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Math Help - Symmetric Group

  1. #1
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    Symmetric Group

    I have been trying to show that
    "Sym(p) is generated by any transposition and any p-cycle..."

    I have showed some examples...
    but I couldnt see the difference between them(actually where should I you that prime ?).

    Thanks
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  2. #2
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    You need the result that conjugating a cycle \sigma by \tau amounts to applying \tau internally to each letter in the cycle \sigma.

    i.e. if \sigma=(1\,2\,3\,\cdots\,k) then \tau\sigma\tau^{-1}=(\tau(1)\,\tau(2)\,\cdots\,\tau(k)).

    Let (1 2) and (1 2 3 ... n) be your generating cycles. Conjugate the transposition by the longer cycle and you get (2 3). Conjugate that guy by the longer cycle and you get (3 4). Keep doing that and you'll have all your transpositions of the form (k k+1).

    That's all you need to generate S_n, because (1 k) = (1 k-1)(k-1 k)(1 k-1), so you can now generate all transpositions that look like (1 k).

    Now remember that (j k) = (1 j)(1 k)(1 j), so now you have enough to generate all possible (j k), which is all possible transpositions. Combine those transpositions to make any cycle you want in S_n.

    BTW you may notice I didn't generate with any old transposition and n-cycle, I chose two specific ones. It doesn't matter-- the proof holds for any arrangement of symbols.
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