# Sylow Theorems

Let be a finite group.

## Theorem 1

For every prime factor with multiplicity of the order of , there exists a Sylow -subgroup of , of order .

## Theorem 2

All Sylow -subgroups of are conjugate to each other, i.e.\ if and are Sylow -subgroups of , then there exists an element with .

## Theorem 3

Let be a prime such that , where . Let be the number of Sylow -subgroups of . Then:

1) , which is the index of the Sylow -subgroup in .

2) .

## Theorem 3b (Proof)

We have , where is any Sylow -subgroup of and denotes the normalizer.

### Proof

Let be a Sylow -subgroup of and let act on by conjugation. We have , .

By the Orbit-Stabilizer Theorem, , thus .

## Orbit-Stabilizer Theorem

Let be a group which acts on a finite set . Then