Results 1 to 3 of 3

Math Help - conjugacy class

  1. #1
    Newbie
    Joined
    Sep 2010
    Posts
    9

    conjugacy class

    what is conjugacy class?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,242
    Thanks
    1795
    Two members of a group, x and y, are said to be "conjugate" if there exists a member of the group, z, such that zxz^{-1}= y. That, of course, is the same as saying that zx= yz. It is easy to prove that that is an equivalence relation (refexive: x is conjugate to itself by taking z equal to the group identity. symmetric: if x is conjugate to y, so that zxz^{-1}= y, then x= z^{-1}yz= uyu^{-1} with u= z^{-1}. transitive: if x is conjugate to y and y is conjugate to z then uxu^{-1}= y and vyv^{-1}= z. Then v(uxu^{-1})v^{-1}= (vu)x(vu)^{-1}= z so x is conjugate to x).

    A "conjugacy class" is the equivalence class using "conjugate" as the equivalence relation.

    Notice that if the group is Abelian (commutative) then zx= xz so that if x and y are conjuate, zx= xz= yz and then x= y. That, is, in Abelian groups, a member of the group is conjugate only to itself and the conjugacy classes consist only of the single elements.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. conjugacy class
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 1st 2010, 08:09 PM
  2. Conjugacy class
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 19th 2009, 08:44 PM
  3. conjugacy class and class equation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 1st 2009, 07:52 PM
  4. conjugacy class
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 2nd 2008, 10:55 AM
  5. Conjugacy Class Equation
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 22nd 2008, 11:01 AM

Search Tags


/mathhelpforum @mathhelpforum