Sylow theorems

Let $G$ be a finite group whose order is divisible by the prime $p$. Suppose $p^m$ is the highest power of $p$ which is a factor of $|G|$ and set

$$k=\frac{|G|}{p^m}.$$

Then

1. the group $G$ contains at least one subgroup of order $p^m$,

2. any two subgroups of $G$ of order $p^m$ are conjugate, and

3. the number of subgroups of $G$ of order $p^m$ is congruent to $1$ modulo $p$ and is a factor of $k$.