Sylow's Third Theorem

Sylow’s Third Theorem. Let $G$ be a finite group with $|G| = p^{a}m$ where $p$ is prime and $p\nmid m$. Let $n_p$ denote the number of Sylow p-subgroups of $G$. Then:

1. $n_p \mid m$, and
2. $n_p \equiv 1 \pmod p$.

This theorem is a counting consequence of Sylow's second theorem together with the normalizer and the conjugation action on the set of Sylow $p$-subgroups.