# Class equation of a group action

From Groupprops

## Statement

Suppose is a group and is a finite set. Suppose we are given a Group action (?) of on .

- Let denote the set of those points in that are fixed under the action of all elements of .
- Let be the orbits of size greater than one under this action. For each orbit let be an element of and let denote the stabilizer of in . In other words, .

The **class equation** for this action is given as follows:

Note that for the special case of a group acting on itself by conjugation, this equation is called the class equation of a group.

## Related facts

- Class equation of a group: The particular case of a group acting on itself by conjuation.
- Fundamental theorem of group actions, which relates the orbit of an element to the coset space of its stabilizer.
- Orbit-counting lemma (also called Burnside's lemma): A closely related fact that counts the number of orbits in a group action using local computations of how many points of the set are fixed by a group element.
- Polya's theorem: A technique for finding the orbits of combinatorial configurations on a set under the action of symmetry groups on that set.