Zassenhaus conjecture
E827066
The Zassenhaus conjecture is a prominent open problem in group theory concerning the structure of units in integral group rings and their relation to the underlying finite group.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Zassenhaus conjecture canonical | 1 |
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
open problem in group theory ⓘ |
| alsoKnownAs |
Zassenhaus conjecture on torsion units
NERFINISHED
ⓘ
Zassenhaus unit conjecture NERFINISHED ⓘ |
| appliesTo | integral group ring ZG of a finite group G ⓘ |
| assumes |
G is a finite group
ⓘ
u is a torsion unit in the normalized unit group of ZG ⓘ |
| claims | every torsion unit in the normalized unit group of ZG is rationally conjugate to an element of G ⓘ |
| concerns |
relation between torsion units and elements of the underlying finite group
ⓘ
structure of units in integral group rings ⓘ |
| difficulty | considered difficult and technically demanding ⓘ |
| field |
group theory
ⓘ
representation theory ⓘ ring theory ⓘ |
| hasVariant |
first Zassenhaus conjecture
NERFINISHED
ⓘ
second Zassenhaus conjecture NERFINISHED ⓘ third Zassenhaus conjecture NERFINISHED ⓘ |
| holdsFor |
finite abelian groups
ⓘ
finite cyclic groups ⓘ finite nilpotent groups ⓘ many classes of solvable groups ⓘ |
| implies | strong restrictions on partial augmentations of torsion units ⓘ |
| influenced | development of methods for studying torsion units in group rings ⓘ |
| involvesConcept |
Wedderburn decomposition of group algebras
ⓘ
augmentation map ⓘ partial augmentation ⓘ rational conjugacy ⓘ |
| isAbout |
normalized unit group V(ZG)
ⓘ
torsion subgroup of V(ZG) ⓘ |
| mainSubject |
finite group
ⓘ
integral group ring ⓘ normalized unit ⓘ torsion unit ⓘ |
| motivatedBy | understanding the correspondence between group elements and units in ZG ⓘ |
| namedAfter | Hans Zassenhaus NERFINISHED ⓘ |
| proposedBy | Hans Zassenhaus NERFINISHED ⓘ |
| relatedTo |
Luthar–Passi method
NERFINISHED
ⓘ
Sehgal’s problem on torsion units NERFINISHED ⓘ group ring units ⓘ integral representation theory of finite groups ⓘ isomorphism problem for integral group rings ⓘ |
| status |
not proved in full generality
ⓘ
open ⓘ |
| studiedIn |
research on integral group rings
ⓘ
research on units of group rings ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.