Cauchy's theorem in group theory
E620659
Cauchy's theorem in group theory is a fundamental result stating that if a finite group’s order is divisible by a prime p, then the group contains an element (and hence a subgroup) of order p.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Cauchy theorem for finite groups | 1 |
| Cauchy's theorem | 1 |
| Cauchy's theorem in group theory canonical | 1 |
| Cauchy’s theorem in group theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6800955 — 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: Cauchy's theorem in group theory Context triple: [Lagrange's theorem in group theory, isGeneralizedBy, Cauchy's theorem in group theory]
-
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.
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B.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
-
C.
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.
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D.
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.
-
E.
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.
- 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: Cauchy's theorem in group theory Target entity description: Cauchy's theorem in group theory is a fundamental result stating that if a finite group’s order is divisible by a prime p, then the group contains an element (and hence a subgroup) of order p.
-
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.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (35)
| Predicate | Object |
|---|---|
| instanceOf | theorem in group theory ⓘ |
| appliesTo | finite groups ⓘ |
| assumes |
group is finite
ⓘ
p is a prime number ⓘ |
| concerns |
orders of elements in finite groups
ⓘ
prime divisors of the order of a group ⓘ |
| conclusion |
there exists a subgroup of order p in the group
ⓘ
there exists an element of order p in the group ⓘ |
| doesNotRequire | group to be abelian ⓘ |
| domain | group theory ⓘ |
| field |
abstract algebra
ⓘ
group theory ⓘ |
| generalizedBy | Sylow theorems ⓘ |
| hasProofTechnique |
counting arguments in finite groups
ⓘ
group action methods ⓘ orbit-stabilizer arguments ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| holdsFor |
finite abelian groups
ⓘ
finite non-abelian groups ⓘ |
| implies |
every finite abelian group has an element of order p for each prime p dividing its order
ⓘ
existence of a subgroup of order p in a finite group whose order is divisible by p ⓘ existence of an element of order p in a finite group whose order is divisible by p ⓘ |
| importance | fundamental result in finite group theory ⓘ |
| isSpecialCaseOf | results about existence of subgroups of given order ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| namedForContribution | Augustin-Louis Cauchy's work on permutation groups and finite groups ⓘ |
| relatedConcept |
order of a group
ⓘ
order of an element ⓘ subgroup of prime order ⓘ |
| relatedTo | Lagrange's theorem in group theory ⓘ |
| statement | If a finite group G has order divisible by a prime p, then G contains an element of order p. ⓘ |
| usedAs | tool to show existence of elements of specific prime order ⓘ |
| usedIn |
basic structure theory of finite groups
ⓘ
classification of finite groups ⓘ proofs of Sylow theorems ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cauchy's theorem in group theory Description of subject: Cauchy's theorem in group theory is a fundamental result stating that if a finite group’s order is divisible by a prime p, then the group contains an element (and hence a subgroup) of order p.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Augustin-Louis
linked to: Cauchy's theorem in group theory
linked to: Cauchy's theorem in group theory
linked to: Cauchy's theorem in group theory