Closed Subgroups of Profinite Groups

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 72d2 + 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 Γ be a finitely generated prosoluble group. Then each term of the lower central series of Γ and each power subgroup Γn is closed. 1991 Mathematics Subject Classification: 20E18, 20D10.