Group theory and generalizations Group actions and permutation groups Theorem

Burnside's lemma

Builds on Fixed points of an action Orbit Size of an orbit times the size of the stabiliser

Statement

Let a finite group $G$ act on a finite set $X$ . For $g \in G$ , let $X^g = \{x \in X : g \cdot x = x\}$ . Then the number of orbits is $|X/G| = \frac{1}{|G|} \sum_{g \in G} |X^g|$ .

Remarks

The proof counts the pairs $(g, x)$ with $g \cdot x = x$ in two ways, and uses $|G_x| = |G| / |G \cdot x|$ . For example, the rotation group $\mathbb{Z}/4\mathbb{Z}$ of a square acts on the $2^4 = 16$ colourings of the corners with two colours. The fixed colourings number $16$ , $2$ , $4$ and $2$ , so there are $(16 + 2 + 4 + 2)/4 = 6$ colourings up to rotation.

History

William Burnside stated and proved the lemma in his 1897 book on finite groups, and he attributed it to Frobenius, 1887. Cauchy already knew the formula in 1845. For this reason, the lemma is sometimes called "the lemma that is not Burnside's".

Reading path

To understand this page from the ground up, read these 55 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

Element definition
Group concept
Left coset concept
Proposition concept
Set definition
Conjugate elements definition
Group homomorphism definition
Order of a group definition
Subgroup definition
Symmetric group definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Conjugate subgroup definition
Group action definition
Left coset definition
Right coset definition
Subgroup criterion statement
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Disjoint sets definition
Cartesian product definition
Normal subgroup definition
Stabiliser definition
Subset definition
Union of a family of sets definition
Orbit definition
Intersection of subgroups is a subgroup statement
Index of a subgroup definition
Cosets have the same cardinality as the subgroup statement
Binary relation definition
Partition of a set definition
Normal core definition
Fixed points of an action definition
Transitive action definition
Order of a group is the index times the order of the subgroup statement
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Action on left cosets example
Equivalence relation definition
Orbit-stabiliser theorem statement
Equivalence class definition
Equivalence classes form a partition statement
Proof of Lagrange's theorem proof
Lagrange's theorem statement
Size of an orbit times the size of the stabiliser statement
Burnside's lemma statement

Tag	`0Z47M1B2VS`
URI	`m2://0Z47M1B2VS`
Aliases	`orbit-counting-theorem`, `cauchy-frobenius-lemma`
Kind	statement, theorem
Writing	W1 stated
MSC	20B05, 05E18
Status	This copy of the site has no build data. Run `m2 status 0Z47M1B2VS`.
Uses	Fixed points of an action, Orbit, Size of an orbit times the size of the stabiliser
Same as	Wikidata Q1330377 Wikipedia Burnside's lemma

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