 Solution

The identity commutes with every group element, so it lies in the center of a group. If $z,w$ are central and $g\in G$, then $gw=wg$ implies $g{w}^{-1}=w^{-1}g$. ConsequentlyThus $z{w}^{-1}$ is central. The subgroup test provesEquivalently, the centre is nonempty, closed under products, and closed under inverses. This proof does not require finiteness. In fact the centre is an Abelian subgroup, and it is normal because $gz{g}^{-1}=z$ for each central element.