Can anyone help me with the question? I am stuck on showing this is one-to-one.
For subgroup $H$ of $G$ and normal subgroup $N$ in $G$, show the function $\phi:H\to HN/N$ defined by $\phi(h)=hN$ is one-to-one.
My work:
Let $\phi(h_{1})=\phi(h_{2})$. Then $h_{1}N=h_{2}N$. Then there is an $n\in N$ such that $h_{1}=h_{2}n$.
I feel like I need to use the fact that $h_{2}\in HN$?