I am studying Dummit & Foote abstract algebra. This question is what I have a problem.
Let $G$ be a finite group, $H$ be a subgroup of $G$, $N$ be a normal subgroup of $G$. If $|H|$ and $|G:N|$ are relatively prime, prove that $H$ is a subgroup of $N$.
I founded that $H$ is a subset of $N$. But I can't find how to prove about subgroup. Does normality implies that subset becomes a subgroup? If not, how can I prove it?