# Hall direct factor

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a finite group is termed a **Hall direct factor** if it satisfies the following equivalent conditions:

- It is a Hall subgroup and is also a direct factor.
- It is a normal Hall subgroup and possesses a normal complement, i.e., is a retract.
- It is a Hall subgroup and is also a central factor of the whole group.

### Equivalence of definitions

`For full proof, refer: Hall and central factor implies direct factor`

## Relation with other properties

### Stronger properties

### Weaker properties

- Normal Hall subgroup
- Fully characteristic subgroup
- Fully characteristic direct factor
- Direct factor
- Central factor
- Conjugacy-closed normal subgroup
- Conjugacy-closed Hall subgroup

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition