Finite index subgroups in profinite groups

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 = fw (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. © 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved