Applying the Classification of Finite Simple Groups: A User's Guide (draft3, as submitted to AMS, 12sept2017) Stephen D. Smith Department of Mathematics (m/c 249), University of Illinois at Chicago, 851 S. Morgan, Chicago, Illinois 60607-7045 E-mail address: [email protected] 2010 Mathematics Subject Classification. 20D05, 20D06, 20D08, 20Bxx, 20Cxx, 20E25, 20E28, 20Jxx, 05E15, 05E18, 55Uxx Abstract. This book surveys a wide range of applications of the Classification of Finite Simple Groups (CFSG): both within finite group theory itself, and in other mathematical areas which make use of group theory. The book is based on the author's lectures at the September 2015 Venice Summer School on Finite Groups; and so may in particular be of use in a graduate course. It should also be more widely useful as an introduction and basic reference; in addition it indicates fuller citations to the appropriate literature, for readers who wish to go on to more detailed sources. v 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 20 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 ) for p 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 211 Preface Most mathematicians are at least aware of the Classification of Finite Simple Groups (CFSG)|a major project involving work by hundreds of researchers. The work was largly completed by about 1983; though final publication of the \qua- sithin" part was delayed until 2004. And since the 1980s, the result has had a huge influence on work in finite group theory, and in very 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 can't hope to systematically cover all applications. Indeed the par- ticular applications chosen were mainly provided by contacting colleagues who are experts in many related areas. So the resulting collection of material might seem somewhat scattered; however, I've tried to present the choices within contexts of wider areas of applications, and I am grateful to the referee for helpful suggestions in that direction. Origin of the book, and structure of the chapters This book began life as a series of 10 two-hour lectures, which I was invited to give during the September 2015 Venice Summer School on Finite Groups. The material in the book has been adapted only fairly lightly from that lecture format: mainly in order to try to preserve the introductory, and comparatively informal, tone of the original. The primary difference in the book form has been to try to bring the cross-references, and the literature citations, up to the more formal level expected in a published reference book. I'll mention briefly one particular feature of the chapters in the book, arising from the lectures. Each original lecture was divided into two parts: typically, the first part introduced some basic theory from a particular aspect of the CFSG; and the second (\applications") part then demonstrated ways in which that theory can be put to use. The 10 lectures have now become the 10 chapters in this book format; and the reader will notice, even from the Table of Contents, that the later sections of each chapter usually reflect the applications-focus of the second part of the original lecture. Some notes on using the book as a course text For those who wish to use the book as a course text: The material is intended to be accessible to an audience with basic mathe- matical training; for example, to beginning graduate students, with at least an undergraduate course in abstract algebra. But preferably the background should also include first-year graduate algebra, especially some experience with examples of the most familiar types of groups; and ideally at least the basics of Lie algebras. xi xii PREFACE It will also be helpful, at times, to have some exposure to various adjacent areas of mathematics: for example, the fundamentals of algebraic topology and homological algebra, and possibly also a little combinatorics.