Lagrange's theorem in group theory
E156184
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lagrange's theorem in group theory canonical | 2 |
| Lagrange theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1358585 — 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: Lagrange's theorem in group theory Context triple: [Joseph-Louis Lagrange, knownFor, Lagrange's theorem in group theory]
-
A.
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.
-
B.
Fermat's little theorem
Fermat's little theorem is a fundamental result in number theory that characterizes how prime numbers interact with integer powers modulo that prime, forming the basis for many modern cryptographic algorithms.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Abelian groups
Abelian groups are algebraic structures in which the group operation is commutative, meaning the order of combining elements does not affect the result.
-
E.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
- 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: Lagrange's theorem in group theory Target entity description: 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.
-
A.
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.
-
B.
Fermat's little theorem
Fermat's little theorem is a fundamental result in number theory that characterizes how prime numbers interact with integer powers modulo that prime, forming the basis for many modern cryptographic algorithms.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Abelian groups
Abelian groups are algebraic structures in which the group operation is commutative, meaning the order of combining elements does not affect the result.
-
E.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in abstract algebra
ⓘ
theorem in group theory ⓘ |
| appliesTo | finite groups ⓘ |
| assumes | group is finite ⓘ |
| category | finite group theory ⓘ |
| definesConcept | index of a subgroup ⓘ |
| doesNotGenerallyHoldFor | infinite groups ⓘ |
| exampleApplication |
in a group of order 12, any subgroup has order dividing 12
ⓘ
in a group of prime order p, every non-identity element generates the group ⓘ |
| field | group theory ⓘ |
| generalizationOf | divisibility of orders in cyclic groups ⓘ |
| hasConsequence | the index of a subgroup divides the order of the group ⓘ |
| historicalPeriod | 18th century mathematics ⓘ |
| holdsIn |
finite abelian groups
ⓘ
finite non-abelian groups ⓘ |
| implies |
the order of a subgroup equals the number of its left cosets times the order of the subgroup
ⓘ
the order of any element of a finite group divides the order of the group ⓘ |G| = [G : H] · |H| for finite group G and subgroup H ⓘ |
| importance | fundamental theorem for structure of finite groups ⓘ |
| isGeneralizedBy |
Cauchy's theorem in group theory
ⓘ
Sylow theorems ⓘ |
| isRelatedTo |
Burnside's lemma
ⓘ
Cayley's theorem ⓘ orbit-stabilizer theorem ⓘ |
| mathematicalDomain |
algebra
ⓘ
discrete mathematics ⓘ |
| namedAfter | Joseph-Louis Lagrange ⓘ |
| proofMethod | partition of the group into left cosets of a subgroup ⓘ |
| relatesConcept |
group order
ⓘ
subgroup order ⓘ |
| requiresPrerequisite |
definition of coset
ⓘ
definition of group ⓘ definition of group order ⓘ definition of subgroup ⓘ |
| statement | For a finite group G and a subgroup H of G, the order of H divides the order of G ⓘ |
| typicalNotation |
[G : H] for index of H in G
ⓘ
|G| for order of group G ⓘ |H| for order of subgroup H ⓘ |
| usedFor |
classifying small finite groups
ⓘ
proving that groups of prime order are cyclic ⓘ showing that certain subgroups cannot exist in a given finite group ⓘ |
| usedIn |
element order computations
ⓘ
finite group actions ⓘ proving non-existence of elements of certain orders ⓘ |
| usesConcept |
equivalence relation induced by a subgroup
ⓘ
left coset ⓘ right coset ⓘ |
How these facts were elicited
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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: Lagrange's theorem in group theory Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lagrange's theorem in group theory