For finite , the bijection gives , or equivalently .

Let and define . If , then for some , so ; hence is well-defined. It is surjective by the definition of . If , then , so and is injective.

For the action on the vertices of a regular -gon, an orbit has size and a vertex stabiliser has size , giving .

For the isometry group of a cube acting on face centres, an orbit has size and a face stabiliser has size , giving .