PSL(2,ℤ/Nℤ)
E656691
PSL(2,ℤ/Nℤ) is the projective special linear group of 2×2 matrices with entries in the ring of integers modulo N, modulo scalar matrices, forming a fundamental example of a finite (or, for composite N, generally non-simple) group in algebra and number theory.
All labels observed (2)
| Label | Occurrences |
|---|---|
| PSL(2,ℤ/Nℤ) canonical | 1 |
| SL(2,ℤ/Nℤ)/Z(SL(2,ℤ/Nℤ)) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338616 — 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: PSL(2,ℤ/Nℤ) Context triple: [PSL(2,ℤ), hasQuotient, PSL(2,ℤ/Nℤ)]
-
A.
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.
-
B.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
-
C.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
D.
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.
-
E.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
- 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: PSL(2,ℤ/Nℤ) Target entity description: PSL(2,ℤ/Nℤ) is the projective special linear group of 2×2 matrices with entries in the ring of integers modulo N, modulo scalar matrices, forming a fundamental example of a finite (or, for composite N, generally non-simple) group in algebra and number theory.
-
A.
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.
-
B.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
-
C.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
D.
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.
-
E.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
finite group
ⓘ
group ⓘ matrix group ⓘ projective linear group ⓘ quotient group ⓘ |
| actsOn |
P¹(ℤ/Nℤ)
ⓘ
projective line over ℤ/Nℤ ⓘ |
| appearsAs | Galois group of some finite extensions in inverse Galois theory ⓘ |
| definedOver | ℤ/Nℤ ⓘ |
| hasActionType | fractional linear transformations on P¹(ℤ/Nℤ) ⓘ |
| hasCongruenceLevel | N ⓘ |
| hasDefinition |
SL(2,ℤ/Nℤ)/Z(SL(2,ℤ/Nℤ))
ⓘ
linked to:
PSL(2,ℤ/Nℤ)
|
| hasDimension | 3 (as an algebraic group over a field) ⓘ |
| hasElements | equivalence classes of 2×2 matrices of determinant 1 over ℤ/Nℤ modulo scalar matrices ⓘ |
| hasExponent | finite ⓘ |
| hasGenerators | images of standard generators of PSL(2,ℤ) ⓘ |
| hasKernel | principal congruence subgroup of level N in PSL(2,ℤ) ⓘ |
| hasNaturalHomomorphismFrom |
PSL(2,ℤ)
ⓘ
linked to:
modular group PSL(2,Z)
|
| hasOrder | finite for every positive integer N ⓘ |
| hasOrderFormula | |PSL(2,ℤ/pℤ)| = p(p²−1)/gcd(2,p−1) for prime p ⓘ |
| hasPresentation | quotient of ⟨S,T | S² = (ST)³ = 1⟩ by congruence relations modulo N ⓘ |
| hasTrivialAbelianizationFor | N = p prime, p ≥ 5 ⓘ |
| isCenterTrivialFor | N = p prime, p ≥ 3 ⓘ |
| isFiniteSimpleGroupFor | N = p prime, p ≥ 5 ⓘ |
| isImageOf |
PSL(2,ℤ) under reduction modulo N
ⓘ
SL(2,ℤ) modulo principal congruence subgroup of level N ⓘ |
| isIsomorphicTo | A₅ when N = 5 ⓘ |
| isNonAbelian | true for N ≥ 3 ⓘ |
| isNotSimpleFor |
N = 2
ⓘ
N = 3 ⓘ composite N in general ⓘ |
| isOfLieType | A₁ over ℤ/Nℤ (for N prime power, via corresponding field) ⓘ |
| isPerfectFor | N = p prime, p ≥ 5 ⓘ |
| isQuotientOf |
PSL(2,ℤ/Mℤ) when M is a multiple of N via reduction
ⓘ
SL(2,ℤ/Nℤ) ⓘ |
| isResiduallyFiniteImageOf |
PSL(2,ℤ)
ⓘ
linked to:
modular group PSL(2,Z)
|
| isSimpleFor | N = p prime, p ≥ 5 ⓘ |
| isUsedIn | construction of Ramanujan graphs (for suitable N) ⓘ |
| isUsedToDefine | congruence quotients of the modular group ⓘ |
| quotientedBy |
center of SL(2,ℤ/Nℤ)
ⓘ
{±I} when 2 is invertible in ℤ/Nℤ ⓘ |
| relatedTo |
congruence subgroups of SL(2,ℤ)
ⓘ
modular group PSL(2,ℤ) ⓘ
linked to:
modular group PSL(2,Z)
|
| usedIn |
Galois representations modulo N
ⓘ
arithmetic geometry ⓘ automorphic forms ⓘ classification of finite simple groups of Lie type A₁ ⓘ finite group theory ⓘ modular forms theory ⓘ |
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: PSL(2,ℤ/Nℤ) Description of subject: PSL(2,ℤ/Nℤ) is the projective special linear group of 2×2 matrices with entries in the ring of integers modulo N, modulo scalar matrices, forming a fundamental example of a finite (or, for composite N, generally non-simple) group in algebra and number theory.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
modular group PSL(2,Z)
linked to: PSL(2,ℤ/Nℤ)