Let $G$ be a finitely generated group and $Tor(G)$ the subgroup generated by all torsion elements of $G$. Then $G/Tor(G)$ is torsion-free. In the abelian case we have
Every finitely generated abelian group is the direct product of its torsion subgroup and of a torsion-free subgroup
Can we say something in the non-abelian case? Specifically, I would like to know if $G$ is embedded in $Tor(G) \rtimes H$ for a subgroup $H \simeq G/Tor(G)$.