$p$-group, Sylow $p$-subgroup

Let $G$ be a group and let $p$ be a prime.

A group of order $p^\alpha$ for some $\alpha \ge 0$ is called a $p$-group. Subgroups of $G$ which are $p$-groups are called $p$-subgroups.
If $G$ is a group of order $p^\alpha m$, where $p \nmid m$, then a subgroup of order $p^\alpha$ is called a Sylow $p$-subgroup of $G$.
The set of Sylow $p$-subgroups of $G$ will be denoted by $Syl_p(G)$ and the number of Sylow $p$-subgroups of $G$ will be denoted by $n_p(G)$, or just $n_p$ when $G$ is clear from the context.

Theorem 18. Sylow’s Theorem

TODO $$\tag*{$\blacksquare$}$$

Let $P \in Syl_p(G)$. If $Q$ is any $p$-subgroup of $G$, then $Q \cap N_G(P) = Q \cap P$.