On the closure of cyclic subgroups of a free group in some profinite topologies

Abstract

We determine the closure of a cyclic subgroup H of a free group for the pro-V topology when V is an extension-closed pseudovariety of finite groups. We show that H is always closed for the pro-nilpotent topology and compute its closure for the pro-G(p) and pro- V-p topologies, where G(p) is the pseudovariety of finite p-groups and V-p superset of G(p) denote the pseudovarieties involved in the join decomposition of the pseudovariety of finite supersolvable groups. More generally, given any nonempty set P of primes, we consider the stated problem for the pseudovariety G(P) of all finite groups having order a product of primes in P.