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} $.

Find a subgroup $\displaystyle J \subset G $ that is not finitely generated.