A characterization of infinite 3-abelian groups

Abstract

In this note we prove that every infinite group G is 3-abelian (i.e. (ab)3 = a3b3 for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ∈ X and y ∈ Y such that (xy)3 = x3y3.