Some Engel conditions on finite subsets of certain groups

Abstract

Let n and k be positive integers. We say that a group G satisfies the condition ε(n) (respectively, εk(n)) if and only if any set with n + 1 elements of G contains two distinct elements x, y such that [x,t y] = 1 for some positive integer t = t(x,y) (respectively, [x,k y] = 1). Here we study certain groups satisfying these conditions. We prove that if G is a finite group satisfying the condition ε(n), then G is nilpotent if n < 3 and G is soluble if n < 16. If G is a finitely generated soluble group satisfying the condition ε(2), then G is nilpotent. If k and n are positive integers and G is a finitely generated residually finite group satisfying the condition εk(n), then G is nilpotent if n < 3 and G is polycyclic if n < 16. In particular, there is a positive integer c depending only on k such that G/Zc(G) is finite, where Zc(G) is the (c + 1)-th term of the upper central series of G. Also these bounds cannot be improved.