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I'm trying to prove If $f:G\to H$ is a homomorphism with Kernel $N$ and $K\lt G$, then $f^{-1}(f(K))=KN$.

Since $N$ is a normal subgroup of G, I know that $N\cap K$ is a normal subgroup of $K$ and $NK=N\vee K=KN$, but I don't how to go further, I need help.

Thanks in advance

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Trying thinking about why $f^{-1}(f(K))= KN$ makes sense (What is $f^{-1}(e_H)$?).

Another thing to think about is what the elements in a particular coset $aN$ have in common with each other (If $a,b\in G$ and $aN=bN$, is there a relation between $a$ and $b$'s image under $f$?).

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