On a pseudovariety of finite supersolvable groups

We introduce the pseudovariety of finite groups [Formula: see text], where [Formula: see text] is the set of all primes. We show that [Formula: see text] consists of all finite supersolvable groups with elementary abelian derived subgroup and abelian Sylow subgroups, and so [Formula: see text] has decidable membership problem. We prove that it is decidable whether or not a finitely generated subgroup of a free group is closed or dense for the pro-[Formula: see text] topology. We consider also the pseudovariety of finite groups [Formula: see text] (where [Formula: see text] is a prime and [Formula: see text] divides [Formula: see text]). We study the pro-[Formula: see text] topology on a free group and construct the unique generator of minimum size of the pseudovariety [Formula: see text]. Finally, we prove that the variety of groups generated by [Formula: see text] is the variety of all metabelian groups, obtaining also results on the varieties generated by a Baumslag-Solitar group of the form [Formula: see text] for [Formula: see text] prime.