# Isomorph-automorphic normal subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: isomorph-automorphic subgroup and normal subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed an **isomorph-automorphic normal subgroup** if it satisfies the following two conditions:

- It is a normal subgroup.
- Every subgroup of the whole group isomorphic to it is an automorphic subgroup, i.e., there is an automorphism sending one subgroup to the other.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Isomorph-free subgroup | No other isomorphic subgroup | (see also list of examples) | ||

Order-automorphic normal subgroup |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Isomorph-normal subgroup | every isomorphic subgroup of whole group is normal | isomorph-normal not implies isomorph-automorphic (see also list of examples) | ||

Isomorph-automorphic subgroup | ||||

Normal subgroup |