# Cyclic groups

1. May 6, 2009

### 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
2. May 6, 2009

### 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.