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 (1)
| Label | Occurrences |
|---|---|
| Lagrange's theorem in group theory canonical | 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
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.