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Conjugacy Classes and Group Order

  1. May 13, 2009 #1
    1. The problem statement, all variables and given/known data
    Is there a proof that the number of elements in a conjugacy class of a group has to divide the order of the group?
    Everyone seems to cite it left and right but I've not seen a proof of it anywhere.


    2. Relevant equations



    3. The attempt at a solution
    I'm guessing Langrange's Theorem may come into play but since conjugacy classes themselves aren't subgroups I'm not sure how.
     
  2. jcsd
  3. May 13, 2009 #2

    matt grime

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    Science Advisor
    Homework Helper

    It is a trivial* consequence of the orbit-stabilizer theorem, or the class equation if you want to be fancy.

    If G is a finite group acting on a set X, then |G|=|Orb(x)||Stab(x)|. Let X be G, and let the action be conjugation.

    * Trivial in the sense of an immediate and obvious corollary, not something that is unimportant and easy.
     
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