Hi

You know that is a group homomorphism, and that has order Let's use this:

(whereis the identity element of G)

Moreover, since is a homomorphism,

So we get What can you deduce?

Now, the second question. In a group, elements orders always divide the group order. If then the only common positive divisor between and will be . For any in we want to prove that i.e. that has order . Some idea? (Using the first result of course)