Center of Group is Normal Subgroup/Proof 2

Proof
We have:
 * $\forall a \in G: x \in \map Z G^a \iff a x a^{-1} = x a a^{-1} = x \in \map Z G$

Therefore:
 * $\forall a \in G: \map Z G^a = \map Z G$

and $\map Z G$ is a normal subgroup of $G$.