http://dx.doi.org/10.1090/surv/040.4 Selected Titles in This Series 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second Edition, 1999 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 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 Preese, 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 40.4 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 4, 1999 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 (Continued in the back of this publication) The Classification of the Finite Simple Groups, Number 4 MATHEMATICAL Surveys and Monographs Volume 40, Number 4 The Classification of the Finite Simple Groups, Number 4 Part II, Chapters 1-4: Uniqueness Theorems Daniel Gorenstein Richard Lyons Ronald Solomon AttEMjJ/ | American Mathematical Society Providence, Rhode Island Editorial Board Georgia M. Benkart Tudor Stefan Ratiu, Chair Peter Landweber Michael Renardy The authors were supported in part by NSF grant #DMS 97-01253. 1991 Mathematics Subject Classification. Primary 20D05; Secondary 20B20. ABSTRACT. In this volume several basic uniqueness theorems underlying the classification of the finite simple groups are proved for a minimal counterexample G to the classification; these include and generalize the Bender-Suzuki Strongly Embedded Theorem and the Aschbacher Component Theorem. Library of Congress Cataloging-in-Publication Data ISBN 0-8218-1379-X (number 4) ISBN 0-8218-1391-3 (number 3) ISBN 0-8218-1390-5 (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]. © 1999 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 04 03 02 01 00 99 To the memory of Michio Suzuki Contents Preface xiii PART II, CHAPTERS 1-4: UNIQUENESS THEOREMS Chapter 1. General Lemmas 1 1. Uniqueness Elements 1 2. Permutation Groups 3 3. Two Theorems on Doubly Transitive Groups 7 4. Miscellaneous Lemmas 12 Chapter 2. Strongly Embedded Subgroups and Related Conditions on Involutions 19 1. Introduction and Statement of Results 19 2. The Subsidiary Theorems 25 3. The Basic Setup and Counting Arguments 27 4. Reduction to the Simple Case: Theorem 1 33 5. p-Subgroups Fixing Two or More Points: Theorem 2 38 6. A Reduction: Corollary 3 42 7. Double Transitivity and Ia^z: Theorem 4 44 8. Reductions for Theorem SE 50 9. Good Subgroups of V Exist 54 10. The Structures of V and D 58 11. Proof of Theorem 6 (and Theorem SE) 62 12. The Group L = (z,t, [02>(D),t]) 65 13. The Unitary Case 72 14. The Suzuki Case 74 15. The Linear Case 79 16. The Structure of D and Cx(u) 82 17. The Proof of Theorem ZD 88 18. The Weak 2-Generated 2-Core 91 19. The J2 Case 95 20. Bender Groups as 2-Components 98 21. Theorem SU: The 2-Central Case 101 22. The Ji Case 103 23. The 2A9 Case 105 24. Theorem SA: Strongly Closed Abelian 2-Subgroups 109 25. Theorem SF: Terminal Bender Components 112 26. Theorem SF: Product Disconnection 119 27. Theorem ,49 127 x CONTENTS Chapter 3. p-Component Uniqueness Theorems 131 1. Introduction 131 2. Theorem PUi: The Subsidiary Results 137 3. Exceptional Triples and 3C-Group Lemmas 138 4. M-Exceptional Subgroups of G 142 5. Centralizers of Elements of XP(K*) 147 6. The Proofs of Theorems 1 and 2 150 7. Theorem 3: The Nonsimple Case 153 8. Theorem 3: The Simple Case 163 9. Theorem 4: Preliminaries 165 10. Normalizers of p-Subgroups of K 168 11. The Case K £ S£(p) 171 12. The S£{p) case 172 13. Theorem 5: The M-Exceptional Case 179 14. The Residual M-Exceptional Cases 188 15. Completion of the Proof of Theorem PUi: The p-Rank 1 Case 193 16. Reductions to Theorem PUi 194 17. Further Reductions: The Non-Normal Case 201 18. Theorem PU2: The Setup 206 19. The Case r < p: A Reduction 208 20. The Case r = 1: Conclusion 211 21. The Case r = p = 2: A. Reduction 214 22. The Case r = p = 2, ~K ^ L2(q) 216 23. Theorem PU3: The Simple Case 220 24. The Residual Simple Cases 223 25. The Nonsimple Case 224 26. Theorem PU4 226 27. Corollaries PU2 and PU4 227 28. Aschbacher's Reciprocity Theorem 228 29. Theorem PU5 232 Chapter 4. Properties of K-Groups 237 1. Automorphisms 237 2. Schur Multipliers and Covering Groups 238 3. Bender Groups 245 4. Groups of Low 2-Rank 253 5. Groups of Low p-Rank, p Odd 256 6. Centralizers of Elements of Prime Order 257 7. Sylow 2-Subgroups of Specified DC-Groups 268 8. Disconnected Groups 271 9. Strongly Closed Abelian Subgroups 276 10. Generation 283 11. Generation and Terminal Components 293 12. Preuniqueness Subgroups and Generation 302 13. 2-Constrained Groups 328 14. Miscellaneous Results 331 Background References 333 Expository References 334 CONTENTS xi Errata for Number 3, Chapter 1^: Almost Simple X-Groups 335 Glossary 338 Index of Terminology 340 Preface With three preliminary volumes under our belt, we at last turn directly to the proof of the Classification Theorem; that is, we begin the analysis of a minimal counterexample G to the theorem. Thus G is a DC-proper simple group. Specifically, the heart of this volume provides a collection of uniqueness and pre-uniqueness theorems for G, proved in Chapters 2 and 3. Chapter 2 is primarily dedicated to the proof of a fundamental 2-uniqueness theorem of Michael Aschbacher, which we call Theorem ZD. Historically there is a sequence of basic papers leading up to Theorem ZD. First is a fundamental paper of Michio Suzuki [Su4] on 2-transitive permutation groups; the result of that paper is one of our Background Results [PI, Part II]. Building on that, Helmut Bender [Be3] established the Strongly Embedded Subgroup Theorem, which we call Theorem SE.