Prove that $H$ is a system of representatives for the cosets of $N$ in $G$, and that $G/N$ is isomorphic to $H$.

Question

Let $G$ be a group and let $N \subseteq G$ be a normal subgroup. Suppose that there is a subgroup $H \subseteq G$ such that

$HN = G$ with $HN = \{hn : h \in H, n \in N\}$,

$H \cap N = \{e\}$.

Prove that $H$ is a system of representatives for the cosets of $N$ in $G$, and that $G/N$ is isomorphic to $H$. ($G/N \cong H$)

I have no clue how to do this

Answer 1

I would start by showing that $$G=\bigcup_{h\in H}hN,$$ and that for any $h_1,h_2\in H,$ we have $h_1N=h_2N$ if and only if $h_1=h_2.$ From there, the isomorphism $G/N\to H$ is almost trivial to define and justify.

Let me know if you're unsure about how to accomplish any of these steps, or if you just want to check if your reasoning is correct.