A question on subgroups of a finite group

Question

Let $G$ be a finite group. Let $K$ and $H$ be subgroups of $G$, where $K$ is normal in $G$ and $\gcd([G:H],|K|)=1$.

Prove that $K$ is a subgroup of $H$.

So far we found that $o(K)$ divides $o(H)$, but we are struggling to show that $K$ is contained in $H$.

Any help would be appreciated, thanks.

Answer 1

This is becuse $H \le KH \le G$, so $|KH:H|$ divides $|G:H|$, but $|KH:H| = |K:K \cap H|$ also divides $|K|$, and $|GH:H|$ and $|K|$ are coprime, so $KH=H$.