Sylow theorems
E620660
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Sylow theorems canonical | 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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Sylow second theorem NERFINISHED ⓘ Sylow third theorem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| language | symbolic mathematics ⓘ |
| namedAfter | Ludvig Sylow NERFINISHED ⓘ |
| relatedTo |
Cauchy theorem for finite groups
NERFINISHED
ⓘ
Lagrange theorem NERFINISHED ⓘ |
| 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 ⓘ |
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Input
Subject: Sylow theorems Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.