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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 showsthat the answer is affirmative for finite groups as well as certain classes ofinfinite groups. In general we show that if $G$ is an infinite group in whichsquares commute and cubes commute then the set of torsion elements of $G$denoted by $T(G)$ has to be an abelian normal subgroup of $G$.

Groups in which Squares and Cubes Commute