$G$ is torsion-free $[G:Z(G)]$ is finite $\implies$ $G$ is abelian ?

Question

If $G$ is a a group having no non-identity element of finite order and $Z(G)$ , the center of the group , has finite index , then is it true that $G$ is abelian ?

Answer 1

By Schur's theorem (well, one of the things called that, at least), $G'$ is finite if $Z(G)$ has finite index. Since $G$ contains no torsion, $G'$ must vanish; that is, $G$ is abelian.