# Isomorph-containing implies characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., isomorph-containing subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about isomorph-containing subgroup|Get more facts about characteristic subgroup

## Definitions used

### Isomorph-containing subgroup

Further information: isomorph-containing subgroup

A subgroup $H$ of a group $G$ is termed an isomorph-containing subgroup if, for every subgroup $K$ of $G$ isomorphic to $H$, $K \le H$.

### Characteristic subgroup

Further information: characteristic subgroup

A subgroup $H$ of a group $G$ is termed a characteristic subgroup if, for every automorphism $\sigma$ of $G$, $\sigma(H) \le H$.

## Proof

Given: An isomorph-containing subgroup $H$ of a group $G$.

To prove: For every automorphism $\sigma$ of $G$, $\sigma(H) \le H$

Proof: Since $\sigma$ is an automorphism, the restriction of $\sigma$ to $H$ is an isomorphism from $H$ to $\sigma(H)$. Hence, $\sigma(H)$ is isomorphic to $H$. Since $H$ is isomorph-containing, we get $\sigma(H) \le H$, completing the proof.