Hurwitz group | GPTKB 2.0

All labels observed (1)

Label	Occurrences
Hurwitz group canonical	3

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.
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
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 ⓘ linked to: Hurwitz surfaces
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 ⓘ linked to: Teichmüller theory automorphism groups of algebraic curves over the complex numbers ⓘ
definedBy	Hurwitz bound ⓘ linked to: Hurwitz bound on automorphism groups of curves
field	Riemann surface theory ⓘ algebraic geometry ⓘ geometric group theory ⓘ group theory ⓘ
hasExample	PSL(2,13) ⓘ PSL(2,7) ⓘ PSL(2,8) ⓘ PSL(2,q) for infinitely many prime powers q ⓘ alternating group A7 ⓘ 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 ⓘ
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 ⓘ
relatedTo	(2,3,7) triangle group ⓘ Fuchsian group of signature (2,3,7) ⓘ Galois coverings of the Riemann sphere branched over three points ⓘ Hurwitz curve ⓘ linked to: Hurwitz surfaces Hurwitz surface ⓘ linked to: Hurwitz surfaces automorphism group of a Riemann surface ⓘ compact Riemann surface ⓘ dessins d’enfants ⓘ 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

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hurwitz bound on automorphism groups of curves → relatedConcept → Hurwitz group ⓘ

Hurwitz surface → hasAutomorphismGroup → Hurwitz group ⓘ

subject linked to: Hurwitz surfaces

Hurwitz surface → relatedTo → Hurwitz group ⓘ