On periodic groups acting freely on abelian groups
Let $\pi$ be some set of primes. A periodic group $G$ is called a $\pi$-group if all prime divisors of the order of each of its elements lie in $\pi$. An action of $G$ on a nontrivial group $V$ is called free if, for any $v\in V$ and $g\in G$ such that $vg=v$, either $v=1$ or $g=1$. We describe $\{2,3\}$-groups that can act freely on an abelian group.