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Math Help - Q\to Q by q\to kq

  1. #1
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    Q\to Q by q\to kq

    Prove that for all fixed k\in\mathbb{Q} such that k isn't 0 the map \varphi :\mathbb{Q}\to\mathbb{Q} defined by \varphi (q)=kq is an automorphism of \mathbb{Q}.

    So I need to show it is an isomorphism. I am having a problem with the homomorphism part though.

    Let q,r\in\mathbb{Q}. \varphi (qr)=kqr but \varphi (q)\varphi(r)=kqkr=k^2qr.

    What do I need to do?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Re: Q\to Q by q\to kq

    Quote Originally Posted by dwsmith View Post
    Prove that for all fixed k\in\mathbb{Q} such that k isn't 0 the map \varphi :\mathbb{Q}\to\mathbb{Q} defined by \varphi (q)=kq is an automorphism of \mathbb{Q}.

    So I need to show it is an isomorphism. I am having a problem with the homomorphism part though.

    Let q,r\in\mathbb{Q}. \varphi (qr)=kqr but \varphi (q)\varphi(r)=kqkr=k^2qr.

    What do I need to do?
    You need to remember that you are looking at the rationals under addition! So,

    \varphi(p+q)=k(p+q)=kp+kq=\varphi(p)+\varphi(q).
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  3. #3
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    Re: Q\to Q by q\to kq

    Quote Originally Posted by Swlabr View Post
    You need to remember that you are looking at the rationals under addition! So,

    \varphi(p+q)=k(p+q)=kp+kq=\varphi(p)+\varphi(q).
    Why are you able to assume it is addition?
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Re: Q\to Q by q\to kq

    Quote Originally Posted by dwsmith View Post
    Why are you able to assume it is addition?
    Because it isn't an automorphism otherwise, as you have already shown...
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  5. #5
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    Re: Q\to Q by q\to kq

    Quote Originally Posted by dwsmith View Post
    Why are you able to assume it is addition?

    Because (\Mathbb{Q}, +) is a group while (\mathbb{Q}, \cdot) is not.
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  6. #6
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    Re: Q\to Q by q\to kq

    in dwsmith's defense, it is bad form to speak of "the group Q". a group is not merely a set, but a set and a binary operation. failure to specify the intended binary operation requires an unwarranted amount of mysticism on the student's part.
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