# Finding Isomorphisms

by Obraz35
Tags: group theory, isomorphic, isomorphism
 P: 31 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.
 Emeritus Sci Advisor PF Gold P: 16,101 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)
 P: 31 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?
Emeritus
(I'm assuming you know the appropriate arithmetic rules, such as $(A \times B)^C \cong A^C \times B^C$)