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Math Help - Isomorphism

  1. #1
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    Isomorphism

    Show that Z*Z/2Z is notisomorphic to Z.(I have to look for order of few elements)
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  2. #2
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    Quote Originally Posted by math.dj View Post
    Show that Z*Z/2Z is notisomorphic to Z.(I have to look for order of few elements)

    I suppose that by Z*Z/2Z you meant the direc product of Z and Z/2Z and not their free product, otherwise it is trivial: the direct product is abelian and the free product isn't.

    And anyway it is very easy: the element (0,a)\in\mathbb{Z}\times \mathbb{Z}\slash 2\mathbb{Z} , with \mathbb{Z}\slash 2\mathbb{Z}=\{1,a\} , has finite order and thus must be mapped to the unit in \mathbb{Z} under any homomorphism ...

    Tonio
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