Normal p-Subgroup contained in All Sylow p-Subgroups

Theorem
Let $G$ be a finite group.

Let $p$ be a prime number.

Let $H$ be a normal subgroup of $G$ which is a $p$-group.

Then $H$ is a subset of every Sylow $p$-subgroup of $G$.

Proof
Let $P$ be a Sylow $p$-subgroup of $G$.

By Second Sylow Theorem, $H$ is a subset of a conjugate of $P$.

Then:


 * $\exists g \in G: H \subseteq g P g^{-1}$.

This implies:

Since $P$ is arbitrary, $H$ is a subset of every Sylow $p$-subgroup of $G$.