1. The problem statement, all variables and given/known data Let G=[a] and G'= be cyclic groups of the same order. Show, that among the isomorphisms [tex] \theta[/tex] from G to G', there is exactly one with [tex] \theta(a)=c[/tex] if and only if c is a generator of G. 2. Relevant equations 3. The attempt at a solution I have managed to show the existence, only i am not sure how to establish the uniqueness of such an isomorphism. I know that a proof by contradiction would work, only that i am not sure what i have to prove. To establish the existence i proceded: => Let o(G)=m=o(G'), and let [tex] \theta:G->G'[/tex] be an isomorphism given with [tex]\theta(a)=c[/tex] then from here i easily showed that c is a generator of G'. <= Let's suppose that c is a generator of G', then i also managed to show that the mapping [tex] \theta(a)=c[/tex] is actually an isomorphism. Here it actually is what i am not sure of. On the second part, should i also show here that such an isomorphism is unique, or i should also on the other side? Also, i am not sure what my claim should state: 1. Let's suppose that there are more than one isomorphisms given with [tex]\theta(a)=c[/tex], that is lets suppose that both [tex]\theta_1(a)=c, and , \theta_2(a)=c[/tex] are such isomorphisms, or whether my claim should be something like this: 2. Let's suppose that there are more than one such isomorphisms, that is lets suppose that both: [tex]\theta(a_1)=c, and, \theta(a_2)=c[/tex] are such isomorphisms, where a_1 and a_2 are generators of G? Or none of these would work? P.S This, indeed, is NOT a homework problem, i only thought since it is a typical textbook problem, i would recieve more answers here. I would really appreciate any help!