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Adjacent Transpositions proof

  1. Jan 18, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that every transposition (i,j)(1≤i≤j≤n) in Sn is expressible as a product of adjacent transpositions.

    Also express the transposition (1,9) as a product of adjacent transpositions.

    2. Relevant equations

    none

    3. The attempt at a solution
    Really struggling to even start the proof.

    Is the transposition (1,9)=(1,2)(2,3)(3,4)(4,5)(5,6)(6,7)(7,8)(8,9)?
     
  2. jcsd
  3. Jan 18, 2013 #2

    CompuChip

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    I assume (i, j) means swapping i and j. You can check your answer if you have 9 small pieces of paper. That should also lead you on to a proof.

    You are definitely on the right way that you have to "string" i through i + 1, i + 2, ... until it reaches j and vice versa.
     
  4. Jan 18, 2013 #3

    Stephen Tashi

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    I suggest you start by working a specific example. Express (1,3) as the product of "adjacent transpositions". By the way, what is the definition of an "adjacent transposition"? A cycle like (1,2) ?
     
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