A characterization of infinite \(3\)-Abelian groups (Q1305351)

A group is 3-abelian if it satisfies the law \((xy)^3=x^3y^3\). The author shows that an infinite group is 3-abelian if, and only if, for every two infinite sets of elements of the group there exists an element \(x\) in the one and an element \(y\) in the other that satisfy the above equation. Similar questions, the first asked by Paul Erdős, with similar solutions, have been considered in a series of papers, by many different authors, since 1976.