On a pseudovariety of finite supersolvable groups

Abstract

We introduce the pseudovariety of finite groups U =boolean OR(p is an element of & Popf;)Ab(p) & lowast;Ab(p - 1), where & Popf; is the set of all primes. We show that U consists of all finite supersolvable groups with elementary abelian derived subgroup and abelian Sylow subgroups, and so U 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-U topology. We consider also the pseudovariety of finite groups Ab(p) & lowast;Ab(d) (where p is a prime and d divides p - 1). We study the pro-(Ab(p) & lowast;Ab(d)) topology on a free group and construct the unique generator of minimum size of the pseudovariety Ab(p) & lowast;Ab(d). Finally, we prove that the variety of groups generated by 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.