# Normal-isomorph-automorphic subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Definition with symbols

A subgroup $H$ of a group $G$ is termed a normal-isomorph-automorphic subgroup if $H$ is a normal subgroup of $G$ and for any normal subgroup $K$ of $G$ isomorphic to $H$, $H$ and $K$ are automorphic subgroups in $G$, i.e., there is an automorphism $\sigma$ of $G$ such that $\sigma(H) = K$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal-isomorph-free subgroup normal, and no other isomorphic normal subgroup |FULL LIST, MORE INFO
Isomorph-free subgroup No other isomorphic subgroup (via normal-isomorph-free) (via normal-isomorph-free) Normal-isomorph-free subgroup|FULL LIST, MORE INFO
Isomorph-automorphic normal subgroup Normal, and automorphic to all isomorphic subgroups |FULL LIST, MORE INFO