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 W. Miller, Jr. Symmetry and separation of variables 6 H. Mine Permanents 11 W. B. Jones and W. J. Thron Continued fractions 12 N. F. G. Martin and J. W. England Mathematical theory of entropy 18 H. O. Fattorini The Cauchy problem 19 G. G. Lorentz, K. Jetter and S. D. Riemenschneider Birkhoff interpolation 21 W. T. Tutte Graph theory 22 J. R. Bastida Field extensions and Galois theory 23 J. R. Cannon The one-dimensional heat equation 25 A. Salomaa Computation and automata 26 N. White (ed.) Theory ofmatroids 27 N. H. Bingham, C. M. Goldie and J. L. Teugels Regular variation 28 P. P. Petrushev and V. A. Popov Rational approximation of real functions 29 N. White (ed.) Combinatorial geometries 30 M. Pohst and H. Zassenhaus Algorithmic algebraic number theory 31 J. Aczel and J. Dhombres Functional equations containing several variables 32 M. Kuczma, B. Chozewski and R. Ger Iterative functional equations 33 R. V. Ambartzumian Factorization calculus and geometric probability 34 G. Gripenberg, S.-O. Londen and O. Staffans Volterra integral and functional equations 35 G. Gasper and M. Rahman Basic hypergeometric series 36 E. Torgersen Comparison of statistical experiments 37 A. Neumaier Intervals methods for systems of equations 38 N. Korneichuk Exact constants in approximation theory 39 R. A. Brualdi and H. J. Ryser Combinatorial matrix theory 40 N. White (ed.) Matroid applications 41 S. Sakai Operator algebras in dynamical systems 42 W. Hodges Model theory 43 H. Stahl and V. Totik General orthogonal polynomials 44 R. Schneider Convex bodies 45 G. Da Prato and J. Zabczyk Stochastic equations in infinite dimensions 46 A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler Oriented matroids 47 E. A. Edgar and L. Sucheston Stopping times and directed processes 48 C. Sims Computation with finitely presented groups 49 T. Palmer Banach algebras and the general theory of *-algebras 50 F. Borceux Handbook of categorical algebra I 51 F. Borceux Handbook of categorical algebra II 52 F. Borceux Handbook of categorical algebra III 54 A. Katok and B. Hassleblatt Introduction to the modern theory of dynamical systems 55 V. N. Sachkov Combinatorial methods in discrete mathematics 56 V. N. Sachkov Probabilistic methods in discrete mathematics 57 P. M. Cohn Skew Fields 58 Richard J. Gardner Geometric tomography 59 George A. Baker, Jr. and Peter Graves-Morris Pade approximants 60 Jan Krajicek Bounded arithmetic, propositional logic, and complex theory 61 H. Gromer Geometric applications of Fourier series and spherical harmonics 62 H. O. Fattorini Infinite dimensional optimization and control theory 63 A. C. Thompson Minkowski geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative matrices and applications 65 K. Engel Sperner theory 66 D. Cvetkovic, P. Rowlinson and S. Simic Eigenspaces of graphs 67 F. Bergeron, G. Labelle and P. Leroux Combinatorial species and tree-like structures 68 R. Goodman and N. Wallach Representations of the classical groups 70 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 Introduction 1 1.1 Basic definitions 2 1.2 Morphisms of geometries 5 1.3 Amalgams 7 1.4 Geometrical amalgams 9 1.5 Universal completions and covers 10 1.6 Tits geometries 1 1.7 ^4/£7-geometry 16 1.8 Symplectic geometries over GF(2) 17 1.9 From clasical to sporadic geometries 19 1.10 The main results 21 1.1 Representations of geometries 23 1.12 The stages of clasification 26 1.13 Consequences and development 3 1.14 Terminology and notation 42 2 Mathieu groups 49 2.1 The Golay code 50 2.2 Constructing a Golay code 51 2.3 The Steiner system 5(5,8,24) 53 2.4 Linear groups 56 2.5 The quad of order (2,2) 59 2.6 The rank 2 T-geometry 62 2.7 The projective plane of order 4 64 2.8 Uniquenes of 5(5,8,24) 71 2.9 Large Mathieu groups 74 2.10 Some further subgroups of Mat24 76 vi Contents 2.11 Little Mathieu groups 81 2.12 Fixed points of a 3-element 85 2.13 Some odd order subgroups in Mat24 87 2.14 Involutions in Mat^ 90 2.15 Golay code and Todd modules 95 2.16 The quad of order (3,9) 97 3 Geometry of Mathieu groups 100 3.1 Extensions of planes 101 3.2 Maximal parabolic geometry of MatiA 102 3.3 Minimal parabolic geometry of Mat24 106 3.4 Petersen geometries of the Mathieu groups 112 3.5 The universal cover of ^(Mat2i) 117 3.6 &(Mat23) is 2-simply connected 122 3.7 Diagrams for Jf (Mat24) 124 3.8 More on Golay code and Todd modules 130 3.9 Diagrams for Jf(Mat22) 132 3.10 Actions on the sextets 138 4 Conway groups 141 4.1 Lattices and codes 141 4.2 Some automorphisms of lattices 147 4.3 The uniqueness of the Leech lattice 150 4.4 Coordinates for Leech vectors 153 4.5 Cou C02 and C03 158 4.6 The action of Co\ on A4 160 4.7 The Leech graph 163 4.8 The centralizer of an involution 169 4.9 Geometries of Co\ and C02 173 4.10 The affine Leech graph 178 4.11 The diagram of A 189 4.12 The simple connectedness of ^{Co2) and ^{Co\) 193 4.13 McL geometry 198 4.14 Geometries of 3 • 1/4(3) 203 5 The Monster 210 5.1 Basic properties 211 5.2 The tilde geometry of the Monster 216 5.3 The maximal parabolic geometry 218 5.4 Towards the Baby Monster 222 2 5.5 £6(2)-subgeometry 224 5.6 Towards the Fischer group M(24) 227 Contents vi 5.7 Identifying M(24) 231 5.8 Fischer groups and their properties 236 5.9 Geometry of the Held group 242 5.10 The Baby Monster graph 244 5.11 The simple connectedness of &(BM) 256 5.12 The second Monster graph 259 5.13 Uniqueness of the Monster amalgam 265 5.14 On existence and uniqueness of the Monster 268 5.15 The simple connectedness of &(M) 271 6 From Cn- to Tn-geometries 272 6.1 On induced modules 273 6.2 A characterization of 0(3 • Sp4(2)) 276 6.3 Dual polar graphs 280 6.4 Embedding the symplectic amalgam 285 6.5 Constructing T -geometries 288 6.6 The rank 3 case 290 6.7 Identification of J(n) 293 6.8 A special class of subgroups in J(n) 295 6.9 The f(ri) are 2-simply connected 297 6.10 A characterization of f(ri) 301 6.11 No tilde analogues of the ^/tygeometry 303 7 2-Covers of P -geometries 307 7.1 On P -geometries 307 7.2 A sufficient condition 313 7.3 Non-split extensions 315 23 7.4 ^(3 • Co2) 318 7.5 The rank 5 case: bounding the kernel 321 4371 7.6 ^(3 • BM) 327 7.7 Some further s-coverings 330 8 7-groups 332 8.1 Some history 333 8.2 The 26-node theorem 335 8.3 From Y-groups to 7-graphs 337 8.4 Some orthogonal groups 340 8.5 Fischer groups as 7-groups 345 8.6 The monsters 351 9 Locally projective graphs 358 9.1 Groups acting on graphs 359 viii Contents 9.2 Clasical examples 362 9.3 Localy projective lines 367 9.4 Main types 370 9.5 Geometrical subgraphs 374 9.6 Further properties of geometrical subgraphs 379 9.7 The structure of P 383 9.8 Complete families of geometrical subgraphs 386 9.9 Graphs of smal girth 389 9.10 Projective geometries 392 9.1 Petersen geometries 394 Bibliography 398 Index 406 Preface Sporadic simple groups are the most fascinating objects in modern alge- bra. The discovery of these groups and especially of the Monster is considered to be one of the most important contributions of the classi- fication of finite simple groups to mathematics. Some of the sporadic simple groups were originally realized as automorphism groups of cer- tain combinatorial-geometrical structures like Steiner systems, distance- regular 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". Build- ings off inite groups 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 res- pect to the maximal parabolic subgroups which are maximal over- groups of the normalizer in G of a Sylow p-subgroup (this normalizer is known as the Borel subgroup). Thus ^(G) can be defined in abs- tract 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 sim- ple 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 at- tracted 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 char- acterize 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 <$ let res(x) denote the subgraph in ^ induced on the set of vertices adjacent to x. For xt e <S\ 1 < i < 3, the following hold: is the incidence graph of vertices and edges of the Petersen graph