Burnside's lemma

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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 include William Burnside, George Pólya, Augustin Louis Cauchy, and Ferdinand Georg Frobenius. Note that this result is certainly not due to Burnside himself, who merely quotes it in his book 'On the Theory of Groups of Finite Order'.

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|:

$|X/G| = \frac{1}{|G|}\sum_{g \in G}|X^g|.$

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).

[edit] 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.

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.

Cube

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

A detailed examination of these automorphisms may be found here.

The average fix size is thus

$\frac{1}{24}\left(3^6+6\times 3^3 + 3 \times 3^4 + 8 \times 3^2 + 6 \times 3^3 \right) = 57.$

Hence there are 57 rotationally distinct colourings of the faces of a cube in three colours. In general, the number of rotationally distinct colorings of the faces of a cube in n colors is given by

$\frac{1}{24}\left(n^6+3n^4 + 12n^3 + 8n^2\right)$

[edit] Proof

The proof uses the orbit-stabilizer theorem and the fact that X is the disjoint union of the orbits:

$\sum_{g \in G}|X^g| = |\{(g,x)\in G\times X \mid g\cdot x = x\}| = \sum_{x \in X} |G_x|$

$= \sum_{x \in X} \frac{|G|}{|Gx|} = |G| \sum_{x \in X}\frac{1}{|Gx|} = |G|\sum_{A\in X/G}\sum_{x\in A} \frac{1}{|A|}$

$= |G| \sum_{A\in X/G} 1 = |G| \cdot |X/G|.$

[edit] History: the lemma that is not Burnside's

William Burnside stated and proved this lemma (without attribution) in his 1897 book 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. In fact, the lemma was apparently so well known that Burnside simply omitted to attribute it to Cauchy. Consequently, this lemma is sometimes referred to as the lemma that is not Burnside's. This isn't as ambiguous as it seems - Burnside was so prolific, that most of the lemmas of this field are Burnside's!

Burnside, William (1897). Theory of groups of finite order. Cambridge University Press. (This is the first edition; the introduction to the second edition contains Burnside's famous volte face regarding the utility of representation theory!)