Class equation of a group

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

 * Class equation of a group relative to a prime power

Facts used

 * 1) uses::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 $$g_i$$.