# Definition:Sylow p-Subgroup

## 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

• Results about Sylow $p$-subgroups can be found here.

## Source of Name

This entry was named for Peter Ludwig Mejdell Sylow.