So, for the sake of notational convenience let's call our homomorphism .

Also, I'll assume these are finite groups.

So, let . We claim that . To see this let be arbitrary. By assumption for some . But, that means that for some and so .

Work with that.

How much group theory do you know? You should know by the FIT that and sob. Prove that o(G') divides o(G), whether G is cyclic or not