Establishing uniqueness of an isomorphism

[sutupidmath]

Homework Statement

Let G=[a] and G'= be cyclic groups of the same order. Show, that among the isomorphisms $$\theta$$ from G to G', there is exactly one with $$\theta(a)=c$$ if and only if c is a generator of G.

Homework Equations

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 $$\theta:G->G'$$ be an isomorphism given with $$\theta(a)=c$$ 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 $$\theta(a)=c$$ 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 $$\theta(a)=c$$, that is let's suppose that both $$\theta_1(a)=c, and , \theta_2(a)=c$$ 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 let's suppose that both:

$$\theta(a_1)=c, and, \theta(a_2)=c$$ 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 receive more answers here.


I would really appreciate any help!

 

[sutupidmath]

Ok, if 1. would be true then:

from ther we would have:$$\theta_1(a)=\theta_2(a)$$ now since a is the generator of G, it would mean that this relationship holds true for any element x in G. So, this would mean that $$\theta_1$$ is the same as $$\theta_2$$,and thus such an isomorphism is unique.

But this seems too easy to be true, and thus correct, right? Is this close to being the right path?

 

[sutupidmath]

any suggestions?

 

[sutupidmath]

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