# Definition:Sylow p-Subgroup

## Contents

## Definition

Let $p$ be prime.

Let $G$ be a finite group whose order is denoted by $\left\vert{G}\right\vert$.

### Definition 1

Let $\order G = k p^n$ where $p \nmid k$.

A **Sylow $p$-subgroup** is a $p$-subgroup of $G$ which has $p^n$ elements.

### Definition 2

A **Sylow $p$-subgroup** of $G$ is a **maximal $p$-subgroup** $P$ of $G$.

In this context, maximality means that if $Q$ is a $p$-subgroup of $G$ and $P \le Q$, then $P = Q$.

### Definition 3

Let $n$ be the largest integer such that:

- $p^n \divides \order G$

where $\divides$ denotes divisibility.

A **Sylow $p$-subgroup** is a $p$-subgroup of $G$ which has $p^n$ elements.

## Also known as

**Sylow $p$-subgroups** are sometimes called **$p$-Sylow subgroups**, or just **Sylow subgroups**.

## Also see

## Source of Name

This entry was named for Peter Ludwig Mejdell Sylow.