The classification of quasithin groups: 1 Structure of strongly quasithin k-groups
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| Language: | English |
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Providence, R.I.
American Mathematical Society
2004
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| Series: | Mathematical surveys and monographs
111 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 477 S. |
| ISBN: | 082183410X |
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| 245 | 1 | 0 | |a The classification of quasithin groups |n 1 |p Structure of strongly quasithin k-groups |c Michael Aschbacher, Stephen D. Smith |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2004 | |
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| 490 | 0 | |a Mathematical surveys and monographs |v ... | |
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| adam_text | Contents
Preface xiii
Volume I: Structure of strongly quasithin ^-groups 1
Introduction to Volume I 3
0.1. Statement of Main Results 3
0.2. An overview of Volume I 5
0.3. Basic results on finite groups 7
0.4. Semisimple quasithin and strongly quasithin /C-groups 7
0.5. The structure of SQTK-groups 7
0.6. Thompson factorization and related notions 8
0.7. Minimal parabolics 10
0.8. Pushing up 10
0.9. Weak closure 11
0.10. The amalgam method 11
0.11. Properties of /C-groups 12
0.12. Recognition theorems 13
0.13. Background References 15
Chapter A. Elementary group theory and the known quasithin groups 19
A.I. Some standard elementary results 19
A.2. The list of quasithin /C-groups: Theorems A, B, and C 32
A.3. A structure theory for Strongly Quasithin /C-groups 41
A.4. Signalizers for groups with X = O2(X) 56
A.5. An ordering on M{T) 61
A.6. A group-order estimate 64
Chapter B. Basic results related to Failure of Factorization 67
B.I. Representations and FF-modules 67
B.2. Basic Failure of Factorization 74
B.3. The permutation module for An and its FF*-offenders 83
B.4. F2-representations with small values of q or q 85
B.5. FF-modules for SQTK-groups 98
B.6. Minimal parabolics 112
B.7. Chapter appendix: Some details from the literature 118
Chapter C. Pushing-up in SQTK-groups 121
C.I. Blocks and the most basic results on pushing-up 121
C.2. More general pushing up in SQTK-groups 143
C.3. Pushing up in nonconstrained 2-locals 148
vii
viii CONTENTS
C.4. Pushing up in constrained 2-locals 151
C.5. Finding a common normal subgroup 154
C.6. Some further pushing up theorems 164
Chapter D. The qrc-lemma and modules with q 2 171
D.I. Stellmacher s grc-Lemma 171
D.2. Properties of q and q: Tl(G, V) and Q(G, V) 177
D.3. Modules with q 2 192
Chapter E. Generation and weak closure 209
E.I. £ -generation and the parameter n(G) 209
E.2. Minimal parabolics under the SQTK-hypothesis 215
E.3. Weak Closure 230
E.4. Values of a for F2-representations of SQTK-groups. 240
E.5. Weak closure and higher Thompson subgroups 242
E.6. Lower bounds on r(G,V) 244
Chapter F. Weak BN-pairs and amalgams 259
F.I. Weak BN-pairs of rank 2 259
F.2. Amalgams, equivalences, and automorphisms 264
F.3. Paths in rank-2 amalgams 269
F.4. Controlling completions of Lie amalgams 273
F.5. Identifying L4(3) via its U4(2)-amalgam 299
F.6. Goldschmidt triples 304
F.7. Coset geometries and amalgam methodology 310
F.8. Coset geometries with b 2 315
F.9. Coset geometries with b 2 and m(Vi) = 1 317
Chapter G. Various representation-theoretic lemmas 327
G.I. Characterizing direct sums of natural SLn(F2«)-modules 327
G.2. Almost-special groups 332
G.3. Some groups generated by transvections 337
G.4. Some subgroups of Sp4(2n) 338
G.5. F2-modules for A6 342
G.6. Modules with m(G,V) 2 345
G.7. Small-degree representations for some SQTK-groups 346
G.8. An extension of Thompson s dihedral lemma 349
G.9. Small-degree representations for more general SQTK-groups 351
G.10. Small-degree representations on extraspecial groups 357
G.ll. Representations on extraspecial groups for SQTK-groups 364
G.12. Subgroups of Sp(V) containing transvections on hyperplanes 370
Chapter H. Parameters for some modules 377
H.I. ft4(2n) on an orthogonal module of dimension 4n (n 1) 378
H.2. SU3(2n) on a natural 6n-dimensional module 378
H.3. Sz(2n) on a natural 4n-dimensional module 379
H.4. (S)L3(2n) on modules of dimension 6 and 9 379
H.5. 7-dimensional permutation modules for La(2) 385
H.6. The 21-dimensional permutation module for £13(2) 386
H.7. Sp4(2n) on natural 4n plus the conjugate 4n*. 388
CONTENTS ix
H.8. A7 on 4 © 4 389
H.9. Aut(Ln(2)) on the natural n plus the dual n* 389
H.10. A foreword on Mathieu groups 392
H.ll. M12 on its 10-dimensional module 392
H.12. 3M22 on its 12-dimensional modules 393
H.13. Preliminaries on the binary code and cocode modules 395
H.14. Some stabilizers in Mathieu groups 396
H.15. The cocode modules for the Mathieu groups 398
H.16. The code modules for the Mathieu groups 402
Chapter I. Statements of some quoted results 407
1.1. Elementary results on cohomology 407
1.2. Results on structure of nonsplit extensions 409
1.3. Balance and 2-components 414
1.4. Recognition Theorems 415
1.5. Characterizations of L4(2) and Sp6(2) 418
1.6. Some results on Tl-sets 424
1.7. Tightly embedded subgroups 425
1.8. Discussion of certain results from the Bibliography 428
Chapter J. A characterization of the Rudvalis group 431
J.I. Groups of type Ru 431
J.2. Basic properties of groups of type Ru 432
J.3. The order of a group of type Ru 438
J.4. A 2F4(2)-subgroup 440
J.5. Identifying G as Ru 445
Chapter K. Modules for SQTK-groups with q(G, V) 2. 451
Notation and overview of the approach 451
K.I. Alternating groups 452
K.2. Groups of Lie type and odd characteristic 453
K.3. Groups of Lie type and characteristic 2 453
K.4. Sporadic groups 457
Bibliography and Index 461
Background References Quoted
(Part 1: also used by GLS) 463
Background References Quoted
(Part 2: used by us but not by GLS) 465
Expository References Mentioned 467
Index 471
Volume II: Main Theorems; the classification of simple QTKE-
groups 479
Introduction to Volume II 481
0.1. Statement of Main Results 481
x CONTENTS
0.2. Context and History 483
0.3. An Outline of the Proof of the Main Theorem 487
0.4. An Outline of the Proof of the Even Type Theorem 495
Part 1. Structure of QTKE-Groups and the Main Case Division 497
Chapter 1. Structure and intersection properties of 2-locals 499
1.1. The collection He 499
1.2. The set C*(G,T) of nonsolvable uniqueness subgroups 503
1.3. The set E*(G,T) of solvable uniqueness subgroups of G 508
1.4. Properties of some uniqueness subgroups 514
Chapter 2. Classifying the groups with M(T) = 1 517
2.1. Statement of main result 518
2.2. Bender groups 518
2.3. Preliminary analysis of the set Fo 521
2.4. The case where Fq is nonempty 527
2.5. Eliminating the shadows with Fq empty 550
Chapter 3. Determining the cases for L € Cj(G,T) 571
3.1. Common normal subgroups, and the grc-lemma for QTKE-groups 571
3.2. The Fundamental Setup, and the case division for C*f(G,T) 578
3.3. Normalizers of uniqueness groups contain Nq(T) 585
Chapter 4. Pushing up in QTKE-groups 605
4.1. Some general machinery for pushing up 605
4.2. Pushing up in the Fundamental Setup 608
4.3. Pushing up L2(2n) 613
4.4. Controlling suitable odd locals 619
Part 2. The treatment of the Generic Case 627
Chapter 5. The Generic Case: L2(2 ) in Cj and n(H) 1 629
5.1. Preliminary analysis of the L2(2n) case 630
5.2. Using weak BN-pairs and the Green Book 646
5.3. Identifying rank 2 Lie-type groups 658
Chapter 6. Reducing L2(2n) to n = 2 and V orthogonal 663
6.1. Reducing L2(2n) to L2(4) 663
6.2. Identifying M22 via L2(4) on the natural module 679
Part 3. Modules which are not FF-modules 693
Chapter 7. Eliminating cases corresponding to no shadow 695
7.1. The cases which must be treated in this part 696
7.2. Parameters for the representations 697
7.3. Bounds on w 698
7.4. Improved lower bounds for r 699
7.5. Eliminating most cases other than shadows 700
7.6. Final elimination of L3(2) on 3 © 3 701
7.7. mini-Appendix: r 2 for L3(2).2 on 3 © 3 703
CONTENTS xi
Chapter 8. Eliminating shadows and characterizing the J4 example 711
8.1. Eliminating shadows of the Fischer groups 711
8.2. Determining local subgroups, and identifying J4 714
8.3. Eliminating L3(2) I 2 on 9 723
Chapter 9. Eliminating fij(2n) on its orthogonal module 729
9.1. Preliminaries 729
9.2. Reducing to n = 2 730
9.3. Reducing to n(H) = 1 732
9.4. Eliminating n(H) = 1 735
Part 4. Pairs in the PSU over F2» for n 1. 739
Chapter 10. The case L G £*f(G,T) not normal in M. 741
10.1. Preliminaries 741
10.2. Weak closure parameters and control of centralizers 742
10.3. The final contradiction 755
Chapter 11. Elimination of L3(2n), Sp4(2n), and G2(2n) for n 1 759
11.1. The subgroups Nc(Vi) for T-invariant subspaces V; of V 760
11.2. Weak-closure parameter values, and (VNg Vi^) 766
11.3. Eliminating the shadow of L4(q) 770
11.4. Eliminating the remaining shadows 775
11.5. The final contradiction 778
Part 5. Groups over F2 785
Chapter 12. Larger groups over F2 in C*f{G,T) 787
12.1. A preliminary case: Eliminating Ln(2) on n© n* 787
12.2. Groups over F2, and the case V a Tl-set in G 794
12.3. Eliminating A7 807
12.4. Some further reductions 812
12.5. Eliminating Ls(2) on the 10-dimensional module 816
12.6. Eliminating A$ on the permutation module 822
12.7. The treatment of Ae on a 6-dimensional module 838
12.8. General techniques for Ln(2) on the natural module 849
12.9. The final treatment of Ln(2), n = 4,5, on the natural module 857
Chapter 13. Mid-size groups over F2 865
13.1. Eliminating L G ££(G, T) with L/O2(L) not quasisimple 865
13.2. Some preliminary results on A5 and Ae 876
13.3. Starting mid-sized groups over F2, and eliminating Ua(3) 884
13.4. The treatment of the 5-dimensional module for A6 896
13.5. The treatment of A5 and A6 when (Vf1) is nonabelian 915
13.6. Finishing the treatment of A5 926
13.7. Finishing the treatment of A6 when (VGl) is nonabelian 935
13.8. Finishing the treatment of A6 946
13.9. Chapter appendix: Eliminating the Ai0-configuration 969
Chapter 14. L3(2) in the FSU, and L2(2) when £f(G,T) is empty 975
xii CONTENTS
14.1. Preliminary results for the case £f (G, T) empty 975
14.2. Starting the L2(2) case of £f empty 981
14.3. First steps; reducing (VGl) nonabelian to extraspecial 989
14.4. Finishing the treatment of (VGl) nonabelian 1005
14.5. Starting the case (VGl) abelian for L3(2) and L2(2) 1013
14.6. Eliminating L2(2) when (VGl) is abelian 1020
14.7. Finishing L3(2) with (VGl) abelian 1042
14.8. The QTKE-groups with C{(G, T) ^ 0 1078
Part 6. The case Cf(G,T) empty 1081
Chapter 15. The case £f(G,T) = 0 1083
15.1. Initial reductions when £f(G,T) is empty 1083
15.2. Finishing the reduction to Mf/CMf(V(Mf)) ~ Oj(2) 1104
15.3. The elimination of Mf/CMf (V(Mf)) = S3 wr Z2 1120
15.4. Completing the proof of the Main Theorem 1155
Part 7. The Even Type Theorem 1167
Chapter 16. Quasithin groups of even type but not even characteristic 1169
16.1. Even type groups, and components in centralizers 1169
16.2. Normality and other properties of components 1173
16.3. Showing L is standard in G 1177
16.4. Intersections of Nq(L) with conjugates of Cg(L) 1182
16.5. Identifying Ji, and obtaining the final contradiction 1194
Bibliography and Index 1205
Background References Quoted
(Part 1: also used by GLS) 1207
Background References Quoted
(Part 2: used by us but not by GLS) 1209
Expository References Mentioned 1211
Index 1215
|
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| spelling | Aschbacher, Michael 1944- Verfasser (DE-588)142400955 aut The classification of quasithin groups 1 Structure of strongly quasithin k-groups Michael Aschbacher, Stephen D. Smith Providence, R.I. American Mathematical Society 2004 XIV, 477 S. txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs 111 Mathematical surveys and monographs ... Smith, Stephen D. Sonstige oth (DE-604)BV039790738 1 Erscheint auch als Online-Ausgabe 978-1-4704-1338-5 Mathematical surveys and monographs 111 (DE-604)BV000018014 111 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024651437&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Aschbacher, Michael 1944- The classification of quasithin groups Mathematical surveys and monographs |
| title | The classification of quasithin groups |
| title_auth | The classification of quasithin groups |
| title_exact_search | The classification of quasithin groups |
| title_full | The classification of quasithin groups 1 Structure of strongly quasithin k-groups Michael Aschbacher, Stephen D. Smith |
| title_fullStr | The classification of quasithin groups 1 Structure of strongly quasithin k-groups Michael Aschbacher, Stephen D. Smith |
| title_full_unstemmed | The classification of quasithin groups 1 Structure of strongly quasithin k-groups Michael Aschbacher, Stephen D. Smith |
| title_short | The classification of quasithin groups |
| title_sort | the classification of quasithin groups structure of strongly quasithin k groups |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024651437&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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