کاربرد طبقه‌بندی گروه‌های ساده محدود: راهنمای کاربر

"Classification of Finite Simple Groups (CFSG) is a major project involving work by hundreds of researchers. The work was largely completed by about 1983, although final publication of the "quasithin" part was delayed until 2004. Since the 1980s, CFSG 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 such topics from the very large and increasingly active research area of applications of CFSG. The book is based on the author's lectures at the September 2015 Venice Summer School on Finite Groups. With about 50 exercises from original lectures, it can serve as a second-year graduate course for students who have had first-year graduate algebra. It may be of particular interest to students looking for a dissertation topic around group theory. It can also be 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"--The publisher. Read more... Abstract: Since the 1980s, Classification of Finite Simple Groups (CFSG) 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 such topics from the very large and increasingly active research area of applications of CFSG. Read more... Cover 1 Title page 4 Contents 8 Preface 12 Origin of the book and structure of the chapters 12 Some notes on using the book as a course text 12 Acknowledgments 13 Chapter 1. Background: Simple groups and their properties 16 Introduction: Statement of the CFSG—the list of simple groups 16 1.1. Alternating groups 17 1.2. Sporadic groups 18 1.3. Groups of Lie type 20 Some easy applications of the CFSG-list 34 1.4. Structure of K-groups: Via components in F*(G) 35 1.5. Outer automorphisms of simple groups 38 1.6. Further CFSG-consequences: e.g. doubly-transitive groups 40 Chapter 2. Outline of the proof of the CFSG: Some main ideas 44 2.0. A start: Proving the Odd/Even Dichotomy Theorem 44 2.1. Treating the Odd Case: Via standard form 51 2.2. Treating the Even Case: Via trichotomy and standard type 53 2.3. Afterword: Comparison with later CFSG approaches 59 Applying the CFSG toward Quillen’s Conjecture on S_{p}(G) 60 2.4. Introduction: The poset S_{p}(G) and the contractibility conjecture 60 2.5. Quillen-dimension and the solvable case 62 2.6. The reduction of the p-solvable case to the solvable case 64 2.7. Other uses of the CFSG in the Aschbacher-Smith proof 67 Chapter 3. Thompson Factorization—and its failure: FF-methods 70 Introduction: Some forms of the Frattini factorization 70 3.1. Thompson Factorization: Using J(T) as weakly-closed “W” 72 3.2. Failure of Thompson Factorization: FF-methods 74 3.3. Pushing-up: FF-modules in Aschbacher blocks 76 3.4. Weak-closure factorizations: Using other weakly-closed “W” 81 Applications related to the Martino-Priddy Conjecture 85 3.5. The conjecture on classifying spaces and fusion systems 85 3.6. Oliver’s proof of Martino-Priddy using the CFSG 87 3.7. Oliver’s conjecture on J(T) for p odd 89 Chapter 4. Recognition theorems for simple groups 92 Introduction: Finishing classification problems 92 4.1. Recognizing alternating groups 95 4.2. Recognizing Lie-type groups 95 4.3. Recognizing sporadic groups 97 Applications to recognizing some quasithin groups 99 4.4. Background: 2-local structure in the quasithin analysis 99 4.5. Recognizing rank-2 Lie-type groups 101 4.6. Recognizing the Rudvalis group Ru 102 Chapter 5. Representation theory of simple groups 104 Introduction: Some standard general facts about representations 104 5.1. Representations for alternating and symmetric groups 106 5.2. Representations for Lie-type groups 107 5.3. Representations for sporadic groups 112 Applications to Alperin’s conjecture 113 5.4. Introduction: The Alperin Weight Conjecture (AWC) 113 5.5. Reductions of the AWC to simple groups 114 5.6. A closer look at verification for the Lie-type case 115 A glimpse of some other applications of representations 117 Chapter 6. Maximal subgroups and primitive representations 120 Introduction: Maximal subgroups and primitive actions 120 6.1. Maximal subgroups of symmetric and alternating groups 121 6.2. Maximal subgroups of Lie-type groups 125 6.3. Maximal subgroups of sporadic groups 128 Some applications of maximal subgroups 129 6.4. Background: Broader areas of applications 129 6.5. Random walks on S_{n} and minimal generating sets 130 6.6. Applications to p-exceptional linear groups 132 6.7. The probability of 2-generating a simple group 134 Chapter 7. Geometries for simple groups 136 Introduction: The influence of Tits’s theory of buildings 136 7.1. The simplex for S_{n}; later giving an apartment for GL_{n}(q) 137 7.2. The building for a Lie-type group 140 7.3. Geometries for sporadic groups 144 Some applications of geometric methods 146 7.4. Geometry in classification problems 146 7.5. Geometry in representation theory 148 7.6. Geometry applied for local decompositions 151 Chapter 8. Some fusion techniques for classification problems 154 8.1. Glauberman’s Z*-theorem 154 8.2. The Thompson Transfer Theorem 158 8.3. The Bender-Suzuki Strongly Embedded Theorem 160 Analogous p-fusion results for odd primes p 164 8.4. The Z_{p}*-theorem for odd p 164 8.5. Thompson-style transfer for odd p 165 8.6. Strongly p-embedded subgroups for odd p 165 Chapter 9. Some applications close to finite group theory 168 9.1. Distance-transitive graphs 168 9.2. The proportion of p-singular elements 169 9.3. Root subgroups of maximal tori in Lie-type groups 171 Some applications more briefly treated 172 9.4. Frobenius’ conjecture on solutions of xn=1 172 9.5. Subgroups of prime-power index in simple groups 173 9.6. Application to 2-generation and module cohomology 174 9.7. Minimal nilpotent covers and solvability 175 9.8. Computing composition factors of permutation groups 175 Chapter 10. Some applications farther afield from finite groups 176 10.1. Polynomial subgroup-growth in finitely-generated groups 176 10.2. Relative Brauer groups of field extensions 177 10.3. Monodromy groups of coverings of Riemann surfaces 178 Some exotic applications more briefly treated 180 10.4. Locally finite simple groups and Moufang loops 180 10.5. Waring’s problem for simple groups 182 10.6. Expander graphs and approximate groups 182 Appendix 184 Appendix A. Some supplementary notes to the text 186 A.1. Notes for 6.1.1: Deducing the structures-list for S_{n} 186 A.2. Notes for 8.2.1: The cohomological view of the transfer map 187 A.3. Notes for (8.3.4): Some details of proofs in Holt’s paper 189 Appendix B. Further remarks on certain exercises 198 B.1. Some exercises from Chapter 1 198 B.2. Some exercises from Chapter 4 199 B.3. Some exercises from Chapter 5 206 B.4. Some exercises from Chapter 6 208 Bibliography 214 Index 228 Back Cover 248 Cover......Page 1 Title page......Page 4 Contents......Page 8 Some notes on using the book as a course text......Page 12 Acknowledgments......Page 13 Introduction: Statement of the CFSG—the list of simple groups......Page 16 1.1. Alternating groups......Page 17 1.2. Sporadic groups......Page 18 1.3. Groups of Lie type......Page 20 Some easy applications of the CFSG-list......Page 34 1.4. Structure of -groups: Via components in F*()......Page 35 1.5. Outer automorphisms of simple groups......Page 38 1.6. Further CFSG-consequences: e.g. doubly-transitive groups......Page 40 2.0. A start: Proving the Odd/Even Dichotomy Theorem......Page 44 2.1. Treating the Odd Case: Via standard form......Page 51 2.2. Treating the Even Case: Via trichotomy and standard type......Page 53 2.3. Afterword: Comparison with later CFSG approaches......Page 59 2.4. Introduction: The poset _{}() and the contractibility conjecture......Page 60 2.5. Quillen-dimension and the solvable case......Page 62 2.6. The reduction of the -solvable case to the solvable case......Page 64 2.7. Other uses of the CFSG in the Aschbacher-Smith proof......Page 67 Introduction: Some forms of the Frattini factorization......Page 70 3.1. Thompson Factorization: Using () as weakly-closed “”......Page 72 3.2. Failure of Thompson Factorization: FF-methods......Page 74 3.3. Pushing-up: FF-modules in Aschbacher blocks......Page 76 3.4. Weak-closure factorizations: Using other weakly-closed “”......Page 81 3.5. The conjecture on classifying spaces and fusion systems......Page 85 3.6. Oliver’s proof of Martino-Priddy using the CFSG......Page 87 3.7. Oliver’s conjecture on () for odd......Page 89 Introduction: Finishing classification problems......Page 92 4.2. Recognizing Lie-type groups......Page 95 4.3. Recognizing sporadic groups......Page 97 4.4. Background: 2-local structure in the quasithin analysis......Page 99 4.5. Recognizing rank-2 Lie-type groups......Page 101 4.6. Recognizing the Rudvalis group ......Page 102 Introduction: Some standard general facts about representations......Page 104 5.1. Representations for alternating and symmetric groups......Page 106 5.2. Representations for Lie-type groups......Page 107 5.3. Representations for sporadic groups......Page 112 5.4. Introduction: The Alperin Weight Conjecture (AWC)......Page 113 5.5. Reductions of the AWC to simple groups......Page 114 5.6. A closer look at verification for the Lie-type case......Page 115 A glimpse of some other applications of representations......Page 117 Introduction: Maximal subgroups and primitive actions......Page 120 6.1. Maximal subgroups of symmetric and alternating groups......Page 121 6.2. Maximal subgroups of Lie-type groups......Page 125 6.3. Maximal subgroups of sporadic groups......Page 128 6.4. Background: Broader areas of applications......Page 129 6.5. Random walks on _{} and minimal generating sets......Page 130 6.6. Applications to -exceptional linear groups......Page 132 6.7. The probability of 2-generating a simple group......Page 134 Introduction: The influence of Tits’s theory of buildings......Page 136 7.1. The simplex for _{}; later giving an apartment for _{}()......Page 137 7.2. The building for a Lie-type group......Page 140 7.3. Geometries for sporadic groups......Page 144 7.4. Geometry in classification problems......Page 146 7.5. Geometry in representation theory......Page 148 7.6. Geometry applied for local decompositions......Page 151 8.1. Glauberman’s *-theorem......Page 154 8.2. The Thompson Transfer Theorem......Page 158 8.3. The Bender-Suzuki Strongly Embedded Theorem......Page 160 8.4. The _{}*-theorem for odd ......Page 164 8.6. Strongly -embedded subgroups for odd ......Page 165 9.1. Distance-transitive graphs......Page 168 9.2. The proportion of -singular elements......Page 169 9.3. Root subgroups of maximal tori in Lie-type groups......Page 171 9.4. Frobenius’ conjecture on solutions of n=1......Page 172 9.5. Subgroups of prime-power index in simple groups......Page 173 9.6. Application to 2-generation and module cohomology......Page 174 9.8. Computing composition factors of permutation groups......Page 175 10.1. Polynomial subgroup-growth in finitely-generated groups......Page 176 10.2. Relative Brauer groups of field extensions......Page 177 10.3. Monodromy groups of coverings of Riemann surfaces......Page 178 10.4. Locally finite simple groups and Moufang loops......Page 180 10.6. Expander graphs and approximate groups......Page 182 Appendix......Page 184 A.1. Notes for 6.1.1: Deducing the structures-list for _{}......Page 186 A.2. Notes for 8.2.1: The cohomological view of the transfer map......Page 187 A.3. Notes for (8.3.4): Some details of proofs in Holt’s paper......Page 189 B.1. Some exercises from Chapter 1......Page 198 B.2. Some exercises from Chapter 4......Page 199 B.3. Some exercises from Chapter 5......Page 206 B.4. Some exercises from Chapter 6......Page 208 Bibliography......Page 214 Index......Page 228 Back Cover......Page 248