# Isomorph-normal coprime automorphism-invariant subgroup of group of prime power order

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: isomorph-normal coprime automorphism-invariant subgroup with a group property imposed on theambient group: group of prime power order

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

A subgroup of a group of prime power order is termed an **isomorph-normal coprime automorphism-invariant subgroup of group of prime power order** if it satisfies *both* the following conditions:

- is an isomorph-normal subgroup of (in particular, it is an isomorph-normal subgroup of group of prime power order): In other words, for any subgroup of isomorphic to , is a normal subgroup of .
- is a coprime automorphism-invariant subgroup of (in particular, it is a coprime automorphism-invariant subgroup of group of prime power order).

## Relation with other properties

### Stronger properties

- Isomorph-free subgroup of group of prime power order
- Isomorph-normal characteristic subgroup of group of prime power order
- Characteristic subgroup of group of prime power order

### Weaker properties

- Fusion system-relatively weakly closed subgroup:
*For proof of the implication, refer Isomorph-normal coprime automorphism-invariant implies weakly closed for any fusion system and for proof of its strictness (i.e. the reverse implication being false) refer Fusion system-relatively weakly closed not implies isomorph-normal*. - Sylow-relatively weakly closed subgroup
- Coprime automorphism-invariant normal subgroup of group of prime power order, coprime automorphism-invariant normal subgroup
- Coprime automorphism-invariant subgroup of group of prime power order, coprime automorphism-invariant subgroup