On a group that acts freely on an Abelian group

A subgroup of $SL_2(C)$ is proven finite whenever it is generated by two elements $x$ and $y$ of order 3 such that the orders of $xy$ and $xy^{-1}$ are finite. It follows that a group acting freely on a nontrivial abelian group is finite whenever it is generated by two elements $x$ and $y$ of order 3 such that the orders of $xy$ and $xy^{-1}$ are finite.