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Math Help - Cyclic groups

  1. #1
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    Cyclic groups

    Let  G and  H be cyclic groups. If  |G| = m and  |H| = n and  (m,n) = 1 , show  G \times H is cyclic. (Hint: show  |(g,h)| = mn .)
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  2. #2
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    Suppose G=<g>, H=<h>, and |(g,h)|=k. Then (e_G,e_H)=(g,h)^k=(g^k,h^k) so g^k=e_G, h^k=e_H. But then from the properties of the order, m,n both divide k,
    so k is a common multiple. Then what we want is the least common multiple of the orders, and that is \dfrac{mn}{(m,n)}=mn=|G\times H| thus we have found the generator, and the result follows
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