This question is from an assignment which I tried earlier. For Background, I have done a graduate course on group theory and have studied sylow theorems in detail.
If $H$ is a normal subgroup of order $p^k$ of a finite group $G$, then $H$ is contained in every Sylow $p$-subgroup of $G$.
I am sorry but I am at loss of ideas on which result should I use.
I think the statement sylow $p$-subgroup is unique that's why it is normal need not necessary be true though the statement if unique sylow $p$-subgroup exists then it is normal holds and is well known.
Can you please give some hints?
Best Answer
We can also reason like this without Sylow's theorem.
If $H$ is a normal $p$-subgroup of $G$ and $P$ is any other p-subgroup, then $HP$ is a $p$-subgroup of $G$ (this follows from the fact that $H$ is normal and from the formula $|HP|=|H|\cdot|P|/|P\cap H|$). It is also clear that $P\leq PH$.
It follows that if $P$ is a Sylow $p$-subgroup then $P=PH$ and hence $H\leq P$.