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Thread: Why is this not an isomorphism?

  1. #1
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    Why is this not an isomorphism?

    $\displaystyle \gamma : G \rightarrow G$ where $\displaystyle G = (\mathbb{R},\times)$
    $\displaystyle \gamma(x) = 3x$

    It is trivially a bijection and $\displaystyle \gamma(x_1x_2) = \gamma(x_1)\gamma(x_2)$ seems to hold
    as $\displaystyle \gamma(x_1 + x_2) = 3(x_1 + x_2) = 3(x_1) + 3(x_2) = \gamma(x_1) +\gamma(x_2)$
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  2. #2
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    $\displaystyle 3x_{1}x_{2}=\gamma(x_{1}x_{2})\not=\gamma(x_{1})\g amma(x_{2})=(3x_{1})(3x_{2})=9x_{1}x_{2}.$
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  3. #3
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    sorry i meant the group was $\displaystyle (\mathbb{R}, +)$
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  4. #4
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    In that case, I'd say the last line of the OP proves that it is an isomorphism.
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