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Finding Isomorphisms

  1. Mar 11, 2009 #1
    I have the group G whose elements are infinite sequences of integers (a1, a2, ...). These sequences combine termwise as such:
    (a1, a2,...)(b1, b2,...) = (a1+b2, a2+b2,...)

    I would like to find an isomorphism from G x Z (the direct product of G and the integers) to G as well as an isomorphism from G x G to G.

    So far, I have found several homomorphisms for both of these but all of them lack the injective property so fail to be isomorphisms. What sorts of functions can I construct that are isomorphisms to G for these two groups?

    Thanks.
     
  2. jcsd
  3. Mar 11, 2009 #2

    Hurkyl

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    In the arithmetic of Abelian groups, your group G is isomorphic to the infinite Cartesian product ZN.... (actually, it's probably equal to, not just isomorphic to, but that doesn't matter)
     
  4. Mar 11, 2009 #3
    Okay. I see that, but I guess I'm not sure how to use that fact to show that G x Z is isomorphic to G. Should I try to show that G x Z is also isomorphic to the infinite direct product?
     
  5. Mar 11, 2009 #4

    Hurkyl

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    That would work. Calculations like this are one of the reasons we learn how to do arithmetic with sets!

    (I'm assuming you know the appropriate arithmetic rules, such as [itex](A \times B)^C \cong A^C \times B^C[/itex])
     
  6. Mar 11, 2009 #5
    Yes, thanks very much.
     
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