Twelve Sporadic Groups
The finite simple groups are basic objects in algebra since many questions about general finite groups can be reduced to questions about the simple groups. Finite simple groups occur naturally in certain infinite families, but not so for all of them: the
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Springer-Verlag Berlin Heidelberg GmbH
Robert L. Griess, Jr.
Twelve Sporadic Groups
Springer
Robert L. Griess, lr. Department of Mathematics University of Michigan 2072 East HaU Ann Arbor, MI 48109-1109 USA e-mail: [email protected]
Llbrary of Congress Cataloglng-ln-Publlcatlon Data Gr1ess. Robert L.• 1945Twelve sporad1c groups / Robert Gr1ess. Jr. p. cm. -- (Spr1nger monographs 1n mathemat1CSJ Includes blbltographtcal references and lndex. 1. F1nlte slmple groups. OAI77.G75 1998 512' .2--dc21
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Tttle.
II. Serles. 98-18097 CIP
Mathematics Subject Classification (1991 ):20B20, 20B25, 20D05, 20E32, 20F29
ISBN 978-3-642-08305-1 DOI 10.1007/978-3-662-03516-0
ISBN 978-3-662-03516-0 (eBook)
This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted on1y under the provisions of the German Copyright Law of September 9, 1965, in its current venion, and permission for use must aiways be obtained from Springer. Verlag. Violations are liable for prosecution under the German Copyright Law.
e Springer-Verlag Berlin Heidelberg 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1998 Softcover reprint ofthe hardcover Ist edition 1998 Typesetting: Data conversion by Adam Leinz,Karlsruhe SPIN 10570780 41/3143-543210 - Printed on acid-free paper
Table of Contents
Introduction
1
Acknowledgements
3
Notation for Simple Groups
.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 1. Background from General Group Theory
.................
4 5
Chapter 2. Assumed Results about Particular Groups
11
Chapter 3. Codes
24
Chapter 4. The Hexacode
30
Chapter 5. The Golay Code
36
Chapter 6. Subgroups of M
24
54
Chapter 7. The Ternary Golay Code and 2·M 12
76
Chapter 8. Lattices
88
Chapter 9. The Leech Lattice and Conway Groups
95
Chapter 10. Subgroups of the Conway Groups; the Simple Groups of Higman-Sims, McLaughlin, Hall-Janko and Suzuki; Local Subgroups; Conjugacy Classes .............................. Appendix lOA. D. G. Higman's Theory of Rank 3 Permutation Groups Appendix lOB. Classification Theorems ......................... Appendix lOC. Tables of Stabilizers ............................ Appendix lOD. Maximal Subgroups ............................ Appendix lOE. Nick Patterson's Thesis ......................... Appendix lOF. Character Tables ...............................
104 123 125 126 127 127 129
Chapter 11. Generation Three of the Happy Family and the Pariahs Appendix. Some Comments on the Atlas ........................
146 153
References
154
Index
161
List of Group Theoretic Notations
168
The Orders of the Finite Simple Groups
169
Introduction
The finite simple groups are basic objects in algebra since many questions about general finit
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