Groups isomorphic to their non-nilpotent subgroups

Abstract

We were concerned in [12] with groups G that are isomorphic to all of their non-abelian subgroups. In order to exclude groups with all proper subgroups abelian, which are well understood in the finite case [7] and which include Tarski groups in the infinite case, we restricted attention to the class X of groups G that are isomorphic to their nonabelian subgroups and that contain proper subgroups of this type; such groups are easily seen to be 2-generator, and a complete classification was given in [12, Theorem 2] for the case G soluble. In the insoluble case, G/Z(G) is infinite simple [12; Theorem 1], though not much else was said in [12] about such groups. Here we examine a property which represents a natural generalisation of that discussed above. Let us say that a group G belongs to the class W if G is isomorphic to each of its non-nilpotent subgroups and not every proper subgroup of G is nilpotent. Firstly, note that finite groups in which all proper subgroups are nilpotent are (again) well understood [9]. In addition, much is known about infinite groups with all proper subgroups nilpotent (see, in particular, [8] and [13] for further discussion) although, even in the locally nilpotent case, there are still some gaps in our understanding of such groups. We content ourselvesin the present paper with discussing finitely generated W-groups— note that a W-group is certainly finitely generated or locally nilpotent. We shall have a little more to say about the locally nilpotent case below.