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Homework Help: Error in my book

  1. Feb 3, 2008 #1
    [SOLVED] error in my book

    1. The problem statement, all variables and given/known data
    My book says that (Z cross Z)/<(1,2)> = Z. I say it equals Z cross Z_2. This is easy to see if you draw it out on a lattice plane. Right?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 3, 2008 #2

    StatusX

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    I think the books right. Can you spell out your reasoning a little more?
     
  4. Feb 3, 2008 #3
    You can choose any lattice point on the line y=0 or the line y=1 and get a unique line with slope 2 that goes through that point.
     
  5. Feb 3, 2008 #4

    Dick

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    Hmm. And I think it's just Z_2. I mean, it has to be a finite group doesn't it?
     
  6. Feb 3, 2008 #5

    StatusX

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    Construct a map [itex]\phi:\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}[/itex] by [itex]\phi(a,b)=2a-b[/itex]. It's easy to check that this is a surjective homomorphism, and its kernel is <(1,2)>, so by the isomorphism theorem:

    [tex]\mathbb{Z} \times \mathbb{Z}/<(1,2)> \cong\mathbb{Z}[/tex]
     
  7. Feb 3, 2008 #6

    Dick

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    Bing! Sure. It's not (Z/Z)x(Z/(2*Z)). Thanks, StatusX.
     
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