if has monomorphism , and N is abelian
then is homomorphism??
$\displaystyle f^{-1}$ only makes sense on f(N), since there's no unique way to define $\displaystyle f^{-1}(g)$ for $\displaystyle g \in G\setminus f(N)$.
but if we are restricting our attention to f(N), then $\displaystyle f^{-1}$ is more than just a homomorphism, it's an isomorphism (because if $\displaystyle f:N \to G$ is a monomorphism, $\displaystyle f:N \to f(N)$ is an isomorphism).