1. Not finding help here? Sign up for a free 30min 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!

Problem with Permutations

  1. Nov 3, 2009 #1
    1. The problem statement, all variables and given/known data
    Let x=(1,2)(3,4) [tex]\in S_{8}[/tex].
    Find an a [tex]\in S_{8}[/tex] such that a-1xa=(5,6)(1,3)


    2. Relevant equations



    3. The attempt at a solution
    I have no idea how you go about finding the a. Help please.
     
  2. jcsd
  3. Nov 3, 2009 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Permutations

    First, notice that [itex]x[/itex] does not have any effect on 5,6,7,8. Therefore, whichever inputs are mapped by [itex]a[/itex] to these numbers will be mapped back where they started by [itex]a^{-1}[/itex]. You want [itex]a^{-1}xa[/itex] to leave 2,4,7,8 where they are, so you could for example define

    [tex]a(2) = 5[/tex]
    [tex]a(4) = 6[/tex]
    [tex]a(7) = 7[/tex]
    [tex]a(8) = 8[/tex]

    Now let's look at the remaining numbers. Suppose we arbitrarily choose

    [tex]a(1) = 1[/tex]

    Then [itex]x[/itex] maps 1 to 2, so [itex]xa[/itex] maps 1 to 2. We want [itex]a^{-1}xa[/itex] to map 1 to 3, therefore [itex]a^{-1}[/itex] must map 2 to 3:

    [tex]a^{-1}(2) = 3[/tex]

    and thus

    [tex]a(3) = 2[/tex]

    Thus far we have defined [itex]a[/itex] for six of the inputs, and it's easy to verify that [itex]a^{-1}xa[/itex] sends these six inputs to the right outputs. So now you have to define [itex]a[/itex] for the remaining two inputs (5 and 6). I'll let you take it from here.

    Note that there are many possible solutions to this problem.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Problem with Permutations
  1. Permutation problem (Replies: 1)

  2. Permutation problem (Replies: 2)

Loading...