# Isomorph-containing implies characteristic

From Groupprops

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)

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## Contents

## Statement

Any isomorph-containing subgroup of a group is a characteristic subgroup.

## Definitions used

### Isomorph-containing subgroup

`Further information: isomorph-containing subgroup`

A subgroup of a group is termed an **isomorph-containing subgroup** if, for every subgroup of isomorphic to , .

### Characteristic subgroup

`Further information: characteristic subgroup`

A subgroup of a group is termed a **characteristic subgroup** if, for every automorphism of , .

## Related facts

### Stronger facts

- Isomorph-containing implies intermediately characteristic
- Isomorph-containing implies injective endomorphism-invariant
- Isomorph-containing implies intermediately injective endomorphism-invariant

## Proof

**Given**: An isomorph-containing subgroup of a group .

**To prove**: For every automorphism of ,

**Proof**: Since is an automorphism, the restriction of to is an isomorphism from to . Hence, is isomorphic to . Since is isomorph-containing, we get , completing the proof.