Class equation of a group action

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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.