TITLE: If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$
QUESTION [5 upvotes]: If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$
Attempt: $|H|=p^k \implies |G|=p^{n_1} q^{n_2} r^{n_3} \cdots~~|~~n_1 \geq k$
Now, $H$ is also a normal sub group of $G$ and we need to show that $H$ is contained in every $p$ sylow subgroup of $G$ which means in all of $H_{p^{n_1}}, H_{q^{n_2}},H_{r^{n_3}},  \cdots $
Unfortunately, I am not able to decide the strategy to move ahead. How do I move forward?
Thank you for your help.

REPLY [4 votes]: Another strategy: if $G$ is a finite group and $N \lhd G,$ then whenever $X$ is a subgroup of $G,$ then $XN = NX$ is a subgroup of $G$ of order $\frac{|X||N|}{|X \cap N|}.$ Now consider the case that $H = N$ and $X \in {\rm Syl}_{p}(G).$