The proposition I want to prove is this: If $H, K$ are normal subgroups of a finite group $G$ such that $G=HK$, and $P$ is a Sylow $p$-subgroup of $G$, then we have that $P = (P\cap H)(P\cap K)$.
Since $P$ is a Sylow $p$-subgroup of $G$, and $H, K$ are normal, we have that $(P\cap H)$ and $(P\cap K)$ are Sylow $p$-subgroups of $H$ and $K$ respectively.
Then we also know that $|(P\cap H)(P\cap K)|= \frac{(|P\cap H|)(|P\cap K|)}{|P\cap H\cap K|}$, so the cardinality of $(P\cap H)(P\cap K)$ is a power of the prime number $p$.
Edit: A not so immediate fact is that actually $(P\cap H)(P\cap K)$ is a group. To see that we could think of the normalizer of $P$ in $G$. Then $P\cap H$ is a normal subgroup of $N_G(P)$ and so is $P\cap K$, hence $(P\cap H)(P\cap K)$ is a group and actually it is a normal $p$-subgroup of $N_G(P) $.
I'm not sure how to continue from there. Thank you in advance for any help!
