If and has order k, prove that has order m, where m|k.
Also, if gcd(|G|,|H|)=1, then for all
You know that is a group homomorphism, and that has order Let's use this:
(where is 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)