Profinite Groups (eBook)

Preface to the Second Edition 6
Preface to the First Edition 7
Table of Contents 10
Inverse and Direct Limits 14
Inverse or Projective Limits 14
Direct or Inductive Limits 27
Notes, Comments and Further Reading 31
Profinite Groups 32
Pro- C Groups 32
Basic Properties of Pro- C Groups 41
Existence of Sections 42
Exactness of Inverse Limits of Profinite Groups 44
The Order of a Profinite Group and Sylow Subgroups 45
Generators 55
Finitely Generated Profinite Groups 57
Generators and Chains of Subgroups 60
Procyclic Groups 64
The Frattini Subgroup of a Profinite Group 65
Pontryagin Duality for Profinite Groups 71
Pullbacks and Pushouts 79
Profinite Groups as Galois Groups 81
Notes, Comments and Further Reading 85
Analytic Pro- p Groups 86
Number of Generators of a Group and of Its Profinite Completion 87
Free Profinite Groups 88
Profinite Topologies 88
The Pro- C Completion 91
The Completion Functor 94
Free Pro- C Groups 98
Free Pro- C Group on a Set Converging to 1 101
Maximal Pro- C Quotient Groups 109
Characterization of Free Pro- C Groups 111
Open Subgroups of Free Pro- C Groups 126
Notes, Comments and Further Reading 129
A Problem of Grothendieck on Completions 130
Some Special Profinite Groups 132
Powers of Elements with Exponents from Z 132
Subgroups of Finite Index in a Profinite Group 133
Profinite Abelian Groups 142
Automorphism Group of a Profinite Group 145
Automorphism Group of a Free Pro-p Group 150
Profinite Frobenius Groups 155
Torsion in the Profinite Completion of a Group 161
Notes, Comments and Further Reading 167
Profinite Torsion Groups 169
Normal Automorphisms 170
Discrete and Profinite Modules 172
Profinite Rings and Modules 172
Duality Between Discrete and Profinite Modules 178
Free Profinite Modules 179
G -modules and Complete Group Algebras 182
The Complete Group Algebra 183
Projective and Injective Modules 185
Complete Tensor Products 190
Profinite G -spaces 193
Free Profinite [-[R G]-]-modules 202
Diagonal Actions 203
Notes, Comments and Further Reading 206
The Magnus Algebra and Free Pro- p Groups 206
Homology and Cohomology of Profinite Groups 208
Review of Homological Algebra 208
Right and Left Derived Functors 212
Bifunctors 213
The Ext Functors 214
The Tor Functors 215
Cohomology with Coefficients in DMod([-[R G]-]) 216
Standard Resolutions 218
The Homogeneous Bar Resolution 218
The Inhomogeneous Bar Resolution 219
Homology with Coefficients in PMod([-[R G]-]) 221
Cohomology Groups with Coefficients in DMod(G) 225
The Functorial Behavior of Hn(G,A) and Hn(G,A) 227
The Inflation Map 228
Hn(G,A) as Derived Functors on DMod(G) 233
Special Mappings 237
The Restriction Map in Cohomology 237
The Corestriction Map in Cohomology 238
The Corestriction Map in Homology 242
The Restriction Map in Homology 242
Homology and Cohomology Groups in Low Dimensions 244
H2(G,A) and Extensions of Profinite Groups 246
Extensions of Profinite Groups with Abelian Kernel 251
Induced and Coinduced Modules 255
The Induced Module IndGH(B) for H Open 260
Notes, Comments and Further Reading 262
Cohomological Dimension 264
Basic Properties of Dimension 264
The Lyndon-Hochschild-Serre Spectral Sequence 269
Cohomological Dimension of Subgroups 274
Cohomological Dimension of Normal Subgroups and Quotients 279
Groups G with cd p(G) < =1
Projective Profinite Groups 284
Free Pro- p Groups and Cohomological Dimension 288
Generators and Relators for Pro- p Groups 291
Cup Products 295
Notes, Comments and Further Reading 301
Pro- p Groups G with one Defining Relator 303
Poincaré Groups 303
Normal Subgroups of Free Pro-C Groups 305
Normal Subgroup Generated by a Subset of a Basis 306
The S-rank 308
Accessible Subgroups 314
Accessible Subgroups H with w0(F/H)< rank(F)
Homogeneous Pro-C Groups 325
Normal Subgroups 338
Proper Open Subgroups of Normal Subgroups 347
The Congruence Kernel of SL2(Z) 352
Sufficient Conditions for Freeness 353
Characteristic Subgroups of Free Pro-C Groups 360
Notes, Comments and Further Reading 363
Free Constructions of Profinite Groups 364
Free Pro-C Products 364
Amalgamated Free Pro-C Products 378
Cohomological Characterizations of Amalgamated Products 385
Pro-C HNN-extensions 393
Notes, Comments and Further Reading 399
Open Questions 404
Spectral Sequences 407
Spectral Sequences 407
Positive Spectral Sequences 409
The Base Terms 410
The Fiber Terms 410
Spectral Sequence of a Filtered Complex 412
Spectral Sequences of a Double Complex 415
Notes, Comments and Further Reading 416
A Different Characterization of Free Profinite Groups 417
Free vs Projective Profinite Groups 417
Notes, Comments and Further Reading 418
Presentations of Profinite Groups 419
Presentations 419
Relation Modules 422
Notes, Comments and Further Reading 426
Wreath Products and Some Subgroup Theorems 428
Permutational Wreath Products 428
The Nielsen-Schreier Theorem for Free Pro-C Groups 432
The Kurosh Subgroup Theorem for Profinite Groups 434
Kurosh Systems 437
Subgroups of Projective Groups 441
Quasifree Profinite Groups 443
Notes, Comments and Further Reading 447
Bibliography 448
Index of Symbols 459
Index of Authors 462
Index of Terms 465