http://dx.doi.org/10.1090/surv/040.3 Selected Titles in This Series 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya? and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, 1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici, Operator theory and arithmetic in H°°, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W. Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 (Continued in the back of this publication) The Classification of the Finite Simple Groups, Number 3 MATHEMATICAL Surveys and Monographs Volume 40, Number 3 The Classification of the Finite Simple Groups, Number 3 Part I, Chapter A: Almost Simple X- Groups Daniel Gorenstein Richard Lyons Ronald Solomon American Mathematical Society Providence, Rhode Island The authors were supported in part by NSF grant #DMS 94-01852 and by NSA grant #MDA-904-95-H-1048. 1991 Mathematics Subject Classification. Primary 20D06, 20D08; Secondary 20D05, 20E32, 20G40. ABSTRACT. This volume examines in detail the internal structure of the finite simple groups of Lie type, the finite alternating groups and the twenty-six sporadic finite simple groups, as well as their almost simple analogues. Emphasis is on the structure of local subgroups and their relationships with one another. The foundation is laid for the development of the many specific properties of 3C-groups to be used in the inductive proof of the classification theorem. Library of Congress Cataloging-in-Publication Data ISBN 0-8218-0391-3 (number 3) ISBN 0-8218-0390-5 (number 2) The first volume was catalogued as follows: Gorenstein, Daniel. The classification of the finite simple groups / Daniel Gorenstein, Richard Lyons, Ronald Solomon. p. cm. — (Mathematical surveys and monographs; v. 40, number 1-) Includes bibliographical references and index. ISBN 0-8218-0334-4 [number 1] 1. Finite simple groups. I. Lyons, Richard, 1945- . II. Solomon, Ronald. III. Title. IV. Series: Mathematical surveys and monographs; no. 40, pt. 1-. QA177.G67 1994 512/.2-dc20 94-23001 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. © 1998 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98 For Lisa and Myriam Contents List of Tables xi Preface xiii PART I, CHAPTER A: ALMOST SIMPLE K-GROUPS Chapter 1. Some Theory of Linear Algebraic Groups 1 1.1. Fundamental Notions 2 1.2. Jordan Decomposition 4 1.3. Unipotent Groups; One-Parameter Groups 5 1.4. Tori 6 1.5. Solvable Algebraic Groups 6 1.6. Borel Subgroups 7 1.7. Radical; Reductive, Semisimple and Simple Groups 7 1.8. Abstract Root Systems 8 1.9. Weights, Roots and Root Subgroups in Reductive Groups 13 1.10. Classification of Semisimple Groups; Isogenics and Versions 15 1.11. £iV-Structure 16 1.12. Chevalley Generators and Relations 17 1.13. Parabolic Subgroups 22 1.14. Weights and Representations 23 1.15. Isomorphisms, Automorphisms and Endomorphisms 27 1.16. Notes 30 Chapter 2. The Finite Groups of Lie Type 31 2.1. Steinberg Endomorphisms and Lang's Theorem 31 2.2. The Finite Groups of Lie Type 36 2.3. Bruhat Structure 40 2.4. Chevalley Relations for Finite Groups of Lie Type 45 2.5. Automorphisms 53 2.6. Subsystem Subgroups and Parabolic Subgroups 63 2.7. The Classical Groups 67 2.8. Equicharacteristic Representations 73 2.9. Presentations; Curtis-Tits Theorem 79 2.10. Standard Notation 89 Chapter 3. Local Subgroups of Groups of Lie Type, I 91 3.1. A Theorem of Borel and Tits 91 3.2. Structure of Parabolic Subgroups 95 3.3. Equicharacteristic Sylow Structure 106 x CONTENTS Chapter 4. Local Subgroups of Groups of Lie Type, II 119 4.1. Semisimple Automorphisms of Algebraic Groups 119 4.2. Semisimple Automorphisms of Finite Groups of Lie Type 134 4.3. Centralizers of Involutions in Auto (if): Statements 143 4.4. Centralizers of Involutions in Auto(K): Proofs 153 4.5. Centralizers of Involutions in Groups in £jie(r), r Odd: Statements 170 4.6. Centralizers of Involutions in Groups in £ie(r), r Odd: Proofs 186 4.7. Inner-Diagonal and Graph Automorphisms of Exceptional Odd Prime Orders 204 4.8. Some Other p-Local Subgroups 219 4.9. The Remaining Types of Non-Inner Automorphisms 227 4.10. Cross-Characteristic Sylow Structure 236 Chapter 5. The Alternating Groups and the Twenty-Six Sporadic Groups 249 5.1. Schur Multipliers and Coverings of Simple Groups 249 5.2. The Alternating Groups 252 5.3. Structure of the Twenty-Six Sporadic Groups 259 5.4. Justification of the Tables: The Simple Case 288 5.5. Justification of the Tables: The Non-Simple Case 294 5.6. Ranks of Sporadic Groups 302 Chapter 6. Coverings and Embeddings of Quasisimple DC-Groups 311 6.1. The Schur Multipliers and Covering Groups of the Simple X-Groups 311 6.2. Some Embeddings of Quasisimple Groups 313 6.3. The Action of Out (if) on M(K) 317 6.4. Splitting and Stable Involutions 324 6.5. Subgroups of Small Simple Groups 328 Chapter 7. General Properties of X-Groups 335 7.1. Verification of the Fundamental X-Group Properties 335 7.2. Generation of Semisimple Groups: Generalities 343 7.3. Generation for Groups in Qhev: Statements of Main Results 346 7.4. Generation for Groups in Qhev: Proofs 349 7.5. Generation for Groups in Alt U Spor 380 7.6. Strongly p-Embedded Subgroups and Proper fc-Generated Cores 383 7.7. Balance and Signalizers 385 7.8. Fusion 399 Background References 403 Expository References 404 Errata for Numbers 1 and 2 409 Glossary 410 Index 414 List of Tables Chapter 1 1.8. Irreducible reduced crystallographic root systems 12 1.12.5. Centers of universal Chevalley groups 19 1.12.6. Generators of Z(K) 20 1.15.2d. Graph automorphisms and their centralizers 28 Chapter 2 2.2. Orders of the finite groups of Lie type 39 2.3.2d. Twisted root systems 42 2.4. Structure of root groups X& 46 2.4.7.