Profinite Groups

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.

• Galois group
• Profinite group
• algebra
• cohomology
• homology
• number theory

“The book has an extensive bibliography, which appears to be remarkably complete, up to the very recent developments. An excellent guide to this vast amount of literature is provided by the closing sections of each chapter … . the many important and current topics dealt with here are treated with admirable completeness and clarity … . The book is very valuable as a reference work, and offers several excellent choices as a textbook for a graduate course.” (A.Caranti, zbMATH 0949.20017, 2022)

From the reviews of the second edition:

“In this book, Ribes and Zalesskii survey the general theory of profinite groups … . They cover all the important examples and do a particularly fine job of explaining the representation theory and the cohomology theory of profinite groups. … Each chapter concludes with an extensive section of notes, including recent developments and open questions. … It will be extremely useful to researchers in field and even more so to those who (like me) use profinite groups in their own work.” (Fernando Q. Gouvêa, The Mathematical Association of America, August, 2010)

“This valuable book works well both as an introduction to the subject of profinite groups, and as a reference for some specific areas. This second edition presents an updated and enlarged bibliography. More open questions have been added, and solutions are provided for the problems from the previous edition that have been settled since.”­­­ (A. Caranti, Zentralblatt MATH, Vol. 1197, 2010)

• Dept. Mathematics & Statistics, Carleton University, Ottawa, Canada

  Luis Ribes

• Depto. Matemática, Universidade de Brasília, Brasilia, Brazil

  Pavel Zalesskii

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