Mathematical Surveys and Monographs Volume 172 The Classifi cation of Finite Simple Groups Groups of Characteristic 2 Type Michael Aschbacher Richard Lyons Stephen D. Smith Ronald Solomon American Mathematical Society surv-172-smith3-cov.indd 1 2/4/11 1:15 PM http://dx.doi.org/10.1090/surv/172 The Classification of Finite Simple Groups Mathematical Surveys and Monographs Volume 172 The Classification of Finite Simple Groups Groups of Characteristic 2 Type Michael Aschbacher Richard Lyons Stephen D. Smith Ronald Solomon American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 20D05; Secondary 20C20. Abstract. We complete an outline, aimed at the non-expert reader, of the original proof of the Classification of the Finite Simple Groups. The first half of such an outline, namely Volume 1 covering groups of noncharacteristic 2 type, had been published much earlier by Daniel Gorenstein in his very detailed 1983 work [Gor83]. Thus the present book, which we regard as “Volume 2” of that project, aims at presenting a reasonably detailed outline of the second half of the Classification: namely the treatment of groups of characteristic 2 type. Aschbacher was supported in part by NSF DMS 0504852 and subsequent grants. Lyons was supported in part by NSF DMS 0401132, NSA H98230-07-1-0003, and subsequent grants. Smith was supported in part by NSA H98230-05-1-0075 and subsequent grants. Solomon was supported in part by NSF DMS 0400533, NSA H98230-07-1-0014, and subsequent grants. For additional information and updates on this book, visit www.ams.org/bookpages/surv-172 Library of Congress Cataloging-in-Publication Data The classification of finite simple groups : groups of characteristic 2 type / Michael Aschbacher ... [et al.]. p. cm. — (Mathematical surveys and monographs ; v. 172) Includes bibliographical references and index. ISBN 978-0-8218-5336-8 (alk. paper) 1. Finite simple groups. 2. Representations of groups. I. Aschbacher, Michael, 1944– QA177.C53 2011 512.2—dc22 2010048011 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 is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 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 http://www.ams.org/ 10987654321 161514131211 To the memory of Danny Gorenstein Contents Preface xi Background and overview 1 Chapter 0. Introduction 3 0.1. The Classification Theorem 3 0.2. Principle I: Recognition via local subgroups 4 0.3. Principle II: Restricted structure of local subgroups 7 0.4. The finite simple groups 16 0.5. The Classification grid 19 Chapter 1. Overview: The classification of groups of Gorenstein-Walter type 25 The Main Theorem for groups of Gorenstein-Walter type 25 1.1. A strategy based on components in centralizers 26 1.2. The Odd Order Theorem 28 1.3. (Level 1) The Strongly Embedded Theorem and the Dichotomy Theorem 29 1.4. The 2-Rank 2 Theorem 33 1.5. (Level 1) The Sectional 2-Rank 4 Theorem and the 2-Generated Core Theorem 35 1.6. The B-Conjecture and the Standard Component Theorem 41 1.7. The Unbalanced Group Theorem, the 2An-Theorem, and the Classical Involution Theorem 44 1.8. Finishing the Unbalanced Group Theorem and the B-Theorem 48 1.9. The Odd Standard Component Theorem and the Aschbacher-Seitz reduction 53 1.10. The Even Standard Component Theorem 55 Summary: Statements of the major subtheorems 59 Chapter 2. Overview: The classification of groups of characteristic 2 type 63 The Main Theorem for groups of characteristic 2 type 63 2.1. The Quasithin Theorem covering e(G) ≤ 265 2.2. The trichotomy approach to treating e(G) ≥ 366 2.3. The Trichotomy Theorem for e(G) ≥ 469 2.4. The e(G) = 3 Theorem (including trichotomy) 75 2.5. The Standard Type Theorem 77 2.6. The GF (2) Type Theorem 77 2.7. The Uniqueness Case Theorem 78 Conclusion: The proof of the Characteristic 2 Type Theorem 80 vii viii CONTENTS Outline of the classification of groups of characteristic 2 type 83 Chapter 3. e(G) ≤ 2: The classification of quasithin groups 85 3.1. Introduction: The Thompson Strategy 86 3.2. Preliminaries: Structure theory for quasithin 2-locals (SQTK-groups) 88 3.3. More preliminaries: Some general techniques 90 3.4. The degenerate case: A Sylow T in a unique maximal 2-local 98 3.5. The Main Case Division (Possibilities for a suitable group L and module V ) 100 n 3.6. The Generic Case—where L = L2(2 ) with n>1 103 3.7. Reducing to V an FF-module for L 106 3.8. Cases with L over F2n for n>1 109 3.9. Cases with L over F2 (but not L3(2)) 111 3.10. Cases with L = L3(2), and analogues for L2(2) 117 3.11. The final case where Lf (G, T ) is empty 120 3.12. Bonus: The Even Type (Quasithin) Theorem for use in the GLS program 123 Chapter 4. e(G) = 3: The classification of rank 3 groups 127 4.1. The case where σ(G) contains a prime p ≥ 5 128 The Signalizer Analysis 128 The Component Analysis 130 4.2. The case σ(G)={3} 133 The Signalizer Analysis 134 The Component Analysis 142 Chapter 5. e(G) ≥ 4: The Pretrichotomy and Trichotomy Theorems 149 5.1. Statements and Definitions 149 5.2. The Signalizer Analysis 152 5.3. The Component Analysis (leading to standard type) 159 Chapter 6. The classification of groups of standard type 173 6.1. The Gilman-Griess Theorem on standard type for e(G) ≥ 4 173 Identifying a large Lie-type subgroup G0 174 The final step: G = G0 177 6.2. Odd standard form problems for e(G) = 3 (Finkelstein-Frohardt) 180 Chapter 7. The classification of groups of GF (2) type 183 Introduction 184 7.1. Aschbacher’s reduction of GF (2) type to the large-extraspecial case 185 7.2. The treatment of some fundamental extraspecial cases 188 7.3. Timmesfeld’s reduction to a list of possibilities for M 192 7.4. The final treatment of the various cases for M 199 7.5. Chapter appendix: The classification of groups of GF (2n) type 204 CONTENTS ix Chapter 8. The final contradiction: Eliminating the Uniqueness Case 213 8.1. Prelude: From the Preuniqueness Case to the Uniqueness Case 215 8.2. Introduction: General strategy using weak closure and uniqueness theorems 223 8.3. Preliminary results and the weak closure setup 226 8.4. The treatment of small n(H) 230 8.5. The treatment of large n(H) 234 Appendices 249 Appendix A. Some background material related to simple groups 251 A.1. Preliminaries: Some notation and results from general group theory 251 A.2. Notation for the simple groups 254 A.3. Properties of simple groups and K-groups 256 A.4. Properties of representations of simple groups 261 A.5. Recognition theorems for identifying simple groups 262 A.6. Transvection groups and transposition-group theory 264 Appendix B. Overview of some techniques used in the classification 267 B.1. Coprime action 267 B.2. Fusion and transfer 269 B.3. Signalizer functor methods and balance 272 B.4. Connectivity in commuting graphs and i-generated cores 280 B.5. Application: A short elementary proof of the Dichotomy Theorem 287 B.6. Failure of factorization 290 B.7. Pushing-up, and the Local and Global C(G, T ) Theorems 292 B.8. Weak closure 299 B.9. Klinger-Mason analysis of bicharacteristic groups 302 B.10. Some details of the proof of the Uniqueness Case Theorem 305 References and Index 313 References used for both GW type and characteristic 2 type 315 References mainly for GW type (see [Gor82][Gor83] for full list) 317 References used primarily for characteristic 2 type 321 Expository references mentioned 329 Index 333 Preface The present book, “The Classification of Finite Simple Groups: Groups of Characteristic 2 Type”, completes a project of giving an outline of the proof of the Classification of the Finite Simple Groups (CFSG). The project was begun by Daniel Gorenstein in 1983 with his book [Gor83]—which he subtitled “Volume 1: Groups of Noncharacteristic 2 Type”. Thus we regard our present discussion of groups of characteristic 2 type as “Volume 2” of that project. The Classification of the Finite Simple Groups (CFSG) is one of the premier achievements of twentieth century mathematics. The result has a history which, in some sense, goes back to the beginnings of proto-group theory in the late eighteenth century. Many classic problems with a long history are important more for the mathematics they inspire and generate, than because of interesting consequences. This is not true of the Classification, which is an extremely useful result, making possible many modern successes of finite group theory, which have in turn been applied to solve numerous problems in many areas of mathematics.