**Mathematical Surveys and Monographs**

Volume: 230;
2018;
231 pp;
Hardcover

MSC: Primary 20;

Print ISBN: 978-1-4704-4291-0

Product Code: SURV/230

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**Electronic ISBN: 978-1-4704-4682-6
Product Code: SURV/230.E**

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

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*Stephen D. Smith*

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.

#### Readership

Graduate students and researchers interested in group theory and its applications.

#### Table of Contents

# Table of Contents

## Applying the Classification of Finite Simple Groups: A User's Guide

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xi12
- Chapter 1. Background: Simple groups and their properties 116
- Introduction: Statement of the CFSGโthe list of simple groups 116
- 1.1. Alternating groups 217
- 1.2. Sporadic groups 318
- 1.3. Groups of Lie type 520
- Some easy applications of the CFSG-list 1934
- 1.4. Structure of ๐ฆ-groups: Via components in โฑ*(๐ข) 2035
- 1.5. Outer automorphisms of simple groups 2338
- 1.6. Further CFSG-consequences: e.g. doubly-transitive groups 2540

- Chapter 2. Outline of the proof of the CFSG: Some main ideas 2944
- 2.0. A start: Proving the Odd/Even Dichotomy Theorem 2944
- 2.1. Treating the Odd Case: Via standard form 3651
- 2.2. Treating the Even Case: Via trichotomy and standard type 3853
- 2.3. Afterword: Comparison with later CFSG approaches 4459
- Applying the CFSG toward Quillenโs Conjecture on ๐ฎ_{๐ }(๐ข) 4560
- 2.4. Introduction: The poset ๐ฎ_{๐ }(๐ข) and the contractibility conjecture 4560
- 2.5. Quillen-dimension and the solvable case 4762
- 2.6. The reduction of the ๐-solvable case to the solvable case 4964
- 2.7. Other uses of the CFSG in the Aschbacher-Smith proof 5267

- Chapter 3. Thompson Factorizationโand its failure: FF-methods 5570
- Introduction: Some forms of the Frattini factorization 5570
- 3.1. Thompson Factorization: Using ๐ฝ(๐) as weakly-closed โ๐โ 5772
- 3.2. Failure of Thompson Factorization: FF-methods 5974
- 3.3. Pushing-up: FF-modules in Aschbacher blocks 6176
- 3.4. Weak-closure factorizations: Using other weakly-closed โ๐โ 6681
- Applications related to the Martino-Priddy Conjecture 7085
- 3.5. The conjecture on classifying spaces and fusion systems 7085
- 3.6. Oliverโs proof of Martino-Priddy using the CFSG 7287
- 3.7. Oliverโs conjecture on ๐ฝ(๐) for ๐ odd 7489

- Chapter 4. Recognition theorems for simple groups 7792
- Introduction: Finishing classification problems 7792
- 4.1. Recognizing alternating groups 8095
- 4.2. Recognizing Lie-type groups 8095
- 4.3. Recognizing sporadic groups 8297
- Applications to recognizing some quasithin groups 8499
- 4.4. Background: 2-local structure in the quasithin analysis 8499
- 4.5. Recognizing rank-2 Lie-type groups 86101
- 4.6. Recognizing the Rudvalis group ๐ ๐ข 87102

- Chapter 5. Representation theory of simple groups 89104
- Introduction: Some standard general facts about representations 89104
- 5.1. Representations for alternating and symmetric groups 91106
- 5.2. Representations for Lie-type groups 92107
- 5.3. Representations for sporadic groups 97112
- Applications to Alperinโs conjecture 98113
- 5.4. Introduction: The Alperin Weight Conjecture (AWC) 98113
- 5.5. Reductions of the AWC to simple groups 99114
- 5.6. A closer look at verification for the Lie-type case 100115
- A glimpse of some other applications of representations 102117

- Chapter 6. Maximal subgroups and primitive representations 105120
- Introduction: Maximal subgroups and primitive actions 105120
- 6.1. Maximal subgroups of symmetric and alternating groups 106121
- 6.2. Maximal subgroups of Lie-type groups 110125
- 6.3. Maximal subgroups of sporadic groups 113128
- Some applications of maximal subgroups 114129
- 6.4. Background: Broader areas of applications 114129
- 6.5. Random walks on ๐_{๐} and minimal generating sets 115130
- 6.6. Applications to ๐-exceptional linear groups 117132
- 6.7. The probability of 2-generating a simple group 119134

- Chapter 7. Geometries for simple groups 121136
- Introduction: The influence of Titsโs theory of buildings 121136
- 7.1. The simplex for ๐_{๐}; later giving an apartment for ๐บ๐ฟ_{๐}(๐) 122137
- 7.2. The building for a Lie-type group 125140
- 7.3. Geometries for sporadic groups 129144
- Some applications of geometric methods 131146
- 7.4. Geometry in classification problems 131146
- 7.5. Geometry in representation theory 133148
- 7.6. Geometry applied for local decompositions 136151

- Chapter 8. Some fusion techniques for classification problems 139154
- 8.1. Glaubermanโs ๐*-theorem 139154
- 8.2. The Thompson Transfer Theorem 143158
- 8.3. The Bender-Suzuki Strongly Embedded Theorem 145160
- Analogous ๐-fusion results for odd primes ๐ 149164
- 8.4. The ๐_{๐}*-theorem for odd ๐ 149164
- 8.5. Thompson-style transfer for odd ๐ 150165
- 8.6. Strongly ๐-embedded subgroups for odd ๐ 150165

- Chapter 9. Some applications close to finite group theory 153168
- 9.1. Distance-transitive graphs 153168
- 9.2. The proportion of ๐-singular elements 154169
- 9.3. Root subgroups of maximal tori in Lie-type groups 156171
- Some applications more briefly treated 157172
- 9.4. Frobeniusโ conjecture on solutions of ๐ฅโฟ=1 157172
- 9.5. Subgroups of prime-power index in simple groups 158173
- 9.6. Application to 2-generation and module cohomology 159174
- 9.7. Minimal nilpotent covers and solvability 160175
- 9.8. Computing composition factors of permutation groups 160175

- Chapter 10. Some applications farther afield from finite groups 161176
- 10.1. Polynomial subgroup-growth in finitely-generated groups 161176
- 10.2. Relative Brauer groups of field extensions 162177
- 10.3. Monodromy groups of coverings of Riemann surfaces 163178
- Some exotic applications more briefly treated 165180
- 10.4. Locally finite simple groups and Moufang loops 165180
- 10.5. Waringโs problem for simple groups 167182
- 10.6. Expander graphs and approximate groups 167182

- Appendix 169184
- Back Cover Back Cover1248