# Group Homomorphisms into an Abelian Group

The following comes from Hungerford's Algebra.

[Prove that if] $$f: G \to H$$ is a homomorphism, $$H$$ is abelian and $$N$$ is a subgroup of $$G$$ containing $$\ker f$$, then $$N$$ is normal in $$G.$$

A solution found here (at I.5.16) begins by saying "since $$H$$ is abelian, $$f(H)$$ is an abelian subgroup." Obviously, this makes sense, but the domain of $$f$$ is $$G.$$ Why does $$f(H)$$ make sense here? This is really the part of the proof I don't understand.

Is there also another way to proceed with the proof?