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 (1)

Label	Occurrences
Sylow theorems canonical	1

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 ⓘ

Referenced by (1)

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

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