Groups in which Squares and Cubes Commute

Abstract

If for all $a, b$ in a group $G$, we have that $a^2b^2 = b^2a^2$ and $a^3b^3 = b^3a^3$ then does the group necessarily have to be abelian? This paper shows that the answer is affirmative for finite groups as well as certain classes of infinite groups. In general we show that if $G$ is an infinite group in which squares commute and cubes commute then the set of torsion elements of $G$ denoted by $T(G)$ has to be an abelian normal subgroup of $G$.