Center of Group is Normal Subgroup/Proof 2

Theorem
The center $Z \left({G}\right)$ of any group $G$ is a normal subgroup of $G$ which is abelian.

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

Therefore:
 * $\forall a \in G: Z \left({G}\right)^a = Z \left({G}\right)$

and $Z \left({G}\right)$ is a normal subgroup of $G$.