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

Abstract

We introduce the pseudovariety of finite groups $\mathbf{U}=\underset{p\in \mathbb{P}}{\bigvee }\mathbf{Ab}(p)\ast \mathbf{Ab}(p-1)$, 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-$\mathbf{U}$ topology. We consider also the pseudovariety of finite groups $\mathbf{Ab}(p)\ast \mathbf{Ab}(d)$ (where $p$ is a prime and $d$ divides $p-1$). We study the pro-$(\mathbf{Ab}(p)\ast \mathbf{Ab}(d))$ topology on a free group and construct the unique generator of minimum size of the pseudovariety $\mathbf{Ab}(p)\ast \mathbf{Ab}(d)$. Finally, we prove that the variety of groups generated by $\mathbf{U}$ 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.