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.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
group-theoretic construction
ⓘ
series of subgroups ⓘ |
| alsoKnownAs |
Jennings series
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| refinementOf | lower exponent-p central series ⓘ |
| refines | lower central series ⓘ |
| relatedConcept |
Jennings–Lazard correspondence
NERFINISHED
ⓘ
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.