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Cyclic groups

  1. Mar 1, 2009 #1
    1. The problem statement, all variables and given/known data
    Let G and H be cyclic groups, with |G| = m and |H| = n. If gcd(m,n) =1, show that G x H is cyclic.


    3. The attempt at a solution
    Let a = (g,h) in G x H. Then |a| = lcm (|g|,|h|).
    Since gcd(m,n)=1, then lcm (m,n) = mn.
    Thus lcm (|g|,|h|) = lcm (m,n) = mn.
    so <a> = G x H has mn elements and a cyclic group.
    Right?
     
  2. jcsd
  3. Mar 1, 2009 #2

    Dick

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    Basically ok. But you want to be a little more careful with your choice of g and h. It's not true that for every element g in G that |g|=|G|. E.g. |e|=1. And G could have nontrivial subgroups. You'd better make sure that g and h are generators.
     
  4. Mar 3, 2009 #3
    so.. you mean I have to make sense that for every element g in G that |g|=|G|?
    If so..
    Actually, the problem has Hint that [ G=<g> and H = <h>, show |(g,h)| = mn].
    If consider this hint, my answer is ok?
    or any other thing needs to prove?
     
    Last edited: Mar 3, 2009
  5. Mar 3, 2009 #4

    Dick

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    The point is that at least one g in G satisfies |g|=|G|. Otherwise, would it be a cyclic group? That's what the proof is missing.
     
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