Fitting subgroup
E283603
The Fitting subgroup is a characteristic subgroup of a finite group formed by the product of all its nilpotent normal subgroups, playing a central role in the structure theory of finite groups.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Fitting subgroup canonical | 7 |
How this entity was disambiguated
This entity first appeared as the object of triple T2636369 — 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: Fitting subgroup Context triple: [Hans Fitting, notableConcept, Fitting subgroup]
-
A.
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.
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B.
Lie subgroup
A Lie subgroup is a subgroup of a Lie group that is itself a Lie group and an embedded submanifold, inheriting compatible smooth and group structures from the ambient Lie group.
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C.
Lie ring
A Lie ring is an algebraic structure consisting of an abelian group equipped with a bilinear, alternating, and Jacobi-identity-satisfying bracket operation, serving as the ring-theoretic analogue of a Lie algebra.
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D.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
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E.
Conway groups
Conway groups are a set of three closely related sporadic simple groups discovered by John H. Conway in the study of symmetries of the Leech lattice in group theory.
- 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: Fitting subgroup Target entity description: The Fitting subgroup is a characteristic subgroup of a finite group formed by the product of all its nilpotent normal subgroups, playing a central role in the structure theory of finite groups.
-
A.
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.
-
B.
Lie subgroup
A Lie subgroup is a subgroup of a Lie group that is itself a Lie group and an embedded submanifold, inheriting compatible smooth and group structures from the ambient Lie group.
-
C.
Lie ring
A Lie ring is an algebraic structure consisting of an abelian group equipped with a bilinear, alternating, and Jacobi-identity-satisfying bracket operation, serving as the ring-theoretic analogue of a Lie algebra.
-
D.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
E.
Conway groups
Conway groups are a set of three closely related sporadic simple groups discovered by John H. Conway in the study of symmetries of the Leech lattice in group theory.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
characteristic subgroup
ⓘ
group theory concept ⓘ subgroup invariant ⓘ |
| alsoKnownAs | Fitting radical ⓘ |
| centralizes | itself ⓘ |
| centralSeriesRelation | F(G) lies in the hypercenter of G in finite solvable case ⓘ |
| contains | every nilpotent normal subgroup of G ⓘ |
| decomposition | in finite solvable groups, F(G) is direct product of its Sylow subgroups ⓘ |
| definedFor |
arbitrary group
ⓘ
finite group ⓘ |
| definition | product of all nilpotent normal subgroups of a group ⓘ |
| equals |
G when G is nilpotent
ⓘ
its own centralizer in a finite solvable group ⓘ |
| field | group theory ⓘ |
| generalization | generalized Fitting subgroup extends F(G) by including components ⓘ |
| hasAbbreviation | Fitting subgp. ⓘ |
| introducedIn | 20th century ⓘ |
| isCharacteristicIn | G ⓘ |
| isCharacteristicInEveryNormalSubgroup | false ⓘ |
| isCharacteristicSubgroup | true ⓘ |
| isContainedIn |
centralizer of F(G) in G
ⓘ
every normal nilpotent subgroup of G only if equal ⓘ solvable radical of G ⓘ |
| isFullyInvariantSubgroup | true ⓘ |
| isIntersectionOf | all normal subgroups N of G such that G/N is semisimple ⓘ |
| isLargestWithProperty | nilpotent normal subgroup of G ⓘ |
| isNilpotent | true ⓘ |
| isNormalIn | G ⓘ |
| isProductOf | all nilpotent normal subgroups of G ⓘ |
| isSelfCentralizingIn | finite solvable group ⓘ |
| isSubgroupOf | G ⓘ |
| isTrivialWhen | G has no nontrivial nilpotent normal subgroups ⓘ |
| namedAfter | Hans Fitting ⓘ |
| property |
image of F(G) under a surjective homomorphism is contained in F(image)
ⓘ
in finite solvable groups, F(G) contains the product of all Sylow subgroups that are normal in G ⓘ in finite solvable groups, F(G) is nontrivial ⓘ intersection of Fitting subgroups of normal subgroups need not equal F(G) ⓘ preimage of F(quotient) contains F(G) ⓘ |
| relatedConcept |
Frattini subgroup
ⓘ
generalized Fitting subgroup ⓘ layer of a group ⓘ solvable radical ⓘ |
| role |
base for the generalized Fitting subgroup F*(G)
ⓘ
controls much of the structure of finite solvable groups ⓘ |
| symbol | F(G) ⓘ |
| usedIn |
classification of finite simple groups
ⓘ
local analysis of finite groups ⓘ theory of solvable groups ⓘ |
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: Fitting subgroup Description of subject: The Fitting subgroup is a characteristic subgroup of a finite group formed by the product of all its nilpotent normal subgroups, playing a central role in the structure theory of finite groups.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.