Let $N$ be a normal subgroup of finite group $G$. Prove that if order of $H$ and order of $G/N$ are relatively prime then $H$ is a subgroup of $N$.

Question

Let $N$ be a normal subgroup of finite group $G$. Prove that if order of $H$ and order of $G/N$ are relatively prime then $H$ is a subgroup of $N$.

Can someone help me to understand the intuition behind this question and why is this happening with example?

I am basically clueless about how to start.

Answer 1

By the first isomorphism theorem, the order of a homomorphic image of a group divides the order of the group.

Consider the canonical projection $π:G\to G/N$. Then $|π(H)|\mid|H|$.

And by Lagrange, $|π(H)|\mid|G/N|$.

So $π(H)=0$. So $H\le N$.