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Showing Isomorphisms in Subgroups

  1. Mar 5, 2009 #1
    1. The problem statement, all variables and given/known data
    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.
     
  2. jcsd
  3. Mar 5, 2009 #2

    Dick

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    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.
     
  4. Mar 5, 2009 #3
    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.
     
  5. Mar 5, 2009 #4

    Dick

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    (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.
     
  6. Mar 5, 2009 #5
    But don't the generators of G have to lie in G? Because 1+0 is odd.
     
  7. Mar 5, 2009 #6

    Dick

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    No, I meant (1,0) and (0,1) generate Z x Z. The point is that you can also find two elements that generate G.
     
  8. Mar 5, 2009 #7
    Okay, I see it now. Thanks.
     
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