1. May 6, 2009 #1

   curiousmuch

   

   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

   curiousmuch, May 6, 2009
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3. May 6, 2009 #2

   VeeEight

   

   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.

    

   VeeEight, May 6, 2009

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