Normal p-Subgroup contained in All Sylow p-Subgroups

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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 $G$.


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

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


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

This implies:

\(\ds H\) \(=\) \(\ds g^{-1} H g\) Subgroup equals Conjugate iff Normal
\(\ds \) \(\subseteq\) \(\ds g^{-1} \paren {g P g^{-1} } g\) Subset Relation is Compatible with Subset Product/Corollary 2
\(\ds \) \(=\) \(\ds P\)

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