The book on abstract algebra that I'm reading uses the fact that, given the groups $G$ and $G'$ and a homomorphism $\phi:G \rightarrow G'$, then
$$ \operatorname{im}( \phi ) \cong G/\ker( \phi )$$
However the author doesn't provide a proof and simply states that it follows from "standard group theory". How could one prove the theorem above?
I would appreciate any help/thoughts!
Best Answer
Here is a roadmap for the proof, which appears in all books:
Define $\pi : G/\ker( \phi ) \to \operatorname{im}( \phi )$ by $\pi(x \bmod \ker \phi) = \phi(x)$. Then prove:
$\pi$ is well defined
$\pi$ is a group homomorphism
$\pi$ is injective
$\pi$ is surjective