## Basic Properties of Characteristic Groups

## Problem 22

### Definition (automorphism).

An isomorphism from a group $G$ to itself is called* *an* automorphism *of $G$.

The set of all automorphism is denoted by $\Aut(G)$.

### Definition (characteristic subgroup).

A subgroup $H$ of a group $G$ is called ** characteristic** in $G$ if for any $\phi \in \Aut(G)$, we have $\phi(H)=H$. In words, this means that each automorphism of $G$ maps $H$ to itself.

Prove the followings.

**(a)** If $H$ is characteristic in $G$, then $H$ is a normal subgroup of $G$.

**(b) **If $H$ is the unique subgroup of $G$ of a given order, then $H$ is characteristic in $G$.

**(c)** Suppose that a subgroup $K$ is characteristic in a group $H$ and $H$ is a normal subgroup of $G$. Then $K$ is a normal subgroup in $G$.