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 |
How this entity was disambiguated
This entity first appeared as the object of triple T9867907 — 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 conjecture Context triple: [Hans Zassenhaus, notableWork, Zassenhaus conjecture]
-
A.
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.
-
B.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
C.
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.
-
D.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
E.
Conway–Norton collaboration
The Conway–Norton collaboration was a joint mathematical effort, led by John Conway and Simon Norton, that played a key role in developing the theory of monstrous moonshine and the construction of the Monster group.
- 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 conjecture Target entity description: 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.
-
A.
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.
-
B.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
C.
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.
-
D.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
E.
Conway–Norton collaboration
The Conway–Norton collaboration was a joint mathematical effort, led by John Conway and Simon Norton, that played a key role in developing the theory of monstrous moonshine and the construction of the Monster group.
- F. None of above. chosen
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 ⓘ |
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: Zassenhaus conjecture Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.