The notion of a profinite group arises out of a beautiful idea of Krull. Let \(E/F\) be a Galois extension of fields, and let \(G\) be its Galois group. If \(E/F\) is finite-dimensional, then every subgroup of \(G\) has the form \(L'=\{g\in G: ag=a\) for all \(a\in L\}\) for some intermediate field \(L\). If \(E/F\) is infinite-dimensional, this is not the case. The idea of Krull was to make \(G\) into a topological group, by taking the normal subgroups \(N'\), as \(N\) ranges over all intermediate fields such that \(N/F\) is finite-dimensional and Galois, as a fundamental system of neighbourhoods for a topology on \(G\). He then showed that the subgroups of the form \(L'\), for some intermediate field \(L\), are exactly the subgroups that are closed in this topology.
With respect to the Krull topology \(G\) is a compact, totally disconnected group. Such a group is said to be profinite. Equivalently, a profinite group is an inverse limit of an inverse system of finite groups. It is a result of H. Leptin (Theorem 2.11.5 in the book under review) that every profinite group arises as the Galois group of a suitable Galois extension. (Here and in the following, we refer to the book under review for detailed bibliographical references.)
The treatment of profinite groups given by J.-P. Serre [in Cohomologie galoisienne. Cours au College de France, 1962-1963 (French) (Lect. Notes Math. 5, Springer-Verlag, Berlin), (1964; Zbl 0128.26303)] has been extremely influential for further developments. Other books, such as that of M. D. Fried and M. Jarden [Field arithmetic (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 11., Springer-Verlag, Berlin), (1986; Zbl 0625.12001)] have developed important aspects of the theory, with field-theoretic applications or motivations in mind.
The first texts expressly devoted to the theory of profinite groups were written by L. Ribes [Introduction to profinite groups and Galois cohomology (Queen’s Papers in Pure and Applied Mathematics 24, Queen’s University, Kingston, Ontario), (1970; Zbl 0221.12013) and Grupos profinitos, grupos libres y productos libres (Spanish) (Monografias del Instituto de Matematicas 5, Universidad Nacional Autonoma de Mexico, Mexico City), (1977; Zbl 0384.20002)], but they both had limited circulation. The exciting developments in the theory of (profinite) \(p\)-adic analytic groups have been recorded in the books by J. D. Dixon, M. P. F du Sautoy, A. Mann and D. Segal [Analytic pro-\(p\) groups (Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, Cambridge), (1999; Zbl 0934.20001)] and by G. Klaas, C. R. Leedham-Green and W. Plesken [Linear pro-\(p\)-groups of finite width (Lect. Notes Math. 1674, Springer, Berlin), (1997; Zbl 0901.20013)]. It was not until 1998 that the first generally available text on the abstract theory of profinite group, by J. S. Wilson, appeared [Profinite groups (Lond. Math. Soc. Monogr., New Ser. 19, Clarendon Press, Oxford), (1998; Zbl 0909.20001)].
The book under review represents a further contribution to the literature on the subject. It is loosely based on Ribes’ previous books, and it is intended to be followed by another book, entitled “Profinite Trees”, on actions of profinite groups on profinite trees. Chapter 1 deals with inverse and direct limits. In Chapter 2 the notion of pro-\(\mathcal C\)-groups is introduced: this encompasses both general profinite groups and special classes such as pro-\(p\) groups. Free profinite groups are introduced in Chapter 3; in Chapter 8, closed normal subgroups of these are studied. Chapter 4 deals with particular classes, such as profinite Abelian groups and Frobenius profinite groups. Chapters 5-7 are devoted to the homological aspects of profinite groups. As in Wilson’s book, both discrete and profinite modules are considered. Finally, Chapter 9 deals with the basic free constructions of the theory, such as free products, and represents the beginning of the theory of profinite trees to be dealt with in the forthcoming book mentioned above. There is an appendix on spectral sequences.
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, that offer useful comments on the history of the subject.
The authors have designed the book under review to be “an introduction to profinite groups and […] a reference for the specialists in some areas of the theory”. Although they “have [not] tried to be encyclopedic”, the many important and current topics dealt with here are treated with admirable completeness and clarity; also, the choice of topics is in part complementary to that of Wilson’s book mentioned above, so that the two texts end up covering a fairly wide area. The book is very valuable as a reference work, and offers several excellent choices as a textbook for a graduate course. It represents another welcome addition to the growing literature on profinite groups. Too bad its price of about 100 Euro, though not unusually high, is going to make it a library-only item for the moment.