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Math Help - product of transpositions

  1. #1
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    Exclamation product of transpositions

    Show that everly element of S(n) (n>=2) is a product of transpositions of the form (k k+1).
    [Hintk k+2) = (k k+1)(k+1 k+2)(k k+1).]
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  2. #2
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    Quote Originally Posted by r7iris View Post
    Show that everly element of S(n) (n>=2) is a product of transpositions of the form (k k+1).
    [Hintk k+2) = (k k+1)(k+1 k+2)(k k+1).]
    Theorem: For n\geq 2 every element of S_n can be expressed as a product of transpositions.


    Thus, given any premutation we can write it as a product of transpositions and we will show that each of these transpositions is a product of the form \prod (k,k+1).

    Say we are working in S_{10} (products are taken from right to left).

    Consider (1,3). We can write it as (1,2)(2,3)(1,2).

    Consider (1,4). We can write it as (1,2)(2,3)(3,4)(2,3)(1,2).

    Consider (2,6). We can write it as (2,3)(3,4)(4,5)(5,6)(4,5)(3,4)(2,3).

    You get the general idea. So that means everything can be expressed in consecutive form by using the theorem.
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