# Sylow direct factor

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Symbol-free definition

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

- It is a Sylow subgroup and is also a direct factor of the group.
- It is a normal Sylow subgroup and possesses a normal p-complement, i.e., it is a retract of the group.
- It is a Sylow subgroup and is also a central factor of the whole group.
- It is a Sylow subgroup and is also a conjugacy-closed normal subgroup of the whole group.

### Equivalence of definitions

`Further information: Equivalence of definitions of Sylow direct factor`

## Relation with other properties

### Weaker properties

- Normal Sylow subgroup
- Hall direct factor
- Normal Hall subgroup
- Fully characteristic subgroup
- Characteristic subgroup
- Direct factor
- Central factor

## Metaproperties

### Join-closedness

YES:This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closedABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

### 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