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Mathematical Surveys and Monographs Volume 230 Applying the Classification of Finite Simple Groups A User’s Guide Stephen D. Smith 10.1090/surv/230 Applying the Classification of Finite Simple Groups A User’s Guide Mathematical Surveys and Monographs Volume 230 Applying the Classification of Finite Simple Groups A User’s Guide Stephen D. Smith EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 20-02, 20D05, 20Bxx, 20Cxx, 20Exx, 20Gxx, 20Jxx. For additional information and updates on this book, visit www.ams.org/bookpages/surv-230 Library of Congress Cataloging-in-Publication Data Names: Smith, Stephen D., 1948– author. Title: Applying the classification of finite simple groups: A user’s guide / Stephen D. Smith. Other titles: Classification of finite simple groups Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Mathe- matical surveys and monographs; volume 230 | Includes bibliographical references and index. Identifiers: LCCN 2017044767 | ISBN 9781470442910 (alk. paper) Subjects: LCSH: Finite simple groups. | Representations of groups. | AMS: Group theory and generalizations – Research exposition (monographs, survey articles). msc | Group theory and generalizations – Abstract finite groups – Finite simple groups and their classification. msc | Group theory and generalizations – Permutation groups – Permutation groups. msc | Group theory and generalizations – Representation theory of groups – Representation theory of groups. msc | Group theory and generalizations – Structure and classification of infinite or finite groups – Structure and classification of infinite or finite groups. msc | Group theory and generalizations – Linear algebraic groups and related topics – Linear algebraic groups and related topics. msc | Group theory and generalizations – Connections with homological algebra and category theory – Connections with homological algebra and category theory. msc Classification: LCC QA177 .S645 2018 | DDC 512/.2–dc23 LC record available at https://lccn.loc.gov/2017044767 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 select pages 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. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2018 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 232221201918 To Judy—for exemplary patience Contents Preface xi Origin of the book and structure of the chapters xi Some notes on using the book as a course text xi Acknowledgments xii Chapter 1. Background: Simple groups and their properties 1 Introduction: Statement of the CFSG—the list of simple groups 1 1.1. Alternating groups 2 1.2. Sporadic groups 3 1.3. Groups of Lie type 5 Some easy applications of the CFSG-list 19 1.4. Structure of K-groups: Via components in F ∗(G)20 1.5. Outer automorphisms of simple groups 23 1.6. Further CFSG-consequences: e.g. doubly-transitive groups 25 Chapter 2. Outline of the proof of the CFSG: Some main ideas 29 2.0. A start: Proving the Odd/Even Dichotomy Theorem 29 2.1. Treating the Odd Case: Via standard form 36 2.2. Treating the Even Case: Via trichotomy and standard type 38 2.3. Afterword: Comparison with later CFSG approaches 44 Applying the CFSG toward Quillen’s Conjecture on Sp(G)45 2.4. Introduction: The poset Sp(G) and the contractibility conjecture 45 2.5. Quillen-dimension and the solvable case 47 2.6. The reduction of the p-solvable case to the solvable case 49 2.7. Other uses of the CFSG in the Aschbacher-Smith proof 52 Chapter 3. Thompson Factorization—and its failure: FF-methods 55 Introduction: Some forms of the Frattini factorization 55 3.1. Thompson Factorization: Using J(T ) as weakly-closed “W ”57 3.2. Failure of Thompson Factorization: FF-methods 59 3.3. Pushing-up: FF-modules in Aschbacher blocks 61 3.4. Weak-closure factorizations: Using other weakly-closed “W ”66 Applications related to the Martino-Priddy Conjecture 70 3.5. The conjecture on classifying spaces and fusion systems 70 3.6. Oliver’s proof of Martino-Priddy using the CFSG 72 3.7. Oliver’s conjecture on J(T )forp odd 74 Chapter 4. Recognition theorems for simple groups 77 Introduction: Finishing classification problems 77 4.1. Recognizing alternating groups 80 vii viii CONTENTS 4.2. Recognizing Lie-type groups 80 4.3. Recognizing sporadic groups 82 Applications to recognizing some quasithin groups 84 4.4. Background: 2-local structure in the quasithin analysis 84 4.5. Recognizing rank-2 Lie-type groups 86 4.6. Recognizing the Rudvalis group Ru 87 Chapter 5. Representation theory of simple groups 89 Introduction: Some standard general facts about representations 89 5.1. Representations for alternating and symmetric groups 91 5.2. Representations for Lie-type groups 92 5.3. Representations for sporadic groups 97 Applications to Alperin’s conjecture 98 5.4. Introduction: The Alperin Weight Conjecture (AWC) 98 5.5. Reductions of the AWC to simple groups 99 5.6. A closer look at verification for the Lie-type case 100 A glimpse of some other applications of representations 102 Chapter 6. Maximal subgroups and primitive representations 105 Introduction: Maximal subgroups and primitive actions 105 6.1. Maximal subgroups of symmetric and alternating groups 106 6.2. Maximal subgroups of Lie-type groups 110 6.3. Maximal subgroups of sporadic groups 113 Some applications of maximal subgroups 114 6.4. Background: Broader areas of applications 114 6.5. Random walks on Sn and minimal generating sets 115 6.6. Applications to p-exceptional linear groups 117 6.7. The probability of 2-generating a simple group 119 Chapter 7. Geometries for simple groups 121 Introduction: The influence of Tits’s theory of buildings 121 7.1. The simplex for Sn; later giving an apartment for GLn(q) 122 7.2. The building for a Lie-type group 125 7.3. Geometries for sporadic groups 129 Some applications of geometric methods 131 7.4. Geometry in classification problems 131 7.5. Geometry in representation theory 133 7.6. Geometry applied for local decompositions 136 Chapter 8. Some fusion techniques for classification problems 139 8.1. Glauberman’s Z∗-theorem 139 8.2. The Thompson Transfer Theorem 143 8.3. The Bender-Suzuki Strongly Embedded Theorem 145 Analogous p-fusion results for odd primes p 149 ∗ 8.4. The Zp -theorem for odd p 149 8.5. Thompson-style transfer for odd p 150 8.6. Strongly p-embedded subgroups for odd p 150 Chapter 9. Some applications close to finite group theory 153 9.1. Distance-transitive graphs 153 CONTENTS ix 9.2. The proportion of p-singular elements 154 9.3. Root subgroups of maximal tori in Lie-type groups 156 Some applications more briefly treated 157 9.4. Frobenius’ conjecture on solutions of xn = 1 157 9.5. Subgroups of prime-power index in simple groups 158 9.6. Application to 2-generation and module cohomology 159 9.7. Minimal nilpotent covers and solvability 160 9.8. Computing composition factors of permutation groups 160 Chapter 10. Some applications farther afield from finite groups 161 10.1. Polynomial subgroup-growth in finitely-generated groups 161 10.2. Relative Brauer groups of field extensions 162 10.3. Monodromy groups of coverings of Riemann surfaces 163 Some exotic applications more briefly treated 165 10.4. Locally finite simple groups and Moufang loops 165 10.5. Waring’s problem for simple groups 167 10.6. Expander graphs and approximate groups 167 Appendix 169 Appendix A. Some supplementary notes to the text 171 A.1. Notes for 6.1.1: Deducing the structures-list for Sn 171 A.2. Notes for 8.2.1: The cohomological view of the transfer map 172 A.3. Notes for (8.3.4): Some details of proofs in Holt’s paper 174 Appendix B. Further remarks on certain exercises 183 B.1. Some exercises from Chapter 1 183 B.2. Some exercises from Chapter 4 184 B.3. Some exercises from Chapter 5 191 B.4. Some exercises from Chapter 6 193 Bibliography 199 Index 213 Preface Most mathematicians are at least aware of the Classification of Finite Sim- ple Groups (CFSG), a major project involving work by hundreds of researchers. The work was largely completed by about 1983, though final publication of the “quasithin” part was delayed until 2004. Since the 1980s, the result has had a huge influence on work in finite group theory and in many adjacent fields of mathematics. This book attempts to survey and sample a number of topics from the very large and increasingly active research area of applications of the CFSG. The book cannot hope to systematically cover all applications.
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