# Hall retract

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A **Hall retract** of a finite group is a Hall subgroup that is also a retract: in other words, it possesses a normal complement. Note that the normal complement must *also* be a Hall subgroup, and in fact, a normal Hall subgroup.

## Relation with other properties

### Stronger properties

### Weaker properties

- Order-conjugate subgroup
- Order-isomorphic subgroup
- Order-automorphic subgroup
- Isomorph-automorphic subgroup
- Isomorph-conjugate subgroup
- Automorph-conjugate subgroup
- Procharacteristic subgroup
- Pronormal subgroup
- Intermediately isomorph-conjugate subgroup
- Intermediately automorph-conjugate subgroup
- Intermediately normal-to-characteristic subgroup
- Intermediately subnormal-to-normal subgroup