In the following, let G be a finite group that acts on a set X. For each g in G let X^{g} denote the set of elements in X that are fixed by g. Burnside's lemma asserts the following formula for the number of orbits, denoted X/G:
Thus the number of orbits (a natural number or infinity) is equal to the average number of points fixed by an element of G (which consequently is also a natural number or infinity).
The number of rotationally distinct colourings of the faces of a cube using three colours can be determined from this formula as follows.
Let X be the set of 3^{6} fixed coloured cubes, and let the rotation group G of the cube act on X in the natural manner. Then two elements of X belong to the same orbit precisely when one is simply a rotation of the other. The number of rotationally distinct colourings is thus the same as the number of orbits and can be found by counting the sizes of the fixed sets for the 24 elements of G.
The proof uses the orbitstabilizer theorem and the fact that X is the disjoint union of the orbits:
History
William Burnside wrote in 1900 about this formula, but mathematical historians have pointed out that he was not the first to discover it; Cauchy in 1845 and Frobenius[?] in 1887 also knew of this formula.
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