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## Homework Statement

Let G be a subset of Z x Z (direct product) where G = {(a,b)|a+b = 2k for some integer k}.

I'd like to show that G is a proper subgroup of Z x Z and determine whether G is isomorphic to Z x Z.

I am pretty sure I have shown that it is a proper subgroup but the isomorphism part is what is giving me trouble. Here the cardinality of the two sets is equal and it seems reasonable to be able to find a bijection between them, but I am not sure on the details of showing this.