Projective symplectic finite simple groups
Exclude (n,q) = (2,2).
The isometries of a nondegenerate alternating form on Fq^(2n), modulo scalar matrices.
Definition sourcesParameter domain
Exclude (n,q) = (2,2).
| Parameter | Allowed values |
|---|---|
| n | integer n ≥ 2 |
| q | q = p^f, with p prime and f ≥ 1 |
Reference properties
Exceptions and identifications
n = 1: PSp(2,q) ≅ PSL(2,q); use the linear family. [sources]
(2,2): Sp(4,2) ≅ S₆; its derived subgroup is A₆. [sources]
(2,3): PSp(4,3) ≅ PSU(4,2). [sources]
Names and aliases
Cₙ(q) · S₂ₙ(q)
Related objects and maps
- A₆ as the derived small case
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
The isometries of a nondegenerate alternating form on Fq^(2n), modulo scalar matrices.
- Classical groups: detailed lecture notesSections 3.2.1–3.2.2Open source
family:finite-symplectic#definitionParameter domain
n: integer n ≥ 2; q: q = p^f, with p prime and f ≥ 1
- Classical groups: detailed lecture notesSections 3.2.1–3.2.2Open source
family:finite-symplectic#parametersScope and restrictions
Exclude (n,q) = (2,2).
- Classical groups: detailed lecture notesSections 3.2.1–3.2.2Open source
family:finite-symplectic#restrictionOrder
q^(n²) · ∏ᵢ₌₁ⁿ(q^(2i)−1) / gcd(2,q−1)
- Classical groups: detailed lecture notesSection 3.2.1, order and scalar quotientOpen source
family:finite-symplectic#order-formulaExceptions and identifications
n = 1: PSp(2,q) ≅ PSL(2,q); use the linear family.
- Classical groups: detailed lecture notesSections 3.2.1–3.2.2Open source
family:finite-symplectic#exceptions-1Exceptions and identifications
(2,2): Sp(4,2) ≅ S₆; its derived subgroup is A₆.
- Classical groups: detailed lecture notesSection 3.2.2Open source
- 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-symplectic#exceptions-2Exceptions and identifications
(2,3): PSp(4,3) ≅ PSU(4,2).
- 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-symplectic#exceptions-3A₆ as the derived small case
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-symplectic#relation-1