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Math Help - Not isomorphic

  1. #1
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    Not isomorphic

    Show that R is NOT isomorphic to R*. (R is real numbers and R* is the group of nonzero reals under multiplication)

    Can someone show this. I can't even get started...

    Thanks.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Not isomorphic

    Quote Originally Posted by jzellt View Post
    Show that R is NOT isomorphic to R*. (R is real numbers and R* is the group of nonzero reals under multiplication)
    Suppose that f:\mathbb{R}\to \mathbb{R}^* is an isomorphism. Then, f(0)=1 and f(a)=-1 for some a\in\mathbb{R} so that f(2a)=f(a)^2=1. Since f is injective, 2a=0. Thus, a=0 and as a consequence 1=f(0)=-1 (contradiction).
    Thanks from jzellt and Siron
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