Some engel conditions on infinite subsets of certain groups

Abstract

Let k be a positive integer. We denote by εk(∞) the class of all groups in which every infinite subset contains two distinct elements cursive Greek chi, y such that [cursive Greek chi,k y] = 1. We say that a group G is an ε*k-group provided that whenever X,Y are infinite subsets of G, there exists cursive Greek chi ∈ X, y ∈ Y such that [cursive Greek chi,k y] = 1. Here we prove that: (1) If G is a finitely generated soluble group, then G ∈ ε3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3. (2) If G is a finitely generated metabelian group, then G ∈ εk(∞) if and only if G/Zk(G) is finite, where Zk(G) is the (k + 1)-th term of the upper central series of G. (3) If G is a finitely generated soluble εk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt(G) is finite. (4) If G is an infinite ε*k-group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.