Equivalent Definitions of Characteristic Subgroups. Center is Characteristic.
Problem 246
Let $H$ be a subgroup of a group $G$. We call $H$ characteristic in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$.
(a) Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$.
(b) Prove that the center $Z(G)$ of $G$ is characteristic in $G$.
Add to solve later