Finding Isomorphisms

  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. Hurkyl

    Hurkyl 16,089
    Staff Emeritus
    Science Advisor
    Gold Member

    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. 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. Hurkyl

    Hurkyl 16,089
    Staff Emeritus
    Science Advisor
    Gold Member

    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. Yes, thanks very much.
     
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