Center of Group is Subgroup/Proof 2

Proof
We have the result Center is Intersection of Centralizers.

That is, $\map Z G$ is the intersection of all the centralizers of $G$.

All of these are subgroups of $G$ by Centralizer of Group Element is Subgroup.

Thus from Intersection of Subgroups is Subgroup, $\map Z G$ is also a subgroup of $G$.