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 $G$ : for every $\varphi\in\operatorname{Aut}(G)$ ,

\varphi(Z(G))=Z(G).

Remarks

Every characteristic subgroup is normal, but the converse need not hold. The center is preserved because automorphisms preserve commutation: if $z\in Z(G)$ , then $\varphi(z)\varphi(g)=\varphi(g)\varphi(z)$ for every $g\in G$ .