## 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 $G$.