The Classification of the Finite Simple Groups

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  • Format: Hardcover
  • Copyright: 2005-02-01
  • Publisher: Amer Mathematical Society

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The classification of finite simple groups is a landmark result of modern mathematics. The original proof is spread over scores of articles by dozens of researchers. In this multivolume book, the authors are assembling the proof with explanations and references. It is a monumental task. The book, along with background from sections of the previous volumes, presents critical aspects of the classification. Continuing the proof of the classification theorem which began in the previous five volumes (Surveys of Mathematical Monographs, Volumes 40.1.E, 40.2, 40.3, 40.4, and 40.5), in this volume, the authors provide the classification of finite simple groups of special odd type (Theorems $\mathcal{C 2$ and $\mathcal{C 3$, as stated in the first volume of the series). The book is suitable for graduate students and researchers interested in group theory.

Table of Contents

Prefacep. ix
The Special Odd Case
General Introduction to the Special Odd Casep. 1
The Goals: Theorems [characters not reproducible] and [characters not reproducible]p. 1
Theorems [characters not reproducible] and [characters not reproducible]p. 2
General Comments on the Proof of Theorem [characters not reproducible]p. 6
Theorem [characters not reproducible] and Its Proofp. 9
General Lemmasp. 13
The Bender/Glauberman Methodp. 13
p-Groups and Coprime Actionp. 18
Transfer and Fusionp. 28
2-Components and 2-Groupsp. 32
Pumpups and 2-Terminalityp. 35
Terminalityp. 37
Semirigidityp. 38
Preuniqueness Subgroupsp. 43
Signalizer Functors and Balancep. 48
Ordinary Character Theoryp. 48
Modular Character Theoryp. 51
BN-Pairsp. 56
Number Theoryp. 58
Miscellaneousp. 58
Theorem [characters not reproducible]: Stage 1p. 61
Introductionp. 61
The 2-Rank 2 Case: Alperin's Theoremp. 64
Theorem 2: 2-Terminal 2-Components of 2-Rank 1p. 67
The Fusion of z and Structure of Qp. 67
Involutory f-Automorphismsp. 71
Elimination of f-Automorphismsp. 75
Theorem 3: The Nonfused Casep. 77
Theorem 3: The Fused Casep. 80
Theorem 4: 2-Terminal 2-Components of 2-Rank [greater than or equal] 2p. 86
The O'N Casep. 90
Theorem 5p. 93
Groups of 2-Rank 3p. 94
2-Groups of Type M[subscript 12]p. 99
The Proof of Theorem 6p. 102
Theorem [characters not reproducible]: Stage 2p. 105
Introductionp. 105
The Structure of Involution Centralizersp. 107
Other K-Subgroups of Gp. 113
Odd Order Signalizersp. 117
An Equivariant K[superscript infinity]-Theoremp. 123
Theorem 1: Uniqueness Properties of Mp. 126
Theorem 1: The Case m[subscript 2](S) [greater than or equal] 3p. 131
Theorem 1: The Case m[subscript 2](S) = 2p. 133
Theorems 2 and 3: Normalizers of z-Invariant Subgroups of O[subscript p](M)p. 135
Theorem 2: {2, p}-Uniqueness Properties of Mp. 143
Theorem 2: Reduction to the Ly Casep. 146
Theorem 2: The Ly Casep. 148
Theorem 3: The 2-Rank 2 Casep. 150
The Identification of L[subscript 3](q)p. 157
Reduction to L[subscript 3](q)-Levi Formp. 160
The L[subscript 3](q)-Levi Form Casep. 164
Theorems 4 and 5: The Structure of Mp. 165
Theorems 4 and 5: Uniqueness Properties of Mp. 167
The Strong B-Propertyp. 172
Theorem 6p. 173
Theorem [characters not reproducible]: Stage 3p. 181
Introductionp. 181
Theorem 1: The Embedding of Fp. 183
Character Theory for the Dihedral Casep. 188
Coherencep. 190
The Case [vertical bar S vertical bar greater than or equal] 8p. 194
Theorem BG[subscript 2]: The Case [vertical bar S vertical bar greater than or equal] 8p. 200
Character Theory for the Case [vertical bar S vertical bar] = 4p. 203
Elimination of the Extruded Casep. 208
The Identification of A[subscript 7]p. 217
The Identification of L[subscript 2](7) and A[subscript 6]p. 221
Reduction to the Brauer-Suzuki-Wall Casep. 223
The Identification of L[subscript 2](q)p. 224
Theorem 3: The Case M = QSp. 230
Theorem 3: The Structure of Cp. 233
Theorem 3: The Final Contradictionp. 240
Theorem 4: Local Analysisp. 245
Regularity and [vertical bar G vertical bar]: Stage 1p. 252
Regularity: The Case B [not equal] 1p. 259
Regularity: The Case Q = 1p. 264
G [approximate] U[subscript 3](q)p. 266
Theorem [characters not reproducible]: Stage 4p. 273
Introductionp. 273
Theorem 1: Field Automorphismsp. 275
The Characteristic of Kp. 286
G [approximate superscript 2]G[subscript 2](q)p. 290
Theorem 2: The Weyl Groupp. 293
Theorem 2: The Torally Singular Casep. 297
Theorem 2: The Case q = 3[superscript n greater than sign] 9p. 300
Theorem 2: The Case q = 9p. 304
Theorem 3: The Structure of a Sylow p-Subgroup of Gp. 309
The (B, N)-Pair Structure of Gp. 313
The Commutator Relationsp. 314
The Action of K[subscript 1]Hp. 318
The Isomorphism Type of Gp. 321
Theorem [characters not reproducible]: Stage 5p. 325
Introductionp. 325
Theorem Sp: 2-Local Structurep. 325
The p-Locals of Gp. 328
The Identification of PSp[subscript 4](q)p. 332
SL[subscript 4 superscript plus or minus 1](q)-Amalgamsp. 333
Theorem LU: Local Structurep. 340
Theorem LU: The General Casep. 343
Theorem LU: The L[subscript 4](5) Casep. 344
Theorem [characters not reproducible]: Stage 1p. 349
Introductionp. 349
Theorem 1p. 353
Theorems 2 and 3p. 354
Terminally Unbalancing Triplesp. 356
Theorem 4: Balance with Respect to Elements of [epsilon subscript *](G)p. 364
Theorem 5: L[subscript 2](G) [subset equals] [characters not reproducible]p. 366
Theorem 5p. 372
Theorem 6p. 378
Theorem 7p. 382
Theorem [characters not reproducible]: Stages 2 and 3p. 389
Introductionp. 389
Theorem 1: The L[subscript 3](4) Casep. 392
Theorem 1: The L[subscript 2](q), A[subscript 7] Casesp. 394
Fusion in the A[subscript n] Casesp. 401
Theorem 2: Elements of V[subscript 2 superscript *](G)p. 408
The Level of Balancep. 413
Signalizer Functorsp. 424
The L-Preuniqueness of Mp. 432
Theorems 3 and 4: Reductionsp. 440
The L[subscript 3](4) Casep. 443
G [approximate] A[subscript n]p. 444
Theorem [characters not reproducible]: Stage 3p. 446
IV[subscript K]: Preliminary Properties of K-Groupsp. 451
The Groups L[subscript 2](q) and A[subscript 7]p. 451
The Groups L[subscript 3](4) and U[subscript 3](4)p. 462
Product Structuresp. 467
Other Groups of Lie Typep. 471
Alternating Groups and Some Sporadic Groupsp. 477
Generation of K-Groupsp. 480
Pumpupsp. 484
2-Signalizers and p-Stabilityp. 494
K-Groups with Prescribed Local Subgroupsp. 509
Signalizers and Balancep. 512
Representationsp. 515
Background Referencesp. 519
Expository Referencesp. 521
Glossaryp. 523
Indexp. 527
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