Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective.

Question

I am trying to show the following:

Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective.

Note that $H\leq G$ and $N$ is normal in $G$.

My attempt so far: Let $\phi(h_{1})=\phi(h_{2})$. Then $h_{1}N=h_{2}N$ or there exists an $n\in N$ such that $h_{1}=h_{2}n$.

This is where I am stuck. I apologize for my elementary knowledge in cosets and quotient groups. Can anyone point me to the right direction, such as a property for cosets, to help me?

Answer 1

It's not true. $\phi$ is a homomorphism and you have $h\in Ker(\phi)$ $\iff$ $hN=N$ $\iff$ $h\in H\cap N$. So the kernel is trivial only when $H\cap N=\{e\}$.