Sylow Theorems

Let $G$ be a finite group.

Theorem 1

For every prime factor $p$ with multiplicity $n$ of the order of $G$ , there exists a Sylow $p$ -subgroup of $G$ , of order $p^n$ .

Theorem 2

All Sylow $p$ -subgroups of $G$ are conjugate to each other, i.e.\ if $H$ and $K$ are Sylow $p$ -subgroups of $G$ , then there exists an element $g\in G$ with $g^{-1}Hg=K$ .

Theorem 3

Let $p$ be a prime such that $|G|=p^nm$ , where $p\nmid m$ . Let $n_p$ be the number of Sylow $p$ -subgroups of $G$ . Then:
1) $n_p\mid m$ , which is the index of the Sylow $p$ -subgroup in $G$ .
2) $n_p\equiv 1\pmod p$ .

Theorem 3b (Proof)

We have $n_p=[G:N_G(P)]$ , where $P$ is any Sylow $p$ -subgroup of $G$ and $N_G$ denotes the normalizer.

Proof

Let $P$ be a Sylow $p$ -subgroup of $G$ and let $G$ act on $\text{Syl}_p(G)$ by conjugation. We have $|\text{Orb}(P)|=n_p$ , $\text{Stab}(P)=\{g\in G:gPg^{-1}=P\}=N_G(P)$ .