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Cyclic groups

  1. May 6, 2009 #1
    1. The problem statement, all variables and given/known data
    Let H be a finite abelian group that has one subgroup of order d for every positive divisor d of the order of H. Prove that H is cyclic


    2. Relevant equations
    We want to show H={a^n|n is an integer}
     
    Last edited: May 6, 2009
  2. jcsd
  3. May 6, 2009 #2
    I haven't done this stuff in a while but since no one is helping I'll give it a shot.

    Assume not. Then by Fundamental theorem of finitely generated abelian groups, H is isomorphic to Z_a x Z_b. Since, by assumption, H is not cyclic, it follows that Z_a x Z_b is not cyclic. This implies that a and b are not relatively prime. Which implies that there exists some prime p that divides both a and b. Perhaps apply Cauchy's theorem here.
     
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