# Showing Isomorphisms in Subgroups

## 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.

Dick
Homework Helper
It may seem reasonable that if the cardinalities are equal there should be an isomorphism, but that's not necessarily true. In this case it is. Z x Z is an abelian group which can be generated by two elements. G is also an abelian group with two generators. Can you find them? Hint: draw Z x Z in the plane and circle the elements belonging to G. It's a lattice. You can get an isomorphism by mapping generators to generators.

I see the visual representation, but I guess I am just not seeing how this gives you the generators. The only way I can think to generate G is with (2,0), (0,2) and (1,1). I'm also not sure how to map the rest of the elements that are not generators.

Dick
Homework Helper
(0,2)=(-1)*(2,0)+2*(1,1). You don't need three generators. Clearly, (a,b)=a*(1,0)+b*(0,1). Only two generators. Your G is not that much different.

But don't the generators of G have to lie in G? Because 1+0 is odd.

Dick