Projective special linear finite simple groups
Exclude (n,q) = (2,2) and (2,3).
SL(n,Fq) modulo its scalar center; the untwisted finite group of Lie type Aₙ₋₁.
Definition sourcesParameter domain
Exclude (n,q) = (2,2) and (2,3).
| Parameter | Allowed values |
|---|---|
| n | integer n ≥ 2 |
| q | q = p^f, with p prime and f ≥ 1 |
Reference properties
Exceptions and identifications
(2,2), (2,3): PSL(2,2) ≅ S₃ and PSL(2,3) ≅ A₄ are not simple. [sources]
(2,4), (2,5): Both identify with A₅. [sources]
(2,7), (3,2): PSL(2,7) ≅ PSL(3,2). [sources]
(2,9): PSL(2,9) ≅ A₆. [sources]
(4,2): PSL(4,2) ≅ A₈. [sources]
Names and aliases
Lₙ(q) · Aₙ₋₁(q)
Related objects and maps
- Canonical A₅ record
- Canonical PSL(2,7) record
- Canonical A₆ record
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
SL(n,Fq) modulo its scalar center; the untwisted finite group of Lie type Aₙ₋₁.
- Finite simple groups: Lecture 2 — Linear groupsDefinitions; more examples; simplicity of PSLₙ(q)Open source
family:finite-linear#definitionParameter domain
n: integer n ≥ 2; q: q = p^f, with p prime and f ≥ 1
- Finite simple groups: Lecture 2 — Linear groupsDefinitions; more examples; simplicity of PSLₙ(q)Open source
family:finite-linear#parametersScope and restrictions
Exclude (n,q) = (2,2) and (2,3).
- Finite simple groups: Lecture 2 — Linear groupsDefinitions; more examples; simplicity of PSLₙ(q)Open source
family:finite-linear#restrictionOrder
q^[n(n−1)/2] · ∏ᵢ₌₂ⁿ(qⁱ−1) / gcd(n,q−1)
- Finite simple groups: Lecture 2 — Linear groupsOrders of linear groups and scalar quotientOpen source
family:finite-linear#order-formulaExceptions and identifications
(2,2), (2,3): PSL(2,2) ≅ S₃ and PSL(2,3) ≅ A₄ are not simple.
- Finite simple groups: Lecture 2 — Linear groupsDefinitions; more examples; simplicity of PSLₙ(q)Open source
family:finite-linear#exceptions-1Exceptions and identifications
(2,4), (2,5): Both identify with A₅.
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-linear#exceptions-2Exceptions and identifications
(2,7), (3,2): PSL(2,7) ≅ PSL(3,2).
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-linear#exceptions-3Exceptions and identifications
(2,9): PSL(2,9) ≅ A₆.
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-linear#exceptions-4Exceptions and identifications
(4,2): PSL(4,2) ≅ A₈.
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-linear#exceptions-5Canonical A₅ record
group:a-5
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-linear#relation-1Canonical PSL(2,7) record
group:psl-2-7
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-linear#relation-2Canonical A₆ record
group:a-6
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-linear#relation-3