Ok, here is something i thought i understood, but it turns out i am having difficulties fully grasping/proving it. Let [tex]\theta:G->G'[/tex] be an isomorphism between G and G', where o(G)=m=o(G'), and both G and G' are cyclic, i.e. G=[a] and G'= So my question is, when we want to find the total number of isomorphisms from G to G', i 'know' that the total number of such isomorphisms is basically the number of generators of G': [tex]\theta(a)=b^k[/tex] where gcd(k,m)=1. But i don't really understand why? That is, how are we sure that by just counting the number of generators on G' we are actually finding the total number of such isomorphisms. ? Any explanations would be appreciated.