Applying the Classification of Finite Simple Groups. A User’s Guide

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