# Sylow retract

From Groupprops

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

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 conjugacy-closed subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Symbol-free definition

A **Sylow retract** of a finite group is any of the following equivalent things:

- A Sylow subgroup that is also a retract. In other words, it is a -Sylow subgroup for some prime , such that there exists a normal p-complement: a normal Hall subgroup that is a permutable complement to it.
- A Sylow subgroup that is also conjugacy-closed: In other words, it is a Sylow subgroup with the property that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.

### Equivalence of definitions

`Further information: Conjugacy-closed and Sylow implies retract`