# Pronormal Hall subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: pronormal subgroup and Hall subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Definition with symbols

A subgroup of a finite group is termed a **pronormal Hall subgroup** if it satisfies the following two conditions:

- is a Hall subgroup of : The order and index of in are relatively prime.
- is a pronormal subgroup of : If is a conjugate subgroup to in , then are conjugate subgroups inside .

## Relation with other properties

### Stronger properties

- Sylow subgroup
- Intermediately isomorph-conjugate Hall subgroup: Also related:

### Weaker properties

- Hall subgroup:
`For full proof, refer: Hall not implies pronormal` - Hall WNSCDIN-subgroup