Let $\displaystyle G = \frac { \mathbb {Q} } { \mathbb {Z} } $ under +, so the elements are the equivalence classes $\displaystyle \hat {r} = \{ s \in \mathbb {Q} : s-r \in \mathbb {Z} \} $. Write $\displaystyle r \equiv s \ (mod \ 1 ) $ if $\displaystyle r - s \in \mathbb {Z} $. If H is a finitely generated subgroup of $\displaystyle G = \frac { \mathbb {Q} } { \mathbb {Z} } $, then H is a finite cyclic subgroup. Also determine a generator for H.

proof so far.

Suppose that H is a finitely generated subgroup, so I need to find an element h such that <h> = H. How should I process with this? Thanks!