Burnside's lemma
Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms are based on William Burnside, George Pólya, Augustin Louis Cauchy, and Ferdinand Georg Frobenius. The result is not due to Burnside himself, who merely quotes it in his book 'On the Theory of Groups of Finite Order', attributing it instead to Frobenius (1887).
In the following, let G be a finite group that acts on a set X. For each g in G let Xg denote the set of elements in X that are fixed by g (also said to be left invariant by g), i.e. Xg = { x ∈ X | g.x = x }. Burnside's lemma asserts the following formula for the number of orbits, denoted |X/G|:
Thus the number of orbits (a natural number or +∞) is equal to the average number of points fixed by an element of G (which is also a natural number or infinity). If G is infinite, the division by |G| may not be well-defined; in this case the following statement in cardinal arithmetic holds:
The number of rotationally distinct colourings of the faces of a cube using three colours can be determined from this formula as follows.
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