Burnside's lemma
E586575
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Burnside's lemma canonical | 2 |
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
NERFINISHED
ⓘ
Cauchy–Frobenius–Burnside lemma NERFINISHED ⓘ |
| appearsIn | Burnside's book "Theory of Groups of Finite Order" NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| relatesConcept |
Polya enumeration theorem
NERFINISHED
ⓘ
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
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Instruction
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Input
Subject: Burnside's lemma Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.