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Math Help - Isomorphism

  1. #1
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    Isomorphism

    Find an isomorphism from Z_12 to Z_4 (direct product) Z_3. (Z_4 circle plus Z_3)

    how many isomorphism are there in total.
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  2. #2
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    Quote Originally Posted by Juancd08 View Post
    Find an isomorphism from Z_12 to Z_4 (direct product) Z_3. (Z_4 circle plus Z_3)
    If \theta : \mathbb{Z}_{12} \to \mathbb{Z}_4 \times \mathbb{Z}_3 is a homomorphism then \theta (n) = \theta (1+...+1) = n\cdot \theta (1) (where 0\leq n\leq 11).

    Now if \theta is an isomorphism then | 1| = |\theta(1)| = 12. Thus, we require that order of \theta (1) to be equal to 12. The elements that have order 12 in \mathbb{Z}_4 \times \mathbb{Z}_3 are (1,1),(1,2),(3,1),(3,2). Now confirm that \phi (1) = (1,1),(1,2),(1,3),(3,2) in each of four cases extends to an isomorphism. Thus, that means there are four isomorphisms.
    --

    Here is another way to do this problem. Note that the number of isomorphisms between is the same as the number of automophisms of \mathbb{Z}_n. Now use the result that |\text{Aut}(\mathbb{Z}_n)| = \phi (n). In this case \phi(12) = \phi(4)\phi(3) = 4.
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  3. #3
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    order?

    how do you get that those elements have order 12
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