Sylow theorems

E620660

theorem in group theory

The Sylow theorems are fundamental results in finite group theory that describe the existence, conjugacy, and number of subgroups whose orders are powers of a prime dividing the group order.

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All labels observed (5)

Label	Occurrences
Sylow theorems canonical	5
Sylow first theorem	1
Sylow p-subgroups	1
Sylow second theorem	1
Sylow third theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T6800956 — 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: Sylow theorems
Context triple: [Lagrange's theorem in group theory, isGeneralizedBy, Sylow theorems]

A. 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.
B. 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.
C. 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.
D. 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.
E. Theorie der Gruppen von endlicher Ordnung
"Theorie der Gruppen von endlicher Ordnung" is a foundational mathematical monograph on finite group theory that helped shape the modern development of abstract algebra.
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: Sylow theorems
Target entity description: The Sylow theorems are fundamental results in finite group theory that describe the existence, conjugacy, and number of subgroups whose orders are powers of a prime dividing the group order.

A. 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.
B. 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.
C. 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.
D. 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.
E. Theorie der Gruppen von endlicher Ordnung
"Theorie der Gruppen von endlicher Ordnung" is a foundational mathematical monograph on finite group theory that helped shape the modern development of abstract algebra.
F. None of above. chosen

Statements (39)

Predicate	Object
instanceOf	theorem in group theory ⓘ
appearsIn	finite simple group classification arguments ⓘ proofs about groups of small order ⓘ
appliesTo	finite groups ⓘ
concerns	Sylow p-subgroups ⓘ linked to: Sylow theorems p-subgroups ⓘ
describes	conjugacy of subgroups of prime power order ⓘ existence of subgroups of prime power order ⓘ number of subgroups of prime power order ⓘ
field	finite group theory ⓘ group theory ⓘ mathematics ⓘ
formalizes	structure of subgroups whose order is a power of a prime dividing the group order ⓘ
hasConsequence	constraints on possible group orders for simple groups ⓘ existence of normal Sylow p-subgroups when unique ⓘ
hasPart	Sylow first theorem ⓘ linked to: Sylow theorems Sylow second theorem ⓘ linked to: Sylow theorems Sylow third theorem ⓘ linked to: Sylow theorems
historicalPeriod	19th century mathematics ⓘ
implies	all Sylow p-subgroups of a finite group are conjugate ⓘ existence of Sylow p-subgroups for each prime divisor of the group order ⓘ the number of Sylow p-subgroups divides the group order ⓘ the number of Sylow p-subgroups is congruent to 1 modulo p ⓘ
introducedBy	Ludvig Sylow ⓘ
language	symbolic mathematics ⓘ
namedAfter	Ludvig Sylow ⓘ
relatedTo	Cauchy theorem for finite groups ⓘ linked to: Cauchy's theorem in group theory Lagrange theorem ⓘ linked to: Lagrange's theorem in group theory
standardReferenceIn	graduate algebra textbooks ⓘ undergraduate abstract algebra courses ⓘ
states	any two Sylow p-subgroups of a finite group G are conjugate in G ⓘ for a finite group G of order n and a prime p dividing n, G has a subgroup of order p^k where p^k is the highest power of p dividing n ⓘ the number of Sylow p-subgroups of a finite group G is congruent to 1 modulo p and divides the order of G ⓘ
topicOf	many research and expository papers in group theory ⓘ
usedFor	analyzing subgroup structure of finite groups ⓘ classification of finite groups ⓘ proving existence of normal subgroups ⓘ proving simplicity or non-simplicity of finite groups ⓘ
yearProposed	1872 ⓘ

How these facts were elicited

Referenced by (9)

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

Lagrange's theorem in group theory → isGeneralizedBy → Sylow theorems ⓘ

Theory of Groups → topic → Sylow theorems ⓘ

Cauchy's theorem in group theory → generalizedBy → Sylow theorems ⓘ

Sylow theorems → concerns → Sylow p-subgroups ⓘ

linked to: Sylow theorems

Sylow theorems → hasPart → Sylow first theorem ⓘ

linked to: Sylow theorems

Sylow theorems → hasPart → Sylow second theorem ⓘ

linked to: Sylow theorems

Sylow theorems → hasPart → Sylow third theorem ⓘ

linked to: Sylow theorems

orbit-stabilizer theorem → relatedTo → Sylow theorems ⓘ

A Survey of Modern Algebra → hasSubject → Sylow theorems ⓘ

All labels observed (5)

How this entity was disambiguated Show

Statements (39)

How these facts were elicited Show

Referenced by (9)

How this entity was disambiguated

How these facts were elicited