# Sylow Theorems

## Theorem

### First Sylow Theorem

Let $p$ be a prime number.

Let $G$ be a group such that:

$\order G = k p^n$

where:

$\order G$ denotes the order of $G$
$p$ is not a divisor of $k$.

Then $G$ has at least one Sylow $p$-subgroup.

### Second Sylow Theorem

Let $P$ be a Sylow $p$-subgroup of the finite group $G$.

Let $Q$ be any $p$-subgroup of $G$.

Then $Q$ is a subset of a conjugate of $P$.

### Third Sylow Theorem

All the Sylow $p$-subgroups of a finite group are conjugate.

### Fourth Sylow Theorem

The number of Sylow $p$-subgroups of a finite group is congruent to $1 \pmod p$.

### Fifth Sylow Theorem

The number of Sylow $p$-subgroups of a finite group is a divisor of their common index.

## Examples

### Sylow $2$-Subgroups in Group of Order $12$

In a group of order $12$, there are either $1$ or $3$ Sylow $2$-subgroups.

### Sylow $3$-Subgroups in Group of Order $12$

In a group of order $12$, there are either $1$ or $4$ Sylow $3$-subgroups.

### Sylow $7$-Subgroups in Group of Order $28$

In a group of order $28$, there exists exactly $1$ Sylow $7$-subgroup.

This Sylow $7$-subgroup is normal in $G$.

### Sylow $2$-Subgroups in Group of Order $32$

The Sylow Theorems tell us nothing about the Sylow $2$-subgroups in a group of order $32$.

### Sylow $2$-Subgroups in Group of Order $48$

In a group of order $48$, there are either $1$ or $3$ Sylow $2$-subgroups.

## Source of Name

This entry was named for Peter Ludwig Mejdell Sylow.

## Historical Note

When cracking open the structure of a group, it is a useful plan to start with investigating the prime subgroups.

The Sylow Theorems are a set of results which provide us with just the sort of information we need.

Ludwig Sylow was a Norwegian mathematician who established some important facts on this subject.

He published what are now referred to as the Sylow Theorems in $1872$.

The name is pronounced something like Soolof.

There is no standard numbering for the Sylow Theorems.

Different authors use different labellings.

Therefore, the nomenclature as defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ is to a greater or lesser extent arbitrary.