Cauchy's theorem in group theory

E620659

theorem in group theory

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.

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All labels observed (2)

Label	Occurrences
Cauchy's theorem in group theory canonical	1
Cauchy’s theorem in group theory	1

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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
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 ⓘ

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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 (2)

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

Lagrange's theorem in group theory → isGeneralizedBy → Cauchy's theorem in group theory ⓘ

Augustin-Louis → notableFor → Cauchy's theorem in group theory ⓘ

subject surface form: Augustin-Louis Cauchy

this entity surface form: Cauchy’s theorem in group theory