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.