Class equation of a group

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Statement

Suppose is a finite group, is the center of , and are all the conjugacy classes in comprising the elements outside the center. Let be an element in for each . Then, we have:

.

Note that this is a special case of the class equation of a group action where the group acts on itself by conjugation.

Related facts

Facts used

  1. Class equation of a group action

Proof

The proof follows directly from fact (1), and the following observations:

  • When a group acts on itself by conjugation, the set of fixed points under the action is precisely the center of the group.
  • The stabilizer of a point under the action by conjugation is precisely the centralizer of .