On the pseudovariety of groups $\mathbf{U} = \displaystyle\bigvee_{p \in \mathbb{P}} {\bf Ab}(p) \ast {\bf Ab}(p-1)$

Abstract: 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.