If $G$ is a group and $H\leq G$ is a subgroup, then a transversal of $H$ is a subset $T\subseteq G$ which meets every coset of $H$ in a unique point.
The axiom of choice clearly implies that every subgroup of a group has a transversal. But does the converse hold? I'm sure somebody must have proved this, but I haven't been able to find a reference.
The first theorem proves that requiring this just for Abelian groups is enough to prove the axiom of choice.