Geometry of Sporadic Groups II: Representations and Amalgams (Encyclopedia of Mathematics and its Applications 91)

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

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Volume 91

Geometry of Sporadic Groups II Representations and amalgams

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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Geometry of Sporadic Groups II Representations and Amalgams

A. A. IVANOV Imperial College, London

S. V. SHPECTOROV Bowling Green State University, Ohio

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 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Cambridge University Press 2002 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 2002 Printed in the United Kingdom at the University Press, Cambridge Typeface Monotype Times 10/12pt

System MJBX

[UPH]

A catalogue record of this book is available from the British Library ISBN 0 521 62349 9 hardback

Contents

Preface 1 1.1 1.2 1.3 1.4 1.5

Preliminaries Geometries and diagrams Coverings of geometries Amalgams of groups Simple connectedness via universal completion Representations of geometries

page ix 1 1 3 5 7 11

Part I. Representations 2 General features 2.1 Terminology and notation 2.2 Collinearity graph 2.3 Geometric hyperplanes 2.4 Odd order subgroups 2.5 Cayley graphs 2.6 Higher ranks 2.7 c-extensions 2.8 Non-split extensions

17 19 19 22 24 27 32 34 35 40

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

43 43 45 47 50 51 55 56 64

Classical geometries Linear groups The Grassmanian 8P\ is uniserial 0(S4(2)) Symplectic groups Orthogonal groups Brouwer's conjecture 0(3 • S4(2))

vi

Contents 9{Alt5) 9(3®* • S2n(2))

66 68

4 4.1 4.2 4.3 4.4 4.5 4.6

Mathieu groups and Held group ^(M 23 )

76 76 77 81 82 88 92

5 5.1 5.2 5.3 5.4 5.5 5.6

Conway groups Leech lattice $(Co2) 9(Cox) Abelianization 0(3 2 3 • Co2) 3(3 • C/4(3))

93 93 97 99 101 103 108

6 6.1 6.2 6.3 6.4 6.5 6.6

Involution geometries General methods J(Alt7) S(M22) •(1/4(3)) S(C2,2B) S{Cou2A)

111 111 115 117 120 122 125

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Large sporadics Existence of the representations A reduction via simple connectedness The structure of N(p) Identifying Ri(p) Ri{p) is normal in R[p\ R\_p\ is isomorphic to G(p) Generation of G(p) n G(q) Reconstructing the rank 3 amalgam ^(3 4 3 7 1 • BM)

128 128 131 134 141 146 151 153 155 159

3.9 3.10