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Math Help - show has no proper subgroups of finite index

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    show has no proper subgroups of finite index

    Show that Q, the group of rational numbers under addition, has no proper subgroups of finite index, but Z has.
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    Quote Originally Posted by mandy123 View Post
    Show that Q, the group of rational numbers under addition, has no proper subgroups of finite index, but Z has.
    let N be a subgroup of \mathbb{Q} with [\mathbb{Q}:N]=n. since (\mathbb{Q},+) is abelian, N is normal and hence \mathbb{Q}/N is a group of order n. so: \mathbb{Q}=n\mathbb{Q} \subseteq N \subseteq \mathbb{Q}. thus: N=\mathbb{Q}. \ \ \Box

    is this really not in your lecture notes that every nonzero subgroup of \mathbb{Z} has finite index?!!!
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