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Establishing uniqueness of an isomorphism!

  1. Nov 21, 2008 #1
    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!
     
  2. jcsd
  3. Nov 21, 2008 #2
    Ok, if 1. would be true then:

    from ther we would have:[tex]\theta_1(a)=\theta_2(a)[/tex] 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 [tex]\theta_1[/tex] is the same as [tex]\theta_2[/tex],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?
     
    Last edited: Nov 21, 2008
  4. Nov 23, 2008 #3
    any suggestions?
     
  5. Nov 23, 2008 #4
    ..............
     
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