Results 1 to 6 of 6

Math Help - Conjugacy

  1. #1
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    Conjugacy

    hey I was having some trouble here and I was wondering if anybody could assist me.

    Show that (1 2 3 4 5) is conjugate to (1 3 5 2 4) in  S_5 but not in  A_5 .

    Clearly (4 5 3 2) will work and this is odd, so not in A5. but why can't we find an even permutation that will work for the conjugation?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    I found an old past paper solution that says the following:

    If  ^{\sigma}(1\ 2\ 3\ 4\ 5) = (1\ 3\ 5\ 2\ 4)

    Then  \sigma = (2\ 3\ 4\ 5)(1\ 2\ 3\ 4\ 5)^{l} .

    Why? And why is the number of elements in  A_5 of order 5 equal to  \frac{5!}{5} ??
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by slevvio View Post
    hey I was having some trouble here and I was wondering if anybody could assist me.

    Show that (1 2 3 4 5) is conjugate to (1 3 5 2 4) in  S_5 but not in  A_5 .

    Clearly (4 5 3 2) will work and this is odd, so not in A5. but why can't we find an even permutation that will work for the conjugation?
    Have you checked to see if the first is odd and the second even?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    first and second what?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by slevvio View Post
    hey I was having some trouble here and I was wondering if anybody could assist me.

    Show that (1 2 3 4 5) is conjugate to (1 3 5 2 4) in  S_5 but not in  A_5 .

    Clearly (4 5 3 2) will work and this is odd, so not in A5. but why can't we find an even permutation that will work for the conjugation?
    Quote Originally Posted by slevvio View Post
    first and second what?
    Well, my first inclination (and this may be incorrect) was to use the fact that the product of any number of even permutations is even (and the odd analogue) to show that g(12345)g^{-1}=(13524)\implies g\text{ is odd}. You could do this by showing that (12345) is even and (13524) is odd. The problem would follow.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    Those 2 permutations can be expressed as an even number of transpositions hence they are both even
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. conjugacy classes
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: March 2nd 2010, 12:12 AM
  2. conjugacy class
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 1st 2010, 07:09 PM
  3. Conjugacy
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: November 23rd 2009, 03:39 AM
  4. Conjugacy
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 24th 2009, 08:52 AM
  5. Conjugacy classes
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 26th 2008, 06:22 AM

Search Tags


/mathhelpforum @mathhelpforum