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Thread: Finitely generated subgroup

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    Finitely generated subgroup

    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.
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    Quote Originally Posted by tttcomrader View Post
    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.
    How about, $\displaystyle S = \left\{ \left[ \frac{1}{p} \right] : p \text{ prime }\right\}$ and now let $\displaystyle J = \left< S \right>$ i.e. subgroup generated by $\displaystyle S$.
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