Results 1 to 7 of 7

Math Help - conjugacy classes

  1. #1
    Banned
    Joined
    Mar 2011
    Posts
    118

    conjugacy classes

    question: Find the order of the folowing elements of s15 and determine which pairs are conjugate.

    i) (1 2 4 8)(3 6 12)(5 10 15)

    ii) (1 2)(4 8)(3 12 11)(5 13 7)

    iii) (1 7 9)(11 12 13)(5 10 8 6)

    I have found the orders using lcm i) 12, ii) 6, iii) 12.

    To be conjugate, elements need to have same order so only i) and iii) are possibilities.
    I tried using ii) as 'g' in a=gbg^-1 but it didn't work. I don't think it expects to trawl through s15 looking for a g so is there an easy way to determine conjugacy?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by poirot View Post
    question: Find the order of the folowing elements of s15 and determine which pairs are conjugate.

    i) (1 2 4 8)(3 6 12)(5 10 15)

    ii) (1 2)(4 8)(3 12 11)(5 13 7)

    iii) (1 7 9)(11 12 13)(5 10 8 6)

    I have found the orders using lcm i) 12, ii) 6, iii) 12.

    To be conjugate, elements need to have same order so only i) and iii) are possibilities.
    I tried using ii) as 'g' in a=gbg^-1 but it didn't work. I don't think it expects to trawl through s15 looking for a g so is there an easy way to determine conjugacy?
    A possibility is to decompose them into disjoint cycles then use the fact that two elements of S_n are conjugate if and only if they have the same cycle structure (same number of disjoint cycles of same length)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Mar 2011
    Posts
    118
    Quote Originally Posted by Drexel28 View Post
    A possibility is to decompose them into disjoint cycles then use the fact that two elements of S_n are conjugate if and only if they have the same cycle structure (same number of disjoint cycles of same length)
    so are (i) and (iii) conjugate?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by poirot View Post
    so are (i) and (iii) conjugate?
    Hell if I know...I'm sorry I don't feel like wading through the cycle deomposition. Did you try it?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Mar 2011
    Posts
    118
    Quote Originally Posted by Drexel28 View Post
    Hell if I know...I'm sorry I don't feel like wading through the cycle deomposition. Did you try it?
    those elelments are as a product of disjoint cycles so there is no further work.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by poirot View Post
    those elelments are as a product of disjoint cycles so there is no further work.
    Hahaha! God bless you sir. Anyways, then I would ask you to inspect them and tell me if they have the same number of disjoint cycles (yes ) and if the cycles have the same length (yes )
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Banned
    Joined
    Mar 2011
    Posts
    118
    Quote Originally Posted by Drexel28 View Post
    Hahaha! God bless you sir. Anyways, then I would ask you to inspect them and tell me if they have the same number of disjoint cycles (yes ) and if the cycles have the same length (yes )
    haha lol. Yes I can count, I was just comfirming thats what you meant.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. conjugacy classes
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 16th 2009, 12:25 PM
  2. conjugacy classes
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 5th 2009, 02:13 AM
  3. Conjugacy Classes
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 9th 2008, 12:29 PM
  4. Conjugacy Classes in A_n
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 1st 2008, 04:53 PM
  5. Conjugacy classes
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 26th 2008, 07:22 AM

Search Tags


/mathhelpforum @mathhelpforum