# Sylow subgroup of normal subgroup

From Groupprops

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: Sylow subgroup and normal subgroup

View other such compositions|View all subgroup properties

## Contents

## Definition

### Symbol-free definition

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

- It is a Sylow subgroup of a normal subgroup of the whole group.
- It is the intersection of a normal subgroup of the whole group with a Sylow subgroup of the whole group.
- It is a Sylow subgroup inside its normal closure.

### Equivalence of definitions

`Further information: Equivalence of definitions of Sylow subgroup of normal subgroup`