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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?

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