Q:

Surjectivity of product map

Let $H$ be a subgroup of a group $G$, let $\phi: G \rightarrow H$ be a homomorphism with kernel $N$, and suppose that the restriction of $\phi$ to $H$ is the identity. 
I am trying to determine whether the product map $f : H \times N \rightarrow G$ defined by $f(h,n) = hn$ is surjective. I've managed to show that it is injective and that it need not be a homomorphism, but I don't know how to prove it is surjective, or come up with a counterexample if it is not. Any help is appreciated!

A:

Fix $g\in G$, and let $h=\varphi(g)$. Then $\varphi(g)=\varphi(h)$, so $gN=hN$, so ... ?

