  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 $P$.
