# Order-automorphic normal subgroup

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

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

- It is a normal subgroup.
- it is an order-automorphic subgroup, i.e., any subgroup of the same order is an automorphic subgroup.

## Relation with other properties

### Stronger properties

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

Order-unique subgroup |

### Weaker properties

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

Order-automorphic subgroup | ||||

Normal subgroup | ||||

Order-normal subgroup | ||||

Order-isomorphic normal subgroup | ||||

Order-isomorphic subgroup | ||||

Isomorph-automorphic normal subgroup | ||||

Isomorph-normal subgroup |