Fitting lemma
E283604
The Fitting lemma is a result in group theory and module theory that characterizes how certain algebraic structures decompose into direct sums of invariant subcomponents, often involving nilpotent and invertible parts.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Fitting lemma canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2636370 — 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 lemma Context triple: [Hans Fitting, notableConcept, Fitting lemma]
-
A.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
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B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
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C.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
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D.
Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
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E.
Schwarz lemma
Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
- 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 lemma Target entity description: The Fitting lemma is a result in group theory and module theory that characterizes how certain algebraic structures decompose into direct sums of invariant subcomponents, often involving nilpotent and invertible parts.
-
A.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
-
D.
Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
-
E.
Schwarz lemma
Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in group theory ⓘ result in module theory ⓘ |
| appliesTo |
finite groups
ⓘ
linear operators on finite-dimensional vector spaces ⓘ modules over a ring ⓘ |
| category | theorems in abstract algebra ⓘ |
| context |
linear algebra
ⓘ
structure theory of finite groups ⓘ structure theory of modules ⓘ |
| describes |
decomposition into invariant submodules
ⓘ
decomposition into nilpotent and invertible parts ⓘ direct sum decomposition ⓘ |
| field |
algebra
ⓘ
group theory ⓘ module theory ⓘ representation theory ⓘ |
| generalizes | decomposition of a linear operator into nilpotent and invertible components on invariant subspaces ⓘ |
| hasConsequence |
classification of endomorphisms up to similarity in finite dimension
ⓘ
decomposition of a module into torsion and torsion-free parts in certain settings ⓘ |
| holdsOver |
Artinian modules
ⓘ
Noetherian modules under suitable hypotheses ⓘ |
| implies | existence of a largest nilpotent normal subgroup in a finite group via the Fitting subgroup ⓘ |
| involvesConcept |
Fitting decomposition
ⓘ
Fitting subgroup ⓘ direct sum ⓘ invariant submodule ⓘ invertible endomorphism ⓘ nilpotent endomorphism ⓘ nilpotent group ⓘ primary decomposition ⓘ |
| namedAfter | Hans Fitting ⓘ |
| relatedTo |
Jordan–Chevalley decomposition
ⓘ
primary decomposition theorem ⓘ rational canonical form ⓘ |
| requires | finite length condition on the module or finite dimension on the vector space ⓘ |
| states |
for a linear operator on a finite-dimensional vector space, the space decomposes into a direct sum of the generalized eigenspaces corresponding to the nilpotent and invertible parts
ⓘ
for an endomorphism of a finite-length module, the module decomposes as a direct sum of the kernel of a power and the image of a power ⓘ |
| usedFor |
analyzing structure of modules via endomorphisms
ⓘ
decomposing representations of finite groups ⓘ defining the Fitting subgroup of a finite group ⓘ |
| usedIn |
module decomposition theorems
ⓘ
proofs in finite group theory ⓘ representation theory of finite 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 lemma Description of subject: The Fitting lemma is a result in group theory and module theory that characterizes how certain algebraic structures decompose into direct sums of invariant subcomponents, often involving nilpotent and invertible parts.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.