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Free abelian group?

  1. Feb 28, 2009 #1

    tgt

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    1. The problem statement, all variables and given/known data
    How many different homomorphisms are there of a free abelian group of rank 2 into S_{3}?

    Where S_{3} is the symmetric group of 3 elements.

    3. The attempt at a solution
    I think 12 but the answers suggest 18. ?????
     
  2. jcsd
  3. Mar 1, 2009 #2
    We can assume the free abelian group of rank 2 is [tex]\mathbb{Z}\times\mathbb{Z}[/tex].
    The homomorphisms from [tex]\mathbb{Z}\times\mathbb{Z}[/tex] to [tex]S_3[/tex], correspond to pairs [tex](a,b)\in S_3\times S_3[/tex] with [tex]ab=ba[/tex] via
    [tex]f:\mathbb{Z}\times\mathbb{Z}\to S_3[/tex] corresponds to [tex](f(1,0),f(0,1))[/tex].
    (Show this!) So you just need to find all such pairs.

    Tip for enumerating these pairs: There are only three types of elements in [tex]S_3[/tex]: the identity, the transpositions and the cyclic permutations.
     
    Last edited: Mar 1, 2009
  4. Mar 6, 2009 #3

    tgt

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    I understand your explanation. But there seem only to be 12 such pairs.
     
  5. Mar 6, 2009 #4
    I found 18 pairs. Did you count the pairs (a,b), (b,a) as the same? They give different homomorphisms.
     
  6. Mar 9, 2009 #5

    tgt

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    I see now. thank you.
     
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