1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook