Closed subgroups of profinite groups. (English) Zbl 1030.20017

Segal, Dan

Proc. Lond. Math. Soc., III. Ser. 81, No. 1, 29-54 (2000).

Theorem 1 asserts that in a finitely generated prosoluble group, every subgroup of finite index is open. This generalises an old result of Serre about pro-\(p\) groups. It follows by a standard argument from Theorem 2: In a \(d\)-generator finite soluble group, every element of the derived group is equal to a product of \(72d^2+46d\) commutators.
This result about finite soluble groups is proved by induction on the order of the group, and is elementary though rather intricate. The essence of the proof lies in reducing the problem to one about the number of solutions of quadratic equations over a finite field.
Corollaries include the following: Let \(\Gamma\) be a finitely generated prosoluble group. Then each term of the lower central series of \(\Gamma\) and each power subgroup \(\Gamma^n\) is closed.

Reviewer: Dan Segal (Oxford)

MSC:

20E18	Limits, profinite groups
20D10	Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20E07	Subgroup theorems; subgroup growth
20F05	Generators, relations, and presentations of groups
20F12	Commutator calculus
20F14	Derived series, central series, and generalizations for groups

Keywords:

subgroups of finite index; derived groups; products of commutators; finitely generated prosoluble groups; finite soluble groups; lower central series; closed subgroups