I have a bunch of properties of group homomorphism and isomorphism that I need to prove:

So far, I'm stuck on a few properties. Hope someone can give me a hand.

Let be a group homomorphism.

1)If then divides

In particular, if is an isomorphism, then

2) If G is cyclic, then is cyclic. In particular, if is an isomorphism and is cyclic, then G is cyclic.