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.