Profinitely closed subgroups of soluble groups of finite rank

Abstract

In his famous paper [6] Malcev proved that all subgroups of a soluble group of finite abelian section rank (as usual, we denote this class by -ip,) with no quasicyclic section are closed with respect to the profinite topology. He also showed that any torsion-free soluble group all subgroups of which are closed belongs to 9, and has no quasicyclic section. But it also seems to be of interest to obtain some information about (profinitely) closed subgroups of Y,-groups possessing quasicyclic sections (e.g., [ 1, Key Lemma]). In this paper we will characterize closed subgroups and investigate the relation between arbitrary subgroups and their closure in a class related to YO. Furthermore, we look at products and homomorphic images of closed subgroups of locally nilpotent groups. This work has mainly been motivated by the need of some results on (prolinitely) closed subgroups in order to prove theorems on conjugacy separability and on groups with the same finite images. But these results are of interest on their own and should therefore be written down together. Before stating our results we need some notation: The spectrum n(G) of a group G is the set of all primes p for which G has a quasicyclic p-section. If X is any class of groups we denote by S= its subclass of all reduced (i.e., with no radicable subgroups) S-groups of finite abelian section rank whose spectrum is contained in rr. If 7c is the set of all primes, we write Sr instead of Xx. We mainly use this definition in the case where 37 denotes the class d, JtT, JCJV and 9’ of all abelian, nilpotent, locally nilpotent and soluble groups, respectively. fl is the class of all finite groups. We put