# Order-automorphic normal subgroup

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

## Definition

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

1. It is a normal subgroup.
2. 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