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Math Help - Finitely generated subgroup

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    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.
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