(adsbygoogle = window.adsbygoogle || []).push({}); 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 isexactlyone 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, isNOTa homework problem, i only thought since it is a typical textbook problem, i would recieve more answers here.

I would really appreciate any help!

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# Homework Help: Establishing uniqueness of an isomorphism!

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