Hurwitz group
E904003
A Hurwitz group is a finite group that attains the maximal possible order of the automorphism group of a compact Riemann surface of given genus, as specified by Hurwitz's bound.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hurwitz group canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11085921 — 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: Hurwitz group Context triple: [Hurwitz bound on automorphism groups of curves, relatedConcept, Hurwitz group]
-
A.
Fuchsian group
A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
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B.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
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C.
Harada–Norton group
The Harada–Norton group is one of the 26 sporadic simple groups in finite group theory, notable for its large order and close relationship to the Monster group.
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D.
Hurwitz theorem
Hurwitz theorem is a fundamental result in Diophantine approximation that gives an optimal bound on how well any irrational real number can be approximated by infinitely many rational numbers.
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E.
modular group PSL(2,Z)
The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
- 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: Hurwitz group Target entity description: A Hurwitz group is a finite group that attains the maximal possible order of the automorphism group of a compact Riemann surface of given genus, as specified by Hurwitz's bound.
-
A.
Fuchsian group
A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
-
B.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
C.
Harada–Norton group
The Harada–Norton group is one of the 26 sporadic simple groups in finite group theory, notable for its large order and close relationship to the Monster group.
-
D.
Hurwitz theorem
Hurwitz theorem is a fundamental result in Diophantine approximation that gives an optimal bound on how well any irrational real number can be approximated by infinitely many rational numbers.
-
E.
modular group PSL(2,Z)
The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
finite group
ⓘ
mathematical concept ⓘ |
| actsOn | Hurwitz surface NERFINISHED ⓘ |
| appearsIn | classification problems of finite simple groups with given generating triples ⓘ |
| characterizedBy | attaining maximal possible order of automorphism group of a compact Riemann surface of given genus ⓘ |
| context |
Teichmüller theory and moduli of curves
NERFINISHED
ⓘ
automorphism groups of algebraic curves over the complex numbers ⓘ |
| definedBy | Hurwitz bound NERFINISHED ⓘ |
| field |
Riemann surface theory
ⓘ
algebraic geometry ⓘ geometric group theory ⓘ group theory ⓘ |
| hasExample |
PSL(2,13)
NERFINISHED
ⓘ
PSL(2,7) NERFINISHED ⓘ PSL(2,8) NERFINISHED ⓘ PSL(2,q) for infinitely many prime powers q ⓘ alternating group A7 NERFINISHED ⓘ some alternating groups An for suitable n ⓘ some sporadic simple groups ⓘ |
| hasMaximalityProperty | maximal order of automorphism group among all compact Riemann surfaces of given genus GENERATED ⓘ |
| hasProperty |
acts as full group of conformal automorphisms of some compact Riemann surface
ⓘ
admits a generating triple of orders 2, 3, and 7 with product 1 ⓘ every Hurwitz group is a quotient of the (2,3,7) triangle group with torsion-free kernel ⓘ infinitely many pairwise non-isomorphic Hurwitz groups exist ⓘ is a quotient of the (2,3,7) triangle group ⓘ is generated by elements of orders 2 and 3 whose product has order 7 ⓘ many Hurwitz groups are non-abelian simple groups ⓘ maximizes symmetry among surfaces of fixed genus ⓘ order is determined by the genus of the associated Hurwitz surface ⓘ realizable as a group of orientation-preserving isometries of a hyperbolic surface ⓘ |
| lowerBoundOnGenus | 2 ⓘ |
| namedAfter | Adolf Hurwitz NERFINISHED ⓘ |
| relatedTo |
(2,3,7) triangle group
NERFINISHED
ⓘ
Fuchsian group of signature (2,3,7) ⓘ Galois coverings of the Riemann sphere branched over three points ⓘ Hurwitz curve NERFINISHED ⓘ Hurwitz surface NERFINISHED ⓘ automorphism group of a Riemann surface ⓘ compact Riemann surface ⓘ dessins d’enfants NERFINISHED ⓘ finite simple group ⓘ hyperbolic geometry ⓘ triangle group ⓘ |
| satisfies | |G| = 84(g-1) for some compact Riemann surface of genus g ≥ 2 ⓘ |
| usedIn |
construction of highly symmetric Riemann surfaces
ⓘ
study of extremal automorphism groups of curves ⓘ |
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: Hurwitz group Description of subject: A Hurwitz group is a finite group that attains the maximal possible order of the automorphism group of a compact Riemann surface of given genus, as specified by Hurwitz's bound.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.