Results 1 to 2 of 2

Math Help - Permutation group conjugates

  1. #1
    Newbie
    Joined
    Oct 2011
    Posts
    24

    Permutation group conjugates

    Hey,

    I just have a small question regarding the conjugation of permutation groups.

    Two permutations are conjugates iff they have the same cycle structure.

    However the conjugation permutation, which i'll call s can be any cycle structure. (s-1 a s = b) where a, b and conjugate permutations

    My question is, how can you find out how many conjugation permutations (s) are within a group which also conjugate a and b.

    So for example (1 4 2)(3 5) conjugates to (1 2 4)(3 5) under s = (2 4), how could you find the number of alternate s's in the group of permutations with 5 objects?

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,397
    Thanks
    760

    Re: Permutation group conjugates

    well, you know that s has to map the set {1,2,4} to the set {1,2,4} and the set {3,5} to the set {3,5}.

    this gives us TWO choices for s on 3 and 5, either:

    s(3) = 3, s(5) = 5 -OR-
    s(3) = 5, s(5) = 3.

    the first set ({1,2,4}) is a little trickier. we have 3 choices for s(1): s(1) = 1, s(1) = 2, or s(1) = 4.

    if s(1) = 1, then s(4) must be 2, and thus s(2) = 4.

    if s(1) = 2, then s(4) must be 4, and s(2) = 1. for example, one can verify that if s = (1 2), that:

    (1 2)[(1 4 2)(3 5)](1 2) = [(1 2)(1 4 2)][(1 2)(3 5)] (since (1 2) and (3 5) commute)

    = [(1 2)(1 4 2)(1 2)](3 5) = (2 4 1)(3 5) = (1 2 4)(3 5) (since (2 4 1) = (1 2 4)).

    if s(1) = 4, then s(4) = 1, and s(2) = 2.

    so s is completely determined by s(1), and s(3). we have 3 choices for s(1), and 2 choices for s(3), giving 6 possible s:

    s = (2 4)
    s = (2 4)(3 5)
    s = (1 2)
    s = (1 2)(3 5)
    s = (1 4)
    s = (1 4)(3 5) is a complete list.

    keep in mind, however, that this is a somewhat artificial example. if we are dealing with S9, for example, s might map any element of {6,7,8,9} to any other (which wouldn't affect the conjugate, since (1 4 2)(3 5) doesn't affect 6,7,8 or 9, so as long as s takes an element of {6,7,8,9} to another element of {6,7,8,9}, s^-1 just takes it back). so in that case, we get 24 permutations for each of the s i listed above. so there's no "general formula", the answer depends on which symmetric group you're conjugating IN, and not just the conjugates themselves.

    keep in mind, as well, that symmetric groups Sn (except for very small values of n), tend to be quite large, and that finding all elements that conjugate a to b, can be a tedious task. usually, finding one that works is "good enough".
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Diagonal elements of a permutation group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 16th 2010, 03:53 PM
  2. Permutation Group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 16th 2010, 03:28 PM
  3. Conjugates of elements in a group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 2nd 2010, 03:17 AM
  4. Conjugates in a finite group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 24th 2008, 05:56 AM
  5. Permutation group order
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 11th 2007, 07:26 PM

Search Tags


/mathhelpforum @mathhelpforum