Jordan–Hölder theorem
E157402
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Jordan–Hölder theorem canonical | 2 |
| Jordan–Hölder theorem for finite length objects in abelian categories | 1 |
| Jordan–Hölder theorem for modules | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382942 — 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: Jordan–Hölder theorem Context triple: [Noether's isomorphism theorems, prerequisiteFor, Jordan–Hölder theorem]
-
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.
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.
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.
-
D.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
E.
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.
- 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: Jordan–Hölder theorem Target entity description: 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.
-
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.
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.
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.
-
D.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem ⓘ theorem in group theory ⓘ |
| appliesTo |
finite groups
ⓘ
finite length modules ⓘ |
| assumes | existence of a composition series ⓘ |
| category | algebraic structure theorem ⓘ |
| concerns | refinement of subnormal series ⓘ |
| ensures |
invariant composition length for a finite group
ⓘ
invariant multiset of composition factors for a finite group ⓘ |
| field |
abstract algebra
ⓘ
group theory ⓘ |
| formalizes | analogy between groups and integers via factorization ⓘ |
| generalizationOf | uniqueness of prime factorization in integers ⓘ |
| guarantees | uniqueness of composition factors up to permutation ⓘ |
| hasConsequence |
classification of finite groups via composition factors
ⓘ
definition of composition length of a group ⓘ |
| hasVersion |
Jordan–Hölder theorem
self-linksurface differs
ⓘ
surface form:
Jordan–Hölder theorem for finite length objects in abelian categories
Jordan–Hölder theorem self-linksurface differs ⓘ
surface form:
Jordan–Hölder theorem for modules
|
| historicalPeriod | late 19th century ⓘ |
| holdsFor |
finite simple groups
ⓘ
finite solvable groups ⓘ |
| implies | uniqueness of composition factors up to order and isomorphism ⓘ |
| involvesConcept |
normal subgroup
ⓘ
quotient group ⓘ simple factor group ⓘ subnormal series of a group ⓘ |
| isPartOf | classical group theory ⓘ |
| isRefinementOf | Schreier refinement theorem in the case of composition series ⓘ |
| namedAfter |
Camille Jordan
ⓘ
Otto Hölder NERFINISHED ⓘ |
| relatedTo |
Schreier refinement theorem
ⓘ
chief series ⓘ composition series ⓘ simple group ⓘ subnormal series ⓘ |
| requires | group to be of finite length for a composition series ⓘ |
| role | fundamental result in group theory ⓘ |
| states |
any two composition series of a finite group have isomorphic composition factors up to order
ⓘ
any two composition series of a finite group have the same length ⓘ |
| subject |
composition series
ⓘ
finite groups ⓘ simple groups ⓘ |
| typeOfUniqueness | uniqueness up to order and isomorphism ⓘ |
| usedIn |
module theory
ⓘ
representation theory ⓘ structure theory of finite groups ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Jordan–Hölder theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Jordan–Hölder theorem for modules
this entity surface form:
Jordan–Hölder theorem for finite length objects in abelian categories