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Math Help - Isomorphism

  1. #1
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    Isomorphism

    Prove that  (\mathbb{R} \ \{0} ) * (\mathbb{R}\{0}) is not isomorphic to  \mathbb{C}\{0} (the group operation in both cases is multiplication).
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  2. #2
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    Quote Originally Posted by Jimmy_W View Post
    Prove that  (\mathbb{R} \ \{0} ) * (\mathbb{R}\{0}) is not isomorphic to  \mathbb{C}\{0} (the group operation in both cases is multiplication).
    suppose there was an isomorphism f. let f(x,y)=i and f(a,b)=-1. then f(a^2,b^2)=1=f(1,1). thus a^2=b^2=1. hence at least one of a,b is equal to -1. why?

    but then f(x^2,y^2)=i^2=-1=f(a,b) and so x^2=a, \ y^2=b. hence at least one of x^2,y^2 is equal to -1, which is obviously impossible!
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  3. #3
    Super Member Gamma's Avatar
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    order considerations

    You can just say these groups cannot be isomorphic due to the fact that \mathbb{C}^{\times} has elements of order 4, namely i, -i while there are no elements of order 4 in \mathbb{R}^{\times}\times \mathbb{R}^{\times}

    everything in \mathbb{R}^{\times}\times \mathbb{R}^{\times} has order
    one: (1,1)
    two: (-1,1), (1,-1) and (-1,-1)
    or
    infinite order: everything else
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