Isomorphism and homomorphism properties

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 $\displaystyle \phi:G \rightarrow H$ be a group homomorphism.

1)If $\displaystyle ord(g)=n$ then $\displaystyle ord(\phi(g))$ divides $\displaystyle n$

In particular, if $\displaystyle \phi$ is an isomorphism, then $\displaystyle ord(\phi(g))=ord(g)$

2) If G is cyclic, then $\displaystyle \phi(G)$ is cyclic. In particular, if $\displaystyle \phi$ is an isomorphism and $\displaystyle \phi(G)$ is cyclic, then G is cyclic.