zbMATH — the first resource for mathematics

Profinite groups. 2nd ed. (English) Zbl 1197.20022
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 40. Berlin: Springer (ISBN 978-3-642-01641-7/hbk; 978-3-642-01642-4/ebook). xvi, 464 p. (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.
Three new Appendices present interesting additions. Appendix B contains a new characterisation of free pro-\(\mathcal C\) groups, based on work of D. Harbater and K. F. Stevenson [Adv. Math. 198, No. 2, 623-653 (2005; Zbl 1104.12003)]. Appendix C is based on work of A. Lubotzky [J. Algebra 242, No. 2, 672-690 (2001; Zbl 0985.20017)], and is related to the material of Section 7.8 on presentations of pro-\(p\) groups. Appendix D is based on a paper of B. Steinberg and the first author [Enseign. Math. (2) 56, No. 1-2, 49-72 (2010; Zbl 1209.20024)]. It extends and generalises some classical results, like the Nielsen-Schreier and the Kurosh theorems, using, to quote from the Preface to this edition, “a self-contained and conceptually simpler approach”.
When it appeared, this book represented a major and welcome addition to the literature. Its usefulness to beginners and experts alike is enhanced by these revisions.
See the review of the first edition (2000) in Zbl 0949.20017.

20E18 Limits, profinite groups
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20J05 Homological methods in group theory
12G05 Galois cohomology
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E05 Free nonabelian groups
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups
Full Text: DOI