Profinite Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Book 40)

The aim of this book is to serve both as an introduction to profinite groups and as a reference for specialists in some areas of the theory. The book is reasonably self-contained. Profinite groups are Galois groups. As such they are of interest in algebraic number theory. Much of recent research on abstract infinite groups is related to profinite groups because residually finite groups are naturally embedded in a profinite group. In addition to basic facts about general profinite groups, the book emphasizes free constructions (particularly free profinite groups and the structure of their subgroups). Homology and cohomology is described with a minimum of prerequisites. This second edition contains three new appendices dealing with a new characterization of free profinite groups, presentations of pro-p groups and a new conceptually simpler approach to the proof of some classical subgroup theorems. Throughout the text there are additions in the form of new results, improved proofs, typographical corrections, and an enlarged bibliography. The list of open questions has been updated; comments and references have been added about those previously open problems that have been solved after the first edition appeared. Preface to the Second Edition Preface to the First Edition Table of Contents Inverse and Direct Limits Inverse or Projective Limits Direct or Inductive Limits Notes, Comments and Further Reading Profinite Groups Pro- C Groups Basic Properties of Pro- C Groups Existence of Sections Exactness of Inverse Limits of Profinite Groups The Order of a Profinite Group and Sylow Subgroups Generators Finitely Generated Profinite Groups Generators and Chains of Subgroups Procyclic Groups The Frattini Subgroup of a Profinite Group Pontryagin Duality for Profinite Groups Pullbacks and Pushouts Profinite Groups as Galois Groups Notes, Comments and Further Reading Analytic Pro- p Groups Number of Generators of a Group and of Its Profinite Completion Free Profinite Groups Profinite Topologies The Pro- C Completion The Completion Functor Free Pro- C Groups Free Pro- C Group on a Set Converging to 1 Maximal Pro- C Quotient Groups Characterization of Free Pro- C Groups Open Subgroups of Free Pro- C Groups Notes, Comments and Further Reading A Problem of Grothendieck on Completions Some Special Profinite Groups Powers of Elements with Exponents from Z Subgroups of Finite Index in a Profinite Group Profinite Abelian Groups Automorphism Group of a Profinite Group Automorphism Group of a Free Pro-p Group Profinite Frobenius Groups Torsion in the Profinite Completion of a Group Notes, Comments and Further Reading Profinite Torsion Groups Normal Automorphisms Discrete and Profinite Modules Profinite Rings and Modules Duality Between Discrete and Profinite Modules Free Profinite Modules G -modules and Complete Group Algebras The Complete Group Algebra Projective and Injective Modules Complete Tensor Products Profinite G -spaces Free Profinite [-[R G]-]-modules Diagonal Actions Notes, Comments and Further Reading The Magnus Algebra and Free Pro- p Groups Homology and Cohomology of Profinite Groups Review of Homological Algebra Right and Left Derived Functors Bifunctors The Ext Functors The Tor Functors Cohomology with Coefficients in DMod([-[R G]-]) Standard Resolutions The Homogeneous Bar Resolution The Inhomogeneous Bar Resolution Homology with Coefficients in PMod([-[R G]-]) Cohomology Groups with Coefficients in DMod(G) The Functorial Behavior of Hn(G,A) and Hn(G,A) The Inflation Map Hn(G,A) as Derived Functors on DMod(G) Special Mappings The Restriction Map in Cohomology The Corestriction Map in Cohomology The Corestriction Map in Homology The Restriction Map in Homology Homology and Cohomology Groups in Low Dimensions H2(G,A) and Extensions of Profinite Groups Extensions of Profinite Groups with Abelian Kernel Induced and Coinduced Modules The Induced Module IndGH(B) for H Open Notes, Comments and Further Reading Cohomological Dimension Basic Properties of Dimension The Lyndon-Hochschild-Serre Spectral Sequence Cohomological Dimension of Subgroups Cohomological Dimension of Normal Subgroups and Quotients Groups G with cd p(G)