On the closure of cyclic subgroups of a free group in pro-V topologies

Abstract

We determine the closure of a cyclic subgroup $H$ of a free group for the pro-{\bf V} topology when {\bf 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-${\mathbf{G}}_p$ and pro-${\mathbf{V}}_p$ topologies, where ${\mathbf{G}}_p$ and ${\mathbf{V}}_p$ denote respectively the pseudovariety of finite $p$-groups and the pseudovariety of finite groups having a normal Sylow $p$-subgroup with quotient an abelian group of exponent dividing $p-1$. More generally, given any nonempty set $P$ of primes, we consider the pseudovariety ${\mathbf{G}}_P$ of all finite groups having order a product of primes in $P$.