Orbit-stabilizer theorem

For a group acting on a set, the orbit of an element has size equal to the group’s order divided by the order of its stabilizer.

Group action: A group action assigns each group element a permutation of a set, consistently with the group operation. The theorem applies to each point in a set acted on by a group.
Burnside's lemma: Burnside's lemma counts orbits of a finite group action by averaging the numbers of fixed points. It uses fixed-point counts to determine how many orbits an action has.
Group (mathematics): A group is a set with an associative operation, an identity element, and inverses for every element. The theorem measures orbits using the order of this algebraic structure.
Dihedral group: The dihedral group consists of the rotations and reflections preserving a regular polygon. Its action on polygon vertices yields orbits and stabilizers with simple sizes.
Orbit: The orbit of a point is the set of all points obtainable from it by a group action. Its size is the quantity the theorem calculates.
Class equation: The class equation partitions a finite group into its center and its noncentral conjugacy classes. Conjugacy-class sizes follow from applying the theorem to conjugation.
Subgroup: A subgroup is a subset of a group that is itself a group under the same operation. Every point stabilizer is a subgroup of the acting group.
Symmetric group: The symmetric group on a finite set consists of all permutations of that set. Its natural action on points illustrates how orbit size depends on the stabilizer.
Stabilizer subgroup: The stabilizer of a point is the subgroup of elements that leave that point fixed. Its size is the divisor in the theorem’s formula.
Conjugacy class: The conjugacy class of a group element consists of all its conjugates. Its size equals the group order divided by the centralizer's order.

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