Assume that the group G' is a homomorphic image of the group G.
a. Prove that G' is cyclic if G is cyclic.
b. Prove that o(G') divides o(G), whether G is cyclic or not
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