Sylow Theorems

Let $p$ be a Prime Number, $G$ a Group, and $\vert G\vert$ the order of $G$.

1. If $p$ divides $\vert G\vert$, then $G$ has a Sylow p-Subgroup.

2. In a Finite Group, all the Sylow p-Subgroup are isomorphic for some fixed $p$.

3. The number of Sylow p-Subgroup for a fixed $p$ is Congruent to 1 (mod $p$).