Zassenhaus conjecture

E827066

mathematical conjecture open problem in group theory

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (7)

Label	Occurrences
Sehgal’s problem on torsion units	1
Zassenhaus conjecture canonical	1
Zassenhaus conjecture on torsion units	1
Zassenhaus unit conjecture	1
first Zassenhaus conjecture	1
second Zassenhaus conjecture	1
third Zassenhaus conjecture	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 ⓘ linked to: Zassenhaus conjecture Zassenhaus unit conjecture ⓘ linked to: Zassenhaus conjecture
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 ⓘ linked to: Zassenhaus conjecture second Zassenhaus conjecture ⓘ linked to: Zassenhaus conjecture third Zassenhaus conjecture ⓘ linked to: Zassenhaus conjecture
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 ⓘ
proposedBy	Hans Zassenhaus ⓘ
relatedTo	Luthar–Passi method ⓘ Sehgal’s problem on torsion units ⓘ linked to: Zassenhaus conjecture 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