Class equation of a group action

Statement
Suppose $$G$$ is a group and $$S$$ is a finite set. Suppose we are given a fact about::group action of $$G$$ on $$S$$.


 * Let $$S_0$$ denote the set of those points in $$S$$ that are fixed under the action of all elements of $$G$$.
 * Let $$\mathcal{O}_1, \mathcal{O}_2, \dots, \mathcal{O}_r$$ be the orbits of size greater than one under this action. For each orbit $$\mathcal{O}_i$$ let $$s_i$$ be an element of $$\mathcal{O}_i$$ and let $$G_i$$ denote the stabilizer of $$s_i$$ in $$G$$. In other words, $$G_i = \{ g \in G \mid g \cdot s_i = s_i \}$$.

The class equation for this action is given as follows:

$$|S| = |S_0| + \sum_{i=1}^r |G|/|G_i|$$

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.