orbit-stabilizer theorem
E620661
The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of a group acting on a set to the sizes of the orbit of an element and its stabilizer subgroup.
All labels observed (1)
| Label | Occurrences |
|---|---|
| orbit-stabilizer theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6800957 — 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: orbit-stabilizer theorem Context triple: [Lagrange's theorem in group theory, isRelatedTo, orbit-stabilizer theorem]
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A.
Burnside's lemma
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.
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B.
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.
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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.
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D.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
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E.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
- 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: orbit-stabilizer theorem Target entity description: The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of a group acting on a set to the sizes of the orbit of an element and its stabilizer subgroup.
-
A.
Burnside's lemma
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.
-
B.
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.
-
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.
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D.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
-
E.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in group theory
ⓘ
theorem ⓘ |
| appliesTo |
finite groups
ⓘ
group actions on sets ⓘ permutation representations ⓘ |
| area | abstract algebra ⓘ |
| assumes | group action of G on X ⓘ |
| classification | fundamental theorem of group actions ⓘ |
| defines |
orbit of x as {g·x : g in G}
ⓘ
stabilizer of x as {g in G : g·x = x} ⓘ |
| field | group theory ⓘ |
| formalStatement | For a group G acting on a set X and x in X, |G| = |Orb(x)| · |Stab(x)| when all sets are finite ⓘ |
| generalizes | Lagrange's theorem for transitive actions ⓘ |
| holdsFor |
actions by conjugation
ⓘ
actions by permutations ⓘ left group actions ⓘ right group actions ⓘ |
| implies |
index of stabilizer equals size of orbit
ⓘ
|G : Stab(x)| = |Orb(x)| for finite groups ⓘ |
| importance |
basic result taught in undergraduate algebra courses
ⓘ
central tool in finite group theory ⓘ |
| involves |
cardinality of sets
ⓘ
coset decomposition ⓘ group action ⓘ orbit ⓘ stabilizer subgroup ⓘ subgroup ⓘ |
| relatedTo |
Burnside's lemma
ⓘ
Cauchy's theorem ⓘ
linked to:
Cauchy's theorem in group theory
Sylow theorems ⓘ class equation ⓘ coset decomposition theorem ⓘ |
| relates |
size of a group
ⓘ
size of a stabilizer subgroup ⓘ size of an orbit ⓘ |
| usedFor |
Burnside's lemma proofs
ⓘ
analyzing group actions on geometric objects ⓘ class equation derivation ⓘ computing orbit sizes ⓘ computing stabilizer sizes ⓘ counting arguments in combinatorics ⓘ proving Lagrange's theorem ⓘ studying symmetry groups ⓘ |
| usedIn |
Galois theory
ⓘ
algebraic combinatorics ⓘ algebraic geometry group actions ⓘ representation theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: orbit-stabilizer theorem Description of subject: The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of a group acting on a set to the sizes of the orbit of an element and its stabilizer subgroup.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.