# Coprime automorphism-invariant normal subgroup

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a finite group is termed a **coprime automorphism-invariant normal subgroup** if it is both normal and coprime automorphism-invariant: it is invariant under all coprime automorphisms of the whole group.

## Relation with other properties

### Stronger properties

### Weaker properties

## Facts

- Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal
- Coprime automorphism-invariant normal of normal Hall implies normal
- Left residual of normal by normal Hall is coprime automorphism-invariant normal
- Hall-relatively weakly closed implies coprime automorphism-invariant normal