Prove that the center of a group is a normal subgroup Let $G$ be a group. We define
$H=\{h\in G\mid \forall g\in G: hg=gh\},$ the center of $G$.
Prove that $H$ is a (normal) subgroup of $G$.

sxd has shown that it is a subgroup. 
That it is normal follows from here:
Let $x\in Z(G)$ (center of $G$).
Then for any $g\in G$, $gxg^{-1}=gg^{-1}x=x\in Z(G)$.
This proves $Z(G)$ is a normal subgroup.
