We are given:

1. h: G -> G' is a group homomorphism.

2. M is a normal subgroup of G'.

3. N is the set of x in G such that h(x) is in M.

Prove that N is a normal subgroup of G.

It's not that hard for me to prove that N is a subgroup of G--that just comes from the fact that h is a group homomorphism. But I really have no idea how to prove that N is not just a subgroup but a normal subgroup. In fact, these normal subgroup proofs are just plain difficult for me right now. Please help.