Odd-dimensional orthogonal finite simple groups
Use odd q and n ≥ 3 to avoid the symplectic and low-rank repetitions.
The projective Ω group of a nondegenerate quadratic form in odd dimension over Fq.
Definition sourcesParameter domain
Use odd q and n ≥ 3 to avoid the symplectic and low-rank repetitions.
| Parameter | Allowed values |
|---|---|
| n | integer n ≥ 3 |
| q | odd prime power |
Reference properties
Exceptions and identifications
dimension 3: PΩ(3,q) ≅ PSL(2,q). [sources]
dimension 5: PΩ(5,q) ≅ PSp(4,q). [sources]
even q: The odd-dimensional orthogonal groups identify with the corresponding symplectic groups. [sources]
Conventions and use
Ω denotes the subgroup required for simplicity; merely taking a projective special orthogonal group does not in general give the simple group. [sources]
Names and aliases
Bₙ(q) · Ω₂ₙ₊₁(q)
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 projective Ω group of a nondegenerate quadratic form in odd dimension over Fq.
- Classical groups: detailed lecture notesSection 3.4, orthogonal groups in odd characteristicOpen source
family:finite-odd-orthogonal#definitionParameter domain
n: integer n ≥ 3; q: odd prime power
- Classical groups: detailed lecture notesSection 3.4, orthogonal groups in odd characteristicOpen source
family:finite-odd-orthogonal#parametersScope and restrictions
Use odd q and n ≥ 3 to avoid the symplectic and low-rank repetitions.
- Classical groups: detailed lecture notesSection 3.4, orthogonal groups in odd characteristicOpen source
family:finite-odd-orthogonal#restrictionExceptions and identifications
dimension 3: PΩ(3,q) ≅ PSL(2,q).
- GAP Reference Manual: GroupsIsomorphismTypeInfoFiniteSimpleGroup: B-type conventionsOpen source
family:finite-odd-orthogonal#exceptions-1Exceptions and identifications
dimension 5: PΩ(5,q) ≅ PSp(4,q).
- Finite simple groups: Lecture 3 — Classical groupsSmall-dimensional orthogonal groupsOpen source
family:finite-odd-orthogonal#exceptions-2Exceptions and identifications
even q: The odd-dimensional orthogonal groups identify with the corresponding symplectic groups.
- GAP ClassicalMaximals: Classical Groups and Aschbacher’s TheoremSection 2.2-4, first paragraphOpen source
family:finite-odd-orthogonal#exceptions-3Conventions and use
Ω denotes the subgroup required for simplicity; merely taking a projective special orthogonal group does not in general give the simple group.
- Classical groups: detailed lecture notesSection 3.4, orthogonal groups in odd characteristicOpen source
family:finite-odd-orthogonal#conventions-1