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. 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 $g_i$.