We introduce the pseudovariety of finite groups $\mathbf{ U} = \displaystyle\bigvee_{ p \in \mathbb{ P}} { \bf Ab}(p) \ast { \bf Ab}(p-1)$, where $\mathbb{ P}$ is the set of all primes. We show that $\mathbf{ U}$ consists of all finite supersolvable groups with elementary abelian derived subgroup and abelian Sylow subgroups, being therefore decidable. We prove that it is decidable whether or not a finitely generated subgroup of a free group is closed or dense for the pro-${ \bf U}$ topology. We consider also the pseudovariety of finite groups ${ \bf Ab}(p) \ast { \bf Ab}(d)$ (where $p$ is a prime and $d$ divides $p-1$). We study the pro-$({ \bf Ab}(p) \ast { \bf Ab}(d))$ topology on a free group and construct the unique generator of minimum size of the pseudovariety ${ \bf Ab}(p) \ast { \bf Ab}(d)$. Finally, we prove that the variety of groups generated by ${ \bf U}$ is the variety of all metabelian groups, obtaining also results on the varieties generated by a Baumslag-Solitar group of the form $BS(1,q)$ for $q$ prime.
Categories: math.GR