# Isomorph-automorphic normal subgroup

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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

## Definition

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

1. It is a normal subgroup.
2. 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