Let $h: G \to H$ an epimorphism. If N is a normal subgroup of H, show that $G/h^{-1}(N) \simeq H/N.$

Question

Let $h: G \to H$ an epimorphism. If N is a normal subgroup of H, show that $$G/h^{-1}(N) \simeq H/N.$$

I have the good intuition that I have to use the first isomorphism theorem.

I'm not able how to continue this problem. Is anyone is able to give me a little hint?

Answer 1

Consider the projection $\pi\colon H\to H/N$; what is the kernel of $\pi\circ h$? What does the homomorphism theorem say? Note that $\pi\circ h$ is surjective.