1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A conjugate of two permutations question

  1. Oct 24, 2011 #1
    1. The problem statement, all variables and given/known data
    Suppose [itex]x_{1} = \begin{pmatrix}
    2 & 9 & 6 \\
    \end{pmatrix}\begin{pmatrix}
    3 & 5 & 8 \\
    \end{pmatrix}\begin{pmatrix}
    4 & 7 \\
    \end{pmatrix}[/itex] and [itex]x_{2} = \begin{pmatrix}
    1 & 5 & 9 \\
    \end{pmatrix}\begin{pmatrix}
    2 & 7 & 6 \\
    \end{pmatrix}\begin{pmatrix}
    3 & 4 \\
    \end{pmatrix}.[/itex]

    Determine the conjugate a, so that x1 = ax2a-1.

    3. The attempt at a solution
    I know the solution is a = (1 6 8)(2 3 7 5), since we did this in class. However, we didn't really explain how we got to this solution. And I can do conjugates where you just line up the cycles one under the other, but this method doesn't work here, because, say 1 does not get sent to 2, and 5 not to 9, as you'd assume if you just wrote (1 5 9) above (2 9 6).

    I really want to figure this out, but this example really puzzles me, as I haven't yet found the general method, and the fact that x2,i = a(x1,i) doesn't really help me here.

    Anyways, any help here would be greatly appreciated.
     
    Last edited: Oct 24, 2011
  2. jcsd
  3. Oct 24, 2011 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I'm not sure what you mean by "line up the cycles" but I think you meant writing [itex]x_1[/itex] as
    [tex]\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\1 & 6 & 5 & 7 & 3 & 9 & 4 & 5 & 2\end{pmatrix}[/tex]
    and [itex]x_2[/itex] as
    [tex]\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\9 & 6 & 4 & 3 & 1 & 7 & 2 & 8 & 5\end{pmatrix}[/tex]

    Do you see how I got that? (I am assuming you are using the convention that your permutations work from right to left.)
     
    Last edited: Oct 24, 2011
  4. Oct 24, 2011 #3
    Yeah, I can follow that, but I can't seem to be able to make the next step then. For example, it isn't obvious to me why a sends 1 to 6 instead of say, to 2 or 9.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: A conjugate of two permutations question
  1. Permutation Question (Replies: 1)

Loading...