Zassenhaus filtration
E827065
Zassenhaus filtration is a descending series of subgroups in group theory that refines the lower central series to study the structure and properties of groups, especially in the context of pro‑p and nilpotent groups.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Zassenhaus filtration canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9867905 — 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: Zassenhaus filtration Context triple: [Hans Zassenhaus, notableWork, Zassenhaus filtration]
-
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.
Hodge filtration
The Hodge filtration is a decreasing sequence of complex subspaces on the cohomology of a complex algebraic variety that encodes its Hodge decomposition and mixed Hodge structure.
-
C.
Schreier refinement theorem
The Schreier refinement theorem is a result in group theory stating that any two subnormal series of a group admit equivalent refinements, serving as a precursor and companion to the Jordan–Hölder theorem.
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D.
Krull–Gabriel dimension
Krull–Gabriel dimension is a refinement of Krull dimension used in the representation theory of rings and abelian categories to measure the complexity of their subobject lattices and module categories.
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E.
The Poincaré-Birkhoff-Witt theorem in ring theory
"The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
- 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: Zassenhaus filtration Target entity description: Zassenhaus filtration is a descending series of subgroups in group theory that refines the lower central series to study the structure and properties of groups, especially in the context of pro‑p and nilpotent 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.
Hodge filtration
The Hodge filtration is a decreasing sequence of complex subspaces on the cohomology of a complex algebraic variety that encodes its Hodge decomposition and mixed Hodge structure.
-
C.
Schreier refinement theorem
The Schreier refinement theorem is a result in group theory stating that any two subnormal series of a group admit equivalent refinements, serving as a precursor and companion to the Jordan–Hölder theorem.
-
D.
Krull–Gabriel dimension
Krull–Gabriel dimension is a refinement of Krull dimension used in the representation theory of rings and abelian categories to measure the complexity of their subobject lattices and module categories.
-
E.
The Poincaré-Birkhoff-Witt theorem in ring theory
"The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
group-theoretic construction
ⓘ
series of subgroups ⓘ |
| alsoKnownAs |
Jennings series
ⓘ
dimension series ⓘ |
| appearsIn | p-adic Lie correspondence for uniform pro-p groups ⓘ |
| appliesTo |
abstract groups
ⓘ
pro-p groups ⓘ profinite groups ⓘ |
| associatedWith | graded Lie algebra of a group ⓘ |
| captures | p-local structure of a group ⓘ |
| compatibleWith | group homomorphisms ⓘ |
| context | local analysis of finite and infinite groups ⓘ |
| definedFor | a group G ⓘ |
| field | group theory ⓘ |
| firstTerm | D_1(G) = G ⓘ |
| generalizes | dimension subgroups of finite p-groups ⓘ |
| hasProperty |
each D_n(G) is a characteristic subgroup of G
ⓘ
intersection over n of D_n(G) can be trivial in residually nilpotent groups ⓘ |
| hasTerm | D_n(G) ⓘ |
| isA | descending series of subgroups ⓘ |
| isDescending | true ⓘ |
| namedAfter | Hans Zassenhaus ⓘ |
| refinementOf | lower exponent-p central series ⓘ |
| refines | lower central series ⓘ |
| relatedConcept |
Jennings–Lazard correspondence
ⓘ
dimension subgroup problem ⓘ p-series of a group ⓘ powerful pro-p group ⓘ |
| relatedTo |
group algebra over a field of characteristic p
ⓘ
lower central series γ_n(G) ⓘ p-power series in a group ⓘ |
| satisfies | D_{n+1}(G) ≤ D_n(G) ⓘ |
| studiedIn | pro-p group cohomology ⓘ |
| termIndex | n ≥ 1 ⓘ |
| toolFor |
analyzing generators and relations in pro-p groups
ⓘ
constructing associated graded Lie rings of groups ⓘ |
| usedIn |
Lie methods in group theory
ⓘ
p-adic analytic group theory ⓘ theory of powerful p-groups ⓘ |
| usedToDefine | dimension subgroups ⓘ |
| usedToStudy |
growth of subgroups in pro-p groups
ⓘ
nilpotency class of groups ⓘ nilpotent groups ⓘ pro-p groups ⓘ residual properties of groups ⓘ structure of groups ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Zassenhaus filtration Description of subject: Zassenhaus filtration is a descending series of subgroups in group theory that refines the lower central series to study the structure and properties of groups, especially in the context of pro‑p and nilpotent groups.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.