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If an Abelian group $G$ has order $n$ and at most one subgroup of order $d$ for all $d$ dividing $n$ then $G$ is cyclic.

I am trying to use the structure theorem for finitely generated abelian groups.

So I write $n=p_1^{\alpha_1}\ldots p_n^{\alpha_n}$.

I am hoping to show each of the alpha's must =1 then I will have that $G$ is isomorphic to $\prod_i^n \mathbb{Z}/p_i \mathbb{Z}$, which is cyclic.

  • 1
    Structure theorem is a 'big' theorem and its use as far as possible must be avoided. So go through the answers below. And, more on this line, look at problem U212 in http://awesomemath.org/wp-content/uploads/reflections/2011_6/MR6.pdf2011-12-22

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