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Profinite Groups
Luis Ribes, Pavel Zalesskii
Verlag Springer-Verlag, 2010
ISBN 9783642016424 , 483 Seiten
2. Auflage
Format PDF
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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
281
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)
318
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
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