Jordan–Hölder theorem

E157402

mathematical theorem theorem theorem in group theory

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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All labels observed (4)

Label	Occurrences
Jordan–Hölder theorem canonical	8
Hölder’s theorem on the isomorphism of composition series	1
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 for finite length objects in abelian categories ⓘ linked to: Jordan–Hölder theorem Jordan–Hölder theorem for modules ⓘ linked to: Jordan–Hölder theorem
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

Referenced by (11)

Full triples — surface form annotated when it differs from this entity's canonical label.

Noether's isomorphism theorems → prerequisiteFor → Jordan–Hölder theorem ⓘ

Jordan–Hölder theorem → hasVersion → Jordan–Hölder theorem for modules ⓘ

linked to: Jordan–Hölder theorem

Jordan–Hölder theorem → hasVersion → Jordan–Hölder theorem for finite length objects in abelian categories ⓘ

linked to: Jordan–Hölder theorem

Camille Jordan → knownFor → Jordan–Hölder theorem ⓘ

Schreier refinement theorem → refines → Jordan–Hölder theorem ⓘ

Schreier refinement theorem → isPrecursorOf → Jordan–Hölder theorem ⓘ

Schreier refinement theorem → isCompanionOf → Jordan–Hölder theorem ⓘ

Otto Hölder → notableFor → Jordan–Hölder theorem ⓘ

Otto Hölder → notableFor → Hölder’s theorem on the isomorphism of composition series ⓘ

linked to: Jordan–Hölder theorem

Zassenhaus lemma → relatedTo → Jordan–Hölder theorem ⓘ

Zassenhaus neighborhood → relatedTo → Jordan–Hölder theorem ⓘ

All labels observed (4)

How this entity was disambiguated Show

Statements (47)

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Referenced by (11)

How this entity was disambiguated

How these facts were elicited