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Me, myself and conjugate permutations

  1. Apr 22, 2005 #1

    Is there a general method, given [tex]\sigma[/tex] and [tex]\rho[/tex] in Sn, for finding a permutation [tex]\tau[/tex] in Sn such that [tex]\rho = \tau ^{-1} \sigma \tau[/tex]? I know how to do it when [tex]\sigma[/tex] and [tex]\rho[/tex] are made of a single k-cycle, but what happens when they are more complex?

    For example, for:
    [tex]\sigma = (1, 2)(3, 4)[/tex]
    [tex]\rho = (5, 6)(1, 3)[/tex]
    In S6.

    Last edited: Apr 22, 2005
  2. jcsd
  3. Apr 22, 2005 #2
    Do you know you've just solved me a problem I've been working on all day? (By reminding me of conjugation).

    Now to current business: I don't remember it precisely, but isn't the conjugation lemma suppose to do the trick? Or does it work only with k-circles?
  4. Apr 22, 2005 #3

    matt grime

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    As a hint you could try thinking of change of basis in a vector space where we realize the permutations as a permutation of basis elements. That ought to work.
  5. Apr 23, 2005 #4
    Glad I could help. :biggrin:

    I figured it out, by trial and error. I needed to find [tex]\tau \in S_6[/tex] such that:
    [tex]\tau (1, 2)(3, 4) \tau ^{-1} = (5, 6)(1, 3)[/tex]
    Which can be written as:
    [tex]\tau (1, 2)\tau ^{-1} \tau (3, 4) \tau ^{-1} = (5, 6)(1, 3)[/tex]
    So I assumed that:
    [tex]\tau (1, 2) \tau ^{-1} = (5, 6)[/tex]
    [tex]\tau (3, 4) \tau ^{-1} = (1, 3)[/tex]
    Which means that:
    [tex]\tau (1) = 5[/tex]
    [tex]\tau (2) = 6[/tex]
    [tex]\tau (3) = 1[/tex]
    [tex]\tau (4) = 3[/tex]
    So I get:
    [tex]\tau = (4, 3, 1)(1, 5)(2, 6)[/tex]

    Hopefully though this wasn't a fluke and this method will work all the time. Matt, unfortunately I don't really know what you're talking about... :blushing: or maybe I know it by a different name. Thanks thought.
  6. Apr 23, 2005 #5
    It should work every time, if I remember my first modern algebra course correctly.

    Oh, and Hag Sameah :)
  7. Apr 24, 2005 #6
    Thank you very much, Happy Passover. :smile:
  8. Apr 24, 2005 #7
    Pessah my friend, I'm from Haifa.
  9. Apr 24, 2005 #8
    Yeah, I thought so. I think I know you from ASAT. :wink:
  10. Apr 24, 2005 #9
    It's a small world after all... :smile:
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