Center is Characteristic Subgroup

Theorem
Let $G$ be a group.

Then its center $\map Z G$ is characteristic in $G$.

Proof
Let $\phi$ be an automorphism of $G$.

Let $x \in \map Z G, y \in G$.

Then:

Hence $\map \phi x \in \map Z G$.

So we have $\phi \sqbrk {\map Z G} \subseteq \map Z G$.

Since $\phi^{-1}$ is also an automorphism:
 * $\phi^{-1} \sqbrk {\map Z G} \subseteq \map Z G$

Since $\phi$ is a bijection:
 * $\map Z G = \phi \sqbrk {\phi^{-1} \sqbrk {\map Z G}} \subseteq \phi \sqbrk {\map Z G}$

Therefore we conclude that $\phi \sqbrk {\map Z G} = \map Z G$.

Hence $\map Z G$ is characteristic in $G$.