# Subset of a normal subgroup is a subgroup?

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?

• Doesn't it suffice to show that $H \subset N$? Since $H$ is a subgroup of $G$, it is itself a group under the operation of $G$, or $N$. ($H \subset N \leq G$). Thus $H$ is a subgroup of $N$. Aug 7, 2020 at 8:23
• But is it not one of the data that $H$ is a (sub)group? Aug 7, 2020 at 8:23