---
tag: 0JWZX6FKSH
kind: definition
name: Sylow p-subgroup
aliases: [sylow-subgroup]
msc: [20D20]
defines: 0053CQE9Y2
uses: [06DAH3VS6M, 0N0PXCTYA7]
sources:
  - url: https://en.wikipedia.org/wiki/Sylow_theorems
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Sylow_theorems
    locator: Statement
---

## Definition

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.
Write $|G| = p^n m$ with $n \geq 0$ and $p \nmid m$.
A Sylow $p$-subgroup of $G$ is a [[subgroup|0X1JXT47GA]] of $G$ of order $p^n$.
The set of the Sylow $p$-subgroups of $G$ is written $\operatorname{Syl}_p(G)$, and its size is written $n_p$.

## Remarks

A Sylow $p$-subgroup is a maximal [[p-subgroup|06QYRTX9KQ]] of $G$.
If $p$ does not divide $|G|$, then the only Sylow $p$-subgroup is $\{e\}$.
In $S_3$, the Sylow $3$-subgroup is $A_3$, and there are three Sylow $2$-subgroups, each of order $2$.

## Explanations

### L1 human

A Sylow subgroup takes all the factors $p$ of the order of the group at once.
It is as large as a $p$-subgroup can be by Lagrange's theorem.
The Sylow theorems say that such subgroups exist and that they are all alike.