Conjugacy separability and separable orbits

Abstract

Two elements (subgroups) of a group G are called conjugacy separable if they are conjugate in G if and only if their images are conjugate in every finite quotient of G. The whole group is termed conjugacy separable if each pair of its elerr,ents is conjugacy separable. Conjugacy separable groups form a rather complicated class of groups. It is closed with respect to forming free products but not under taking subgroups, forming extensions and wreath products [S]. Restriction to the class of soluble groups does not help either. The best known result, the Theorem of Formanek [l] and Remeslennikov [7] yields that this class contains all polycyclic-by-finite groups. These groups also have all subgroups conjugacy separable (Grunewald, Segal [2]). In this paper we will prove conjugacy separability of elements and subgroups for a class of not necessarily finitely generated nilpotent groups of finite abelian section rank. Some examples will show that there is nearly no general way to extend these results to a wider class of groups. Regarding conjugacy as an operation of a group on itself yields another way how to extend theorems on conjugacy separability, which I found in a recent paper of Hilton and Roitberg [3]. Let Q be a group operating on a further group G. Two elements (subgroups) of G are said to bl: Q-conjugate, if there exists an element qE Q mapping the first element (subgr/Jup) onto the second. They are termed Q-separable if they are Q-conjugate or if there e!xists a finite Q-quotient of G in which their images are net Q-conjugate. G has separable Q-orbits if each pair of its elements is Q-separable. As usual Q is said to act nh’potently on G if it acts identically on the factors of a finite Q-invariant series of G and almost nilpotently if it conta:lns a subgroup of finite index acting nilpoltently on a Q-invariant subgroup of finite index of G. Hilton and Roitberg proved orbit separability for finitely generated nilpotent groups on which a finitely generated nilpotent group acts nilpotently. We will extend