I have the group G whose elements are infinite sequences of integers (a1, a2, ...). These sequences combine termwise as such:(adsbygoogle = window.adsbygoogle || []).push({});

(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.

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# Finding Isomorphisms

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