简介
This is the second volume of a two-volume set, which take up where Geoff Mason left off in the 1980x on the issue of quasithin groups of even characteristics. V.2 gives the proof that the groups listed in the Main Theorem are the simple quasithin groups of even characteristics--all of whose proper simple sections are known simple groups. This lively and comprehensive proof includes the structure of QTKE-groups and the main case division, treatments of the generic case and modules which are not FF-modules, and certain pairs in the FSU. While the two volumes address one issue of mathematics, they also serve as models of presentation for analyses Annotation 漏2004 Book News, Inc., Portland, OR (booknews.com)
目录
Preface p. xiii
Structure of strongly quasithin K-groups p. 1
Introduction to Volume I p. 3
Statement of Main Results p. 3
An overview of Volume I p. 5
Basic results on finite groups p. 7
Semisimple quasithin and strongly quasithin K-groups p. 7
The structure of SQTK-groups p. 7
Thompson factorization and related notions p. 8
Minimal parabolics p. 10
Pushing up p. 10
Weak closure p. 11
The amalgam method p. 11
Properties of K-groups p. 12
Recognition theorems p. 13
Background References p. 15
Elementary group theory and the known quasithin groups p. 19
Some standard elementary results p. 19
The list of quasithin K-groups: Theorems A, B, and C p. 32
A structure theory for Strongly Quasithin K-groups p. 41
Signalizers for groups with X = O[superscript 2] (X) p. 56
An ordering on M(T) p. 61
A group-order estimate p. 64
Basic results related to Failure of Factorization p. 67
Representations and FF-modules p. 67
Basic Failure of Factorization p. 74
The permutation module for A[subscript n] and its FF*-offenders p. 83
F[subscript 2]-representations with small values of q or q p. 85
FF-modules for SQTK-groups p. 98
Minimal parabolics p. 112
Chapter appendix: Some details from the literature p. 118
Pushing-up in SQTK-groups p. 121
Blocks and the most basic results on pushing-up p. 121
More general pushing up in SQTK-groups p. 143
Pushing up in nonconstrained 2-locals p. 148
Pushing up in constrained 2-locals p. 151
Finding a common normal subgroup p. 154
Some further pushing up theorems p. 164
The qrc-lemma and modules with q [less than or equal] 2 p. 171
Stellmacher's qrc-Lemma p. 171
Properties of q and q: R(G, V) and Q(G, V) p. 177
Modules with q [less than or equal] 2 p. 192
Generation and weak closure p. 209
[epsilon]-generation and the parameter n(G) p. 209
Minimal parabolics under the SQTK-hypothesis p. 215
Weak Closure p. 230
Values of a for F[subscript 2]-representations of SQTK-groups p. 240
Weak closure and higher Thompson subgroups p. 242
Lower bounds on r(G, V) p. 244
Weak BN-pairs and amalgams p. 259
Weak BN-pairs of rank 2 p. 259
Amalgams, equivalences, and automorphisms p. 264
Paths in rank-2 amalgams p. 269
Controlling completions of Lie amalgams p. 273
Identifying L[subscript 4](3) via its U[subscript 4](2)-amalgam p. 299
Goldschmidt triples p. 304
Coset geometries and amalgam methodology p. 310
Coset geometries with b [greater than sign] 2 p. 315
Coset geometries with b [greater than sign] 2 and m (V[subscript 1]) = 1 p. 317
Various representation-theoretic lemmas p. 327
Characterizing direct sums of natural SL[subscript n](F[subscript 2 superscript e])-modules p. 327
Almost-special groups p. 332
Some groups generated by transvections p. 337
Some subgroups of Sp[subscript 4](2[superscript n]) p. 338
F[subscript 2]-modules for A[subscript 6] p. 342
Modules with m(G, V) [less than or equal] 2 p. 345
Small-degree representations for some SQTK-groups p. 346
An extension of Thompson's dihedral lemma p. 349
Small-degree representations for more general SQTK-groups p. 351
Small-degree representations on extraspecial groups p. 357
Representations on extraspecial groups for SQTK-groups p. 364
Subgroups of Sp(V) containing transvections on hyperplanes p. 370
Parameters for some modules p. 377
[Omega superscript epsilon subscript 4](2[superscript n]) on an orthogonal module of dimension 4n (n [greater than sign] 1) p. 378
SU[subscript 3](2[superscript n]) on a natural 6n-dimensional module p. 378
Sz(2[superscript n]) on a natural 4n-dimensional module p. 379
(S)L[subscript 3](2[superscript n]) on modules of dimension 6 and 9 p. 379
7-dimensional permutation modules for L[subscript 3](2) p. 385
The 21-dimensional permutation module for L[subscript 3](2) p. 386
Sp[subscript 4](2[superscript n]) on natural 4n plus the conjugate 4n[superscript t] p. 388
A[subscript 7] on 4 [plus sign in circle] 4 p. 389
Aut(L[subscript n](2)) on the natural n plus the dual n* p. 389
A foreword on Mathieu groups p. 392
M[subscript 12] on its 10-dimensional module p. 392
3M[subscript 22] on its 12-dimensional modules p. 393
Preliminaries on the binary code and cocode modules p. 395
Some stabilizers in Mathieu groups p. 396
The cocode modules for the Mathieu groups p. 398
The code modules for the Mathieu groups p. 402
Statements of some quoted results p. 407
Elementary results on cohomology p. 407
Results on structure of nonsplit extensions p. 409
Balance and 2-components p. 414
Recognition Theorems p. 415
Characterizations of L[subscript 4](2) and Sp[subscript 6](2) p. 418
Some results on TI-sets p. 424
Tightly embedded subgroups p. 425
Discussion of certain results from the Bibliography p. 428
A characterization of the Rudvalis group p. 431
Groups of type Ru p. 431
Basic properties of groups of type Ru p. 432
The order of a group of type Ru p. 438
A [superscript 2]F[subscript 4](2)-subgroup p. 440
Identifying G as Ru p. 445
Modules for SQTK-groups with q(G, V) [less than or equal] 2 p. 451
Notation and overview of the approach p. 451
Alternating groups p. 452
Groups of Lie type and odd characteristic p. 453
Groups of Lie type and characteristic 2 p. 453
Sporadic groups p. 457
Bibliography and Index p. 461
Background References Quoted (Part 1: also used by GLS) p. 463
Background References Quoted (Part 2: used by us but not by GLS) p. 465
Expository References Mentioned p. 467
Index p. 471
Structure of strongly quasithin K-groups p. 1
Introduction to Volume I p. 3
Statement of Main Results p. 3
An overview of Volume I p. 5
Basic results on finite groups p. 7
Semisimple quasithin and strongly quasithin K-groups p. 7
The structure of SQTK-groups p. 7
Thompson factorization and related notions p. 8
Minimal parabolics p. 10
Pushing up p. 10
Weak closure p. 11
The amalgam method p. 11
Properties of K-groups p. 12
Recognition theorems p. 13
Background References p. 15
Elementary group theory and the known quasithin groups p. 19
Some standard elementary results p. 19
The list of quasithin K-groups: Theorems A, B, and C p. 32
A structure theory for Strongly Quasithin K-groups p. 41
Signalizers for groups with X = O[superscript 2] (X) p. 56
An ordering on M(T) p. 61
A group-order estimate p. 64
Basic results related to Failure of Factorization p. 67
Representations and FF-modules p. 67
Basic Failure of Factorization p. 74
The permutation module for A[subscript n] and its FF*-offenders p. 83
F[subscript 2]-representations with small values of q or q p. 85
FF-modules for SQTK-groups p. 98
Minimal parabolics p. 112
Chapter appendix: Some details from the literature p. 118
Pushing-up in SQTK-groups p. 121
Blocks and the most basic results on pushing-up p. 121
More general pushing up in SQTK-groups p. 143
Pushing up in nonconstrained 2-locals p. 148
Pushing up in constrained 2-locals p. 151
Finding a common normal subgroup p. 154
Some further pushing up theorems p. 164
The qrc-lemma and modules with q [less than or equal] 2 p. 171
Stellmacher's qrc-Lemma p. 171
Properties of q and q: R(G, V) and Q(G, V) p. 177
Modules with q [less than or equal] 2 p. 192
Generation and weak closure p. 209
[epsilon]-generation and the parameter n(G) p. 209
Minimal parabolics under the SQTK-hypothesis p. 215
Weak Closure p. 230
Values of a for F[subscript 2]-representations of SQTK-groups p. 240
Weak closure and higher Thompson subgroups p. 242
Lower bounds on r(G, V) p. 244
Weak BN-pairs and amalgams p. 259
Weak BN-pairs of rank 2 p. 259
Amalgams, equivalences, and automorphisms p. 264
Paths in rank-2 amalgams p. 269
Controlling completions of Lie amalgams p. 273
Identifying L[subscript 4](3) via its U[subscript 4](2)-amalgam p. 299
Goldschmidt triples p. 304
Coset geometries and amalgam methodology p. 310
Coset geometries with b [greater than sign] 2 p. 315
Coset geometries with b [greater than sign] 2 and m (V[subscript 1]) = 1 p. 317
Various representation-theoretic lemmas p. 327
Characterizing direct sums of natural SL[subscript n](F[subscript 2 superscript e])-modules p. 327
Almost-special groups p. 332
Some groups generated by transvections p. 337
Some subgroups of Sp[subscript 4](2[superscript n]) p. 338
F[subscript 2]-modules for A[subscript 6] p. 342
Modules with m(G, V) [less than or equal] 2 p. 345
Small-degree representations for some SQTK-groups p. 346
An extension of Thompson's dihedral lemma p. 349
Small-degree representations for more general SQTK-groups p. 351
Small-degree representations on extraspecial groups p. 357
Representations on extraspecial groups for SQTK-groups p. 364
Subgroups of Sp(V) containing transvections on hyperplanes p. 370
Parameters for some modules p. 377
[Omega superscript epsilon subscript 4](2[superscript n]) on an orthogonal module of dimension 4n (n [greater than sign] 1) p. 378
SU[subscript 3](2[superscript n]) on a natural 6n-dimensional module p. 378
Sz(2[superscript n]) on a natural 4n-dimensional module p. 379
(S)L[subscript 3](2[superscript n]) on modules of dimension 6 and 9 p. 379
7-dimensional permutation modules for L[subscript 3](2) p. 385
The 21-dimensional permutation module for L[subscript 3](2) p. 386
Sp[subscript 4](2[superscript n]) on natural 4n plus the conjugate 4n[superscript t] p. 388
A[subscript 7] on 4 [plus sign in circle] 4 p. 389
Aut(L[subscript n](2)) on the natural n plus the dual n* p. 389
A foreword on Mathieu groups p. 392
M[subscript 12] on its 10-dimensional module p. 392
3M[subscript 22] on its 12-dimensional modules p. 393
Preliminaries on the binary code and cocode modules p. 395
Some stabilizers in Mathieu groups p. 396
The cocode modules for the Mathieu groups p. 398
The code modules for the Mathieu groups p. 402
Statements of some quoted results p. 407
Elementary results on cohomology p. 407
Results on structure of nonsplit extensions p. 409
Balance and 2-components p. 414
Recognition Theorems p. 415
Characterizations of L[subscript 4](2) and Sp[subscript 6](2) p. 418
Some results on TI-sets p. 424
Tightly embedded subgroups p. 425
Discussion of certain results from the Bibliography p. 428
A characterization of the Rudvalis group p. 431
Groups of type Ru p. 431
Basic properties of groups of type Ru p. 432
The order of a group of type Ru p. 438
A [superscript 2]F[subscript 4](2)-subgroup p. 440
Identifying G as Ru p. 445
Modules for SQTK-groups with q(G, V) [less than or equal] 2 p. 451
Notation and overview of the approach p. 451
Alternating groups p. 452
Groups of Lie type and odd characteristic p. 453
Groups of Lie type and characteristic 2 p. 453
Sporadic groups p. 457
Bibliography and Index p. 461
Background References Quoted (Part 1: also used by GLS) p. 463
Background References Quoted (Part 2: used by us but not by GLS) p. 465
Expository References Mentioned p. 467
Index p. 471
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