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Math Help - isomorphisms between certain groups

  1. #1
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    isomorphisms between certain groups

    It is possible to show easily that \mathbb{Z} / 2 is isomorphic with the group of units denoted U_3 that have a multiplicative inverse modulo 3 .
    defined by the isomorphic map : \varphi : \mathbb{Z}/2 \rightarrow \ U_3
    \varphi (0) = 1 , \varphi (1) = 2

    Furthermore we can find another isomorphic map to show that \phi : U_5 \rightarrow \mathbb{Z}/2 \times \mathbb{Z}/2

    I cannot really prove that the map  \phi : U_5 \rightarrow \mathbb{Z}/4 cannot be an isomorphism. However i can define the map \phi (m) = m-1 and by counter-example to prove that this is not isomorphic as
    \phi(4)=\phi(2) + \phi(2) if it is an isomorphism
    but = 1 + 1 = 2 which does equal 3

    However this is not very rigorous and I would like to know if there a deeper/more rigorous way of proving that there cannot be an isomorphism
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  2. #2
    Senior Member roninpro's Avatar
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    An easy way is to look at the orders of the elements, keeping in mind that isomorphisms preserve orders. In particular, the order of every element in U_5 is at most two, whereas some elements in \mathbb{Z}_4 have order four. Therefore, the two groups cannot be isomorphic.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by FGT12 View Post
    It is possible to show easily that \mathbb{Z} / 2 is isomorphic with the group of units denoted U_3 that have a multiplicative inverse modulo 3 .
    defined by the isomorphic map : \varphi : \mathbb{Z}/2 \rightarrow \ U_3
    \varphi (0) = 1 , \varphi (1) = 2

    Furthermore we can find another isomorphic map to show that \phi : U_5 \rightarrow \mathbb{Z}/2 \times \mathbb{Z}/2

    I cannot really prove that the map  \phi : U_5 \rightarrow \mathbb{Z}/4 cannot be an isomorphism. However i can define the map \phi (m) = m-1 and by counter-example to prove that this is not isomorphic as
    \phi(4)=\phi(2) + \phi(2) if it is an isomorphism
    but = 1 + 1 = 2 which does equal 3

    However this is not very rigorous and I would like to know if there a deeper/more rigorous way of proving that there cannot be an isomorphism
    To add on to roninpro's explanation it's a common fact that every group of order p^2 is abelian and thus by the structure theorem isomorphic to either C_p\oplus C_p or C_{p^2}. So, just noting that \left|U_5\right|=\phi\left(5\right)=5-1=4 narrows it down to two groups up to isomorphism. You can then extend what roninpro's one-hundred percent correct explanation to show (and I'm urging you to do this!) that C_{mn}\cong C_m\oplus C_n if and only if (m,n)=1!
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