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.
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)) NERFINISHED ⓘ |
| 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) NERFINISHED ⓘ |
| 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) NERFINISHED ⓘ |
| 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) NERFINISHED ⓘ |
| usedIn |
Galois representations modulo N
ⓘ
arithmetic geometry ⓘ automorphic forms ⓘ classification of finite simple groups of Lie type A₁ ⓘ finite group theory ⓘ modular forms theory ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
PSL(2,ℤ)