Class equation of a group action

Statement

Suppose $G$ is a group and $S$ is a finite set. Suppose we are given a 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.