Finite index subgroups in profinite groups

Abstract

We prove that every subgroup of finite index in a ( topologically) finitely generated profinite group is open. This implies that the topology in such a group is uniquely determined by the group structure. The result follows from a ‘uniformity theorem’ about finite groups: given a group word w that defines a locally finite variety and a natural number d, there exists f= f w ( d) such that in every finite d-generator group G, each element of the verbal subgroup w( G) is a product of f w-values. Similar methods show that in a finite d-generator group, each element of the derived group is a product of g( d) commutators; this implies that the (abstract) derived group in any finitely generated profinite group is closed. To cite this article: N. Nikolov, D. Segal, C. R. Acad. Sci. Paris, Ser. I 337 (2003).