Burnside's lemma

E586575

lemma in group theory result in combinatorics

Burnside's lemma is a result in group theory and combinatorics that counts distinct configurations under symmetries by averaging the number of fixed points of group actions.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (4)

Label	Occurrences
Burnside's lemma canonical	7
Burnside’s theorem	1
Cauchy–Frobenius lemma	1
Cauchy–Frobenius–Burnside lemma	1

How this entity was disambiguated

This entity first appeared as the object of triple T6327843 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Burnside's lemma
Context triple: [enumerative combinatorics, usesConcept, Burnside's lemma]

A. Lagrange's theorem in group theory
Lagrange's theorem in group theory is a fundamental result stating that the order of any subgroup of a finite group divides the order of the group.
B. The Twelvefold Way
The Twelvefold Way is a framework in combinatorics that systematically classifies twelve fundamental ways of counting functions between finite sets under various labeling and structural constraints.
C. Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
D. Alexander–Briggs notation
Alexander–Briggs notation is a classical system for naming and classifying knots in knot theory, assigning each distinct knot a unique label based on its crossing number and order in knot tables.
E. Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Burnside's lemma
Target entity description: Burnside's lemma is a result in group theory and combinatorics that counts distinct configurations under symmetries by averaging the number of fixed points of group actions.

A. Lagrange's theorem in group theory
Lagrange's theorem in group theory is a fundamental result stating that the order of any subgroup of a finite group divides the order of the group.
B. The Twelvefold Way
The Twelvefold Way is a framework in combinatorics that systematically classifies twelve fundamental ways of counting functions between finite sets under various labeling and structural constraints.
C. Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
D. Alexander–Briggs notation
Alexander–Briggs notation is a classical system for naming and classifying knots in knot theory, assigning each distinct knot a unique label based on its crossing number and order in knot tables.
E. Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	lemma in group theory ⓘ result in combinatorics ⓘ
alsoKnownAs	Cauchy–Frobenius lemma ⓘ linked to: Burnside's lemma Cauchy–Frobenius–Burnside lemma ⓘ linked to: Burnside's lemma
appearsIn	Burnside's book "Theory of Groups of Finite Order" ⓘ linked to: Theory of Groups
appliesTo	finite groups ⓘ group actions on finite sets ⓘ
assumption	group action is well-defined ⓘ group is finite ⓘ
category	enumerative combinatorics ⓘ group actions in algebra ⓘ
coreIdea	counts orbits by averaging fixed points of group elements ⓘ
field	combinatorics ⓘ group theory ⓘ
formula	\|X/G\| = (1/\|G\|) * Σ_{g∈G} \|Fix(g)\| ⓘ
generalizedBy	Polya enumeration theorem ⓘ linked to: Pólya enumeration theorem
historicalAttribution	ideas developed by Frobenius ⓘ ideas trace back to Cauchy ⓘ often misattributed solely to Burnside ⓘ
implies	number of orbits equals average of fixed point counts ⓘ
mathematicalDomain	abstract algebra ⓘ discrete mathematics ⓘ
namedAfter	William Burnside ⓘ
relatesConcept	Polya enumeration theorem ⓘ linked to: Pólya enumeration theorem class equation ⓘ orbit-stabilizer principle ⓘ
requires	knowledge of basic group theory ⓘ understanding of permutations ⓘ
statementInformal	the number of distinct configurations up to symmetry equals the average number of configurations fixed by each group element ⓘ
symbolDefinition	Fix(g) is the subset of X fixed by group element g ⓘ G is a finite group acting on X ⓘ X is a finite set with a group action of G ⓘ X/G denotes the set of orbits of X under G ⓘ
typeOfCounting	orbit counting ⓘ
typicalExample	counting distinct bead necklaces under rotation ⓘ counting distinct colorings of faces of a cube ⓘ counting symmetrically distinct vertex colorings of polygons ⓘ
usedFor	counting colorings up to symmetry ⓘ counting combinatorial objects modulo group actions ⓘ counting unlabeled graphs ⓘ enumeration under dihedral symmetry ⓘ enumeration under rotational symmetry ⓘ
usesConcept	fixed point ⓘ group ⓘ group action ⓘ orbit ⓘ set ⓘ

How these facts were elicited

Referenced by (10)

Full triples — surface form annotated when it differs from this entity's canonical label.

enumerative combinatorics → usesConcept → Burnside's lemma ⓘ

Lagrange's theorem in group theory → isRelatedTo → Burnside's lemma ⓘ

William Burnside → knownFor → Burnside's lemma ⓘ

William Burnside → notableIdea → Burnside's lemma ⓘ

Pólya enumeration theorem → usesConcept → Burnside's lemma ⓘ

Pólya enumeration theorem → generalizes → Burnside's lemma ⓘ

Burnside's lemma → alsoKnownAs → Cauchy–Frobenius lemma ⓘ

linked to: Burnside's lemma

Burnside's lemma → alsoKnownAs → Cauchy–Frobenius–Burnside lemma ⓘ

linked to: Burnside's lemma

orbit-stabilizer theorem → relatedTo → Burnside's lemma ⓘ

Schur’s lemma → relatedTo → Burnside’s theorem ⓘ

linked to: Burnside's lemma

All labels observed (4)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (10)

How this entity was disambiguated

How these facts were elicited