# Math Help - homomorphism

1. ## homomorphism

if $f:N \to G$ has monomorphism , and N is abelian
then $f^{-1}$ is homomorphism??

2. ## Re: homomorphism

$f^{-1}$ only makes sense on f(N), since there's no unique way to define $f^{-1}(g)$ for $g \in G\setminus f(N)$.

but if we are restricting our attention to f(N), then $f^{-1}$ is more than just a homomorphism, it's an isomorphism (because if $f:N \to G$ is a monomorphism, $f:N \to f(N)$ is an isomorphism).