You are here

Profinite Groups

Luis Ribes and Pavel Zalesskii
Publication Date: 
Number of Pages: 
Ergebnisse der Mathematik und ihrer Grenzgebiete
[Reviewed by
Fernando Q. Gouvêa
, on

Profinite groups first arose in Galois Theory. The classical theory applied only to finite field extensions, and the corresponding Galois groups were finite. When one decides to think about infinite algebraic extensions and the corresponding infinite groups, however, one finds that the Galois correspondence doesn't work correctly. Specifically, if F/K is an infinite extension, there may be two different subgroups of Gal(F/K) that have the same fixed field. 

In 1928, W. Krull explained how to fix the problem. He used the standard trick: when passing from the finite situation to the infinite, we need to topologize our objects if we want them to remain well-behaved. Krull put a topology on the infinite group Gal(F/K) and showed that the correspondence worked correctly if restricted to closed subgroups. In the finite case the Krull topology is just the discrete topology and we recover standard Galois theory.

With the Krull topology, Gal(F/K) becomes a compact, Hausdorff, totally disconnected group. For a while such groups were known as "groups of Galois type," but now they are known as profinite, because any such group G has a family of open normal subgroups U such that G can be obtained as the inverse limit of the finite groups G/U. Of course, finite groups (with the discrete topology) are profinite as well. This can then be immediately generalized: a group G is "pro-something" if it can be obtained as the inverse limit of finite "something" groups. So we can talk about procyclic groups, prosolvable groups, etc.

Profinite groups are interesting because they are so similar to finite groups. For example, there is a Sylow theorem describing pro-p subgroups of profinite groups. A first moment in the theory (and in this book) happens when one tries to prove as many extensions of this sort as possible. But things become more interesting as one gets into studying group actions and modules over profinite groups. These occur all over Galois theory, of course, since Galois groups act naturally on various things. So we get into representation theory and cohomology.

In this book, Ribes and Zalesskii survey the general theory of profinite groups (and more generally of pro-C groups, where C is a sufficiently nice class of finite 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. It is by no means light reading, but it covers the ground clearly and efficiently, with full proofs. Each chapter concludes with an extensive section of notes, including recent developments and open questions. The open questions are helpfully collected in a special section at the end.

The original edition of Profinite Groups appeared in 2000. The new edition differs by the addition of three new appendices (labeled B, C, D) and many corrections. Most notable, however, is that all the notes have been updated. One of the old open questions has been completely solved and now appears as a Theorem, and new open questions have been added.

The Ergebnisse series describes itself as "a series of modern surveys in mathematics." That's exactly what this book provides for the theory of profinite groups. 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 is Professor of Mathematics at Colby College in Waterville, ME.