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Math Help - Isomorphism of abelian groups

  1. #1
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    Isomorphism of abelian groups

    Let a, b be positive integers and let x = gcd(a,b) and y=lcm(a,b)

    Show that

    Z_{a} \oplus Z_{b} is isomorphic to Z_{x} \oplus Z_{y}.
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  2. #2
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    Quote Originally Posted by aliceinwonderland View Post
    Let a, b be positive integers and let x = gcd(a,b) and y=lcm(a,b)

    Show that

    Z_{a} \oplus Z_{b} is isomorphic to Z_{x} \oplus Z_{y}.
    Use prime power decompositions.
    If N = \prod_{j}p_j^{a_j} then \mathbb{Z}_N \simeq \bigoplus_j \mathbb{Z}_{p_j^{a_j}}
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