# Permutably complemented Hall subgroup

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

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

- is a Hall subgroup of : the order and index of are relatively prime.
- is a permutably complemented subgroup of : there exists a subgroup of such that is trivial and (in other words, and are permutable complements). Note that also must be a Hall subgroup of : if is -Hall, is -Hall.

## Relation with other properties

### Stronger properties

- Hall retract
- Normal Hall subgroup:
`For full proof, refer: Normal Hall implies permutably complemented` - Hall direct factor