Class equation of a group

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Statement

Suppose G is a finite group, Z(G) is the center of G, and c_1, c_2, \dots, c_r are all the conjugacy classes in G comprising the elements outside the center. Let g_i be an element in c_i for each 1 \le i \le r. Then, we have:

|G| = |Z(G)| + \sum_{i=1}^r |G:C_G(g_i)|.

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 g_i under the action by conjugation is precisely the centralizer of g_i.