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Math Help - \mathbb{Q}\times\mathbb{Q} is not cyclic

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    \mathbb{Q}\times\mathbb{Q} is not cyclic

    Prove that \mathbb{Q}\times\mathbb{Q} is not cyclic.

    How do I do this my book says nothing.
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    MHF Contributor Drexel28's Avatar
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    Re: \mathbb{Q}\times\mathbb{Q} is not cyclic

    Quote Originally Posted by dwsmith View Post
    Prove that \mathbb{Q}\times\mathbb{Q} is not cyclic.

    How do I do this my book says nothing.
    If \mathbb{Q}^2 was cyclic, then \mathbb{Q} being the image of \mathbb{Q}^2 under the projection mapping would be cyclic, so it suffices to show that \mathbb{Q} is not cyclic. To see this suppose that there was an isomorphism f:\mathbb{Q}\to\mathbb{Z} we see then f(1)=f(m\frac{1}{m})=mf(\frac{1}{m}) so that f(1) is divisible by m for every m\in\mathbb{Z} and so f(1)=0, but since f(0)=0 this contradicts injectivity of f.
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    MHF Contributor Swlabr's Avatar
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    Re: \mathbb{Q}\times\mathbb{Q} is not cyclic

    Quote Originally Posted by Drexel28 View Post
    If \mathbb{Q}^2 was cyclic, then \mathbb{Q} being the image of \mathbb{Q}^2 under the projection mapping would be cyclic, so it suffices to show that \mathbb{Q} is not cyclic.
    Can;t you just say "subgroups of cyclic groups are cyclic"...
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    Senior Member roninpro's Avatar
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    Re: \mathbb{Q}\times\mathbb{Q} is not cyclic

    To show that \mathbb{Q} is not cyclic, it seems conceptually more straightforward to take an arbitrary number m/n (say, in lowest terms) and say that it can never meet 1/p, where p is some prime that is not a factor of n. (For example, no multiples of 3/16 will ever be 1/5.) So no single number can generate all of \mathbb{Q}.
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    MHF Contributor Drexel28's Avatar
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    Re: \mathbb{Q}\times\mathbb{Q} is not cyclic

    Quote Originally Posted by Swlabr View Post
    Can;t you just say "subgroups of cyclic groups are cyclic"...
    Sure, either way works.

    Quote Originally Posted by roninpro View Post
    To show that \mathbb{Q} is not cyclic, it seems conceptually more straightforward to take an arbitrary number m/n (say, in lowest terms) and say that it can never meet 1/p, where p is some prime that is not a factor of n. (For example, no multiples of 3/16 will ever be 1/5.) So no single number can generate all of \mathbb{Q}.
    Another good way to look at it! My method shows more generally that any divisible group is not cyclic.
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