where H is normal in G (written H < G), by G/H we also denote the corresponding factor group. The core of H in G, denoted by coreG(H) is the largest normal subgroup of G contained in H, so that coreG(H) is the intersection of the conjugates of H in G. The smallest normal subgroup in G which contains H is the normal closure of if in G. Let p and g be different primes. Then OP(G) is the largest normal subgroup in G which is a p-group (i.e. whose order is pk for some fe), Op,). In Section 2.3 we show that a minimal non-empty subset in a Golay code has size 8 (called an octad). Moreover the set of all octads in a Golay code forms the block set of a Steiner system of type S(5,8,24). The residue of a 3-element subset of elements in a Steiner system of type 5(5,8,24) is a projective plane of order 4. In Section 2.4 we review some basic properties of the linear groups and in Sections 2.5 and 2 6 we define the generalized quadrangle of order (2,2) and its triple cover which is the tilde geometry of rank 2. In Section 2.7 we prove uniqueness of the projective plane of order 4 and analyse some properties of the plane and its automorphism group. This analysis enables us to establish the uniqueness of the Steiner system of type S(5,8,24) in Section 2.8. The Mathieu group Mat24 of degree 24 is defined in Section 2.9 as the automorphism group of the unique Golay code. The uniqueness proof implies rather detailed information about Mat24 and two other large Mathieu groups Mat^ and Mat22- In Section 2.10 we study the stabilizers in Mat24 of an octad, a trio and a sextet. In Section 2.11, analysing dodecads in the Golay code and their stabilizers in Mat24, we introduce the little Mathieu groups. In Sections 2.12, 2.13 and 2.14 we classify the subgroups in Mat24 of order 2 and 3 and determine octads, trios and sextets stabilized by such a subgroup. In Section 2.15 we study the action of the large Mathieu groups on the Golay code and on its cocode. Finally in Section 2.16 we describe the generalized quadrangle of order (3,9) in terms of the projective plane of order 4. 49

50

Mathieu groups 2.1 The Golay code

Let X be a finite set of elements. A binary linear code %> based on X is a subspace of the power set 2X (considered as a GF(2)-vector-space). In general a linear code over GF(q) is a triple (V,X,^) where V is a GF(g)-vector-space, X is a basis of V and ^ is a subspace in F. It is obvious that in the case q = 2 this definition is equivalent to the above one. In what follows unless explicitly stated otherwise, when talking about codes we always mean binary linear codes. The size of X is called the length of a code ^ based on X. A code is even (respectively doubly even) if the number of elements in every non-empty subset in ^ is even (respectively divisible by 4). The minimal weight of # is the number of elements in a smallest non-empty subset in ^ . The dual code , and the elementwise stabilizer in L of every such subset is trivial; (iv) the setwise stabilizer in L of a 3-element subset of 8P is cyclic of order 3 acting fixed-point freely on 0*.

Proof. Since L(oo) acts transitively on 9 \ {oo} and T does not fix oo we have (i) and (ii). By the double transitivity every L-orbit on 3-element subsets of 0 contains a triple {oo,0,a} for some a G GF(23)*. Under the action of S the triples of this shape split into two orbits depending on whether a e Q or a e N. Since T stabilizes {oo, 0} and permutes Q and JV, these two orbits are fused and we obtain (iii). Finally (iv) follows from (iii) and the order of L. • For a e GF(23) put Na = {n + a \ n e N} U {a} and let Jf = {Na \ a e GF(23)}. Lemma 2.2.3 (i) L(oo) preserves Jf as a whole and acts on Jf as it acts on GF(23)f in particular L(oo) acts transitively on the set of unordered pairs of subsets in Jf'; (ii) every element b 6 GF(23) is contained in exactly 12 subsets from (iii) any two subsets from Jf have intersection of size 6. Proof. It is straightforward that Nla = iVa+i and N^ = N2a, which imply (i). Since L(oo) acts transitively on GF(23) and preserves Jf, the number of subsets in Jf containing a given element b e GF(23) is independent of the choice of b and (ii) follows. Now counting in two ways the number of configurations (a, {A, B}) where a e GF(23), A, B e JT and a G A n B, we obtain (iii). • Let ^ be the code based on 0 generated by Jf and the whole set 0. Lemma 2.2.4 ^ is stable under L. Proof. By (2.2.3 (i)) # is stable under L(oo). Hence by (2.2.2) it is sufficient to show that ^ is stable under T. Clearly 0>x = 0 and N% = 0>ANo. Now one can check directly or consult [MS77], p. 492 for a general argument that for a ^ 0 we have Nl =

N_1/aANoA0>.

2.3 The Steiner system S(5,8,24)

53

Lemma 2.2.5 (i) (ii) (iii) (iv)

i = °> 1>2>3 and 4, given by (2.3.2), we have the following. Lemma 2.3.3 * ^ is the disjoint union of the into six subsets S{ = S, S2,..., S6 from 0>4 such that St U Sj € * /or 1 < 1 < j < 6. • By the above lemma the minimal weight of a Golay code is exactly 8. A subset of size 8 in a Golay code will be called an octad. A partition of SP into six 4-element subsets such that the union of any two is an octad will be called a sextet. The elements from ^ 4 will be called tetrads. In these terms by (2.3.3) every tetrad is a member of a unique sextet. Lemma 2.3.4 Every element F e ^ 5 is contained in a unique octad. Proof. Let S be a tetrad contained in F and {Si = S, S2,..., S^} be the unique sextet containing S. Let {x} = F \ S and let j , 2 < j < 6, be such that x e Sj. Then 0 = Si U Sj is the octad containing F. If there were

23 The Steiner system S(5,8,24)

55

another octad Or containing F then OAO' would be a non-empty subset in #12 of size at most 6, which is impossible. • Definition 2.3.5 Let t, k, v be integers with 1