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Homework Help: Permutations (last question of sheet, yay )

  1. Feb 17, 2008 #1
    Permutations (last question of sheet, yay!!)

    1. The problem statement, all variables and given/known data[/b]

    [tex]\eta[/tex]:=
    (1 2 ... n-1 n)
    (n n-1 ... 2 1)
    [tex]\in[/tex]S[tex]_{n}[/tex] for any n[tex]\in[/tex]N
    n.b That should be 2 lines all in one large bracket btw
    a.) Determine its sign.

    b.) Let n [tex]\geq[/tex]1. Let <a1,...,as> [tex]\in[/tex]Sn be a cycle and let [tex]\sigma[/tex][tex]\in[/tex]Sn be arbitrary. Show that

    [tex]\sigma\circ[/tex] <a1,...,as> [tex]\circ[/tex][tex]\sigma^{-1}[/tex] = <[tex]\sigma[/tex](a1),...,[tex]\sigma[/tex](as)> in Sn.

    2. Relevant equations



    3. The attempt at a solution

    I get the sign of the permutation to be (-1)^n/2

    I don;t know how to do the second part, any ideas?
     
    Last edited: Feb 17, 2008
  2. jcsd
  3. Feb 17, 2008 #2
    Actually, i thought i had done the first part, but i havent because im stuck on how to show that for negative numbers, i want the n/2 to be taken as the rounded down value. For example if n=7 i want n/2 to be taken as 3. Is there is a simple way to do this for odd numbers but also keep the same form for positive values of n.

    Also it should be (-1)^(n-2/2)

    So it should be (-1)^(n-2/2) for even numbers of n and (-1)^(n-3/2) for odd values of n, is there a neater way to do this?
     
    Last edited: Feb 17, 2008
  4. Feb 17, 2008 #3
    I have just said for even numbers of n, that is, n/2 [tex]\in[/tex]Z and for odd numbers, that is n/2[tex]\notin[/tex]Z to distinguish between the two cases.

    Im still thinking through part b.) so any help is welcomed.
     
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