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Math Help - Abelian groups

  1. #1
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    Abelian groups

    Let G be an abelian group and let g,h Є G.

    (i) Assuming that |g| and |h| are both finite, with hcf (|g| , |h|) = 1. Prove that

    |g + h| = |g| |h|

    (ii) prove that the direct sum of Zm and Zn is isomorphic to Zmn if and only if m and n are relatively prime.

    any help would be appreciated. thanks
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  2. #2
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    Quote Originally Posted by davidmccormick View Post
    Let G be an abelian group and let g,h Є G.

    (i) Assuming that |g| and |h| are both finite, with hcf (|g| , |h|) = 1. Prove that

    |g + h| = |g| |h|
    Let |g|=n,|h|=m. Then g^n = h^m = 0.
    This means (g+h)^{nm} = g^{nm}+h^{nm} = (g^n)^m + (h^m)^n = 0 + 0 = 0
    Now argue that nm is the least such exponent for g+h.

    (ii) prove that the direct sum of Zm and Zn is isomorphic to Zmn if and only if m and n are relatively prime.

    any help would be appreciated. thanks
    The group \mathbb{Z}_n\times \mathbb{Z}_m has order nm. If you can show it is cyclic i.e. find a generator then by isomorphism proporties of cyclic groups it would mean that it has to be isomorphic to \mathbb{Z}_{nm}.
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  3. #3
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    We obtain Zn x Zm by the component-wise addition of Zm and Zn. So, it is definitely cyclic with generator 1 for instance. But that only guarantees isomorphism with either Z or Zn. How can we extend that to Zmn. Also what about the reverse implication?
    thanks
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  4. #4
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    Quote Originally Posted by davidmccormick View Post
    We obtain Zn x Zm by the component-wise addition of Zm and Zn. So, it is definitely cyclic with generator 1 for instance. But that only guarantees isomorphism with either Z or Zn. How can we extend that to Zmn. Also what about the reverse implication?
    thanks
    Because the order of \mathbb{Z}_n\times \mathbb{Z}_m is nm. The group \mathbb{Z} has infinite order therefore it cannot be the isomorphic group, while, the groups \mathbb{Z}_k all have order k and therefore k=nm.
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