Burnside's lemma, sometimes also called Burnside's counting theorem, Pólya's formula, 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 include William Burnside, George Pólya, Augustin Louis Cauchy, and Ferdinand Georg Frobenius. Group theory is that branch of mathematics concerned with the study of groups. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... William Burnside (July 2, 1852 - August 21, 1927) was an English mathematician. ... George Pólya (December 13, 1887 - September 7, 1985, in Hungarian Pólya György) was a mathematician, who was born in Budapest, Hungary and died in Palo Alto, USA. He worked on a great variety of mathematical topics, including series, number theory, combinatorics, and probability. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ...

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|: In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... This article is about the mathematical concept. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ... In mathematics, groups are often used to describe symmetries of objects. ...

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). Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Infinite redirects here, For the album by Eminem, see Infinite (album). ... In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. ...

Example application

The number of rotationally distinct colourings of the faces of a cube using three colours can be determined from this formula as follows. Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...

Let X be the set of 3⁶ 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.

• one identity element fixing all 3⁶ elements of X
• six 90-degree face rotations fixing 3³ elements of X
• three 180-degree face rotations fixing 3⁴ elements of X
• eight 120-degree vertex rotations fixing 3² elements of X
• six 180-degree edge rotations fixing 3³ elements of X

The average fix size is thus Hexahedron (sometimes called cube), rendered by Java applet I wrote. ...

Hence there are 57 rotationally distinct colourings of the faces of a cube in three colours.

Proof

The proof uses the orbit-stabilizer theorem and the fact that X is the disjoint union of the orbits: In mathematics, groups are often used to describe symmetries of objects. ...

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. Hence it is sometimes referred to by witty mathematicians as Not Burnside's lemma. William Burnside (July 2, 1852 - August 21, 1927) was an English mathematician. ... 1900 is a common year starting on Monday. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... 1845 was a common year starting on Wednesday (see link for calendar). ... 1887 is a common year starting on Saturday (click on link for calendar). ...