Finding Isomorphisms

Obraz35

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.

Hurkyl

In the arithmetic of Abelian groups, your group G is isomorphic to the infinite Cartesian product Z^N.... (actually, it's probably equal to, not just isomorphic to, but that doesn't matter)

Obraz35

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?

Hurkyl

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])

Obraz35

Yes, thanks very much.