TITLE: Every finite subgroup of $\mathbb{Q}/\mathbb{Z}$ is cyclic
QUESTION [14 upvotes]: 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.

REPLY [3 votes]: Following your characterisation of cyclic groups it suffices to show that $\mathbb{Q}/\mathbb{Z}$ contains only $p-1$ elements of order $p$ for any given prime $p$. Those are easy to find and it is straightforward to show that there cannot be any more elements of this order.