Center is characteristic

Proposition (Center is characteristic). Let $G$ be a group and let $Z(G)$ denote its center . Then $Z(G)$ is a characteristic subgroup of $G$; that is, for every automorphism $\varphi\in \mathrm{Aut}(G)$ one has $\varphi(Z(G))=Z(G)$.

Context. “Characteristic” is stronger than normal: every characteristic subgroup is normal, but not conversely. The center is the basic example of a subgroup defined purely by the intrinsic multiplication structure of $G$, so it must be preserved by all automorphisms.