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Symplectic Amalgams Book

Symplectic Amalgams
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Symplectic Amalgams, The account is for the most part self-contained and the wealth of detail makes this book an excellent introduction to these recent developments for graduate students, as well as a valuable resource and reference for specialists in the area., Symplectic Amalgams
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  • Symplectic Amalgams
  • Written by author Christopher Parker
  • Published by Springer-Verlag New York, LLC, 1/28/2013
  • The account is for the most part self-contained and the wealth of detail makes this book an excellent introduction to these recent developments for graduate students, as well as a valuable resource and reference for specialists in the area.
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1 Introduction 1
1.1 Symplectic Amalgams 2
1.2 Goldchmidt G[subscript 4]-amalgam Again 19
2 Preliminaries 21
2.1 Some Group Theory Results 22
2.2 Some Representation Theory Results 29
2.3 Sesquilinear Forms 35
2.4 Two Theorems of McLaughlin 40
2.5 Ultraspecial and Extraspecial Groups 42
2.6 Tensor Products and Group Actions on p-Groups 45
2.7 The Goldschmidt Amalgams 48
3 The Structure of SL[subscript 2](q) and its Modules 53
3.1 Group Theoretic Properties of SL[subscript 2](q) 53
3.2 Modules for SL[subscript 2](q) 56
4 Elementary Properties of Symplectic Amalgams 67
4.1 The Coset Graph 67
4.2 Proof of Theorem 1.6 69
5 The Structure of Q[subscript [alpha]] 81
6 The L[subscript [beta]]-Chief Factors in V[subscript [beta]] 93
7 Reduced Symplectic Amalgams 113
7.1 A Reduced Symplectic Subamalgam 113
7.2 Reduced Amalgams and Consequences of Theorem 6.1 118
8 The Largest Normal p'-Subgroup of L[subscript [beta]]/Q[subscript [beta]] 125
9 The Components of L[subscript [beta]]/Q[subscript [beta]] 141
9.1 The Action of L[subscript [beta]] on Comp[subscript p](L[subscript [beta]]) 142
9.2 Two or more Normal Components in L[subscript [beta]]/Q[subscript [beta]] 148
10 The Reduction to Quasisimple when C[subscript U[alpha]](U[subscript [alpha]]/Z[subscript [alpha]]) [neither less than, nor equal to] Q[subscript [beta]] 163
11 A First Look at the Amalgams with |V[subscript [beta]]/Z(V[subscript [beta]])| = q[superscript 4] 177
11.1 A Characteristic 3 Amalgam 178
11.2 The Proof of Theorem 11.1 180
12 The Story so Far 187
13 Groups of Lie Type 189
13.1 Weyl Groups and Parabolic Subgroups 193
13.2 Sylow p-subgroups of Lie Type Groups 195
13.3 Automorphisms and Centres 197
13.4 The Order of Abelian p-subgroups 199
13.5 Extremal Subgroups 202
14 Modules for Groups of Lie Type 215
14.1 Modules in Characteristic p 215
14.2 Module Results for Low Rank Groups of Lie Type 220
14.3 Modules for Lie Type Groups and (2,q)-Transvections 224
14.4 Natural Modules for Orthogonal Groups 231
14.5 Natural Modules for the Symplectic Groups 234
14.6 Natural Modules for G[subscript 2](q) 237
14.7 Some Spin Modules 239
14.8 Modules for Lie Type Groups in Non-defining Characteristic 240
14.9 Some Non-containments 247
15 Sporadic Simple Groups and Their Modules 249
16 Alternating Groups and Their Modules 257
17 Rank One Groups 265
18 Lie Type Groups in Characteristic p and Rank [greater than or equal to] 2 271
18.1 A Subamalgam of A 272
18.2 The Examples 285
18.3 L[subscript [beta]]/Q[subscript [beta]] a Symplectic Group and V[subscript [beta]]/Z(V[subscript [beta]]) a Spin Module 290
19 Lie Type Groups and Natural Modules 293
19.1 The Symplectic and Orthogonal Groups 295
19.2 Sp[subscript 4](2) - A Special Case 305
19.3 Groups of Type G[subscript 2](q) 310
20 Lie Type Groups in Characteristic not p 313
21 Alternating Groups 315
21.1 Large Alternating Groups 315
21.2 Small Alternating Groups 319
22 Sporadic Simple Groups 325
23 The Proof of the Main Theorems 327
24 A Brief Survey of Amalgam Results 331
24.1 Amalgam Results 331
24.2 Pushing-up 340
References 345
Index 357


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