if http://latex.codecogs.com/gif.latex?f:N%20%5Cto%20G has monomorphism , and N is abelian

then http://latex.codecogs.com/gif.latex?f%5E%7B-1%7D is homomorphism??

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- Mar 1st 2012, 11:26 PMvernalhomomorphism
if http://latex.codecogs.com/gif.latex?f:N%20%5Cto%20G has monomorphism , and N is abelian

then http://latex.codecogs.com/gif.latex?f%5E%7B-1%7D is homomorphism?? - Mar 2nd 2012, 07:50 PMDevenoRe: 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).