Show that every finite subgroup of the quotient group $\mathbb{Q}/\mathbb{Z}$ (under addition) is cyclic.
Note: there is a related problem which I just proved: "Let $G$ be a finite abelian group, then $G$ is non-cyclic iff $G$ has a subgroup isomorphic to $C_p \times C_p$ for some prime $p$."
Since $\mathbb{Q} /\mathbb{Z}$ is abelian, so based on the related problem it suffices to show it has no elementary abelian subgroup group. I tried to start prove by contradiction: Let $\mathbb{Z} <A<\mathbb{Q}$ such that
$A$/$\mathbb{Z} \simeq C_p \oplus C_p$ for some prime p, but I can't proceed further.
Best Answer
Not in the direction you wish but you can identify $\mathbb{Q}/\mathbb{Z}$ as a subgroup of the complex numbers: $\mathbb{Q}/\mathbb{Z} < \mathbb{R}/\mathbb{Z} \cong S^1 \subset \mathbb C^\times$, and then use that every finite multiplicative subgroup of a field is cyclic.