# Class equation of a group

(Redirected from Class equation)

## 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.

## 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 .