# Math Help - Finitely generated subgroup

1. ## Finitely generated subgroup

Let $G = \frac { \mathbb {Q} } { \mathbb {Z} }$ under +, so the elements are the equivalence classes $\hat {r} = \{ s \in \mathbb {Q} : s-r \in \mathbb {Z} \}$. Write $r \equiv s \ (mod \ 1 )$ if $r - s \in \mathbb {Z}$.
Find a subgroup $J \subset G$ that is not finitely generated.

Let $G = \frac { \mathbb {Q} } { \mathbb {Z} }$ under +, so the elements are the equivalence classes $\hat {r} = \{ s \in \mathbb {Q} : s-r \in \mathbb {Z} \}$. Write $r \equiv s \ (mod \ 1 )$ if $r - s \in \mathbb {Z}$.
Find a subgroup $J \subset G$ that is not finitely generated.
How about, $S = \left\{ \left[ \frac{1}{p} \right] : p \text{ prime }\right\}$ and now let $J = \left< S \right>$ i.e. subgroup generated by $S$.