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Conjugacy Classes and Group Order

  • Thread starter Kalinka35
  • Start date
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.
 

matt grime

Science Advisor
Homework Helper
9,394
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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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