Finite groups · Family

$\mathrm P\Omega^{-}(2n,q)$ Minus-type even orthogonal finite simple groups

n ≥ 4; minus type has Witt index n−1.

Classification coverageFinite simple classification

Definition

The projective Ω group of a nonsplit nondegenerate quadratic form on Fq^(2n).

Parameter domain

n ≥ 4; minus type has Witt index n−1.

Parameter	Allowed values
n	integer n ≥ 4
q	q = p^f, with p prime and f ≥ 1

Classification sources and conventions

Reference properties

Property	Value	Evidence
Parameter domain	n: integer n ≥ 4; q: q = p^f, with p prime and f ≥ 1	Sources
Scope and restrictions	n ≥ 4; minus type has Witt index n−1.	Sources

Exceptions and identifications

dimension 4: PΩ⁻(4,q) ≅ PSL(2,q²); use the linear family. [sources]

dimension 6: PΩ⁻(6,q) ≅ PSU(4,q); use the unitary family. [sources]

Names and aliases

²Dₙ(q) · O₂ₙ⁻(q)

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Claims and sources

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Definition

The projective Ω group of a nonsplit nondegenerate quadratic form on Fq^(2n).

Classical groups: detailed lecture notesSections 3.4–3.5, minus type and projective Ω groupsOpen source

family:finite-even-orthogonal-minus#definition

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Parameter domain

n: integer n ≥ 4; q: q = p^f, with p prime and f ≥ 1

Classical groups: detailed lecture notesSections 3.4–3.5, minus type and projective Ω groupsOpen source

family:finite-even-orthogonal-minus#parameters

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Scope and restrictions

n ≥ 4; minus type has Witt index n−1.

Classical groups: detailed lecture notesSections 3.4–3.5, minus type and projective Ω groupsOpen source

family:finite-even-orthogonal-minus#restriction

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Exceptions and identifications

dimension 4: PΩ⁻(4,q) ≅ PSL(2,q²); use the linear family.

Finite simple groups: Lecture 3 — Classical groupsSmall-dimensional orthogonal groupsOpen source

family:finite-even-orthogonal-minus#exceptions-1

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Exceptions and identifications

dimension 6: PΩ⁻(6,q) ≅ PSU(4,q); use the unitary family.

Finite simple groups: Lecture 3 — Classical groupsSmall-dimensional orthogonal groupsOpen source

family:finite-even-orthogonal-minus#exceptions-2

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PΩ−(2n,q)\mathrm P\Omega^{-}(2n,q)PΩ−(2n,q)Minus-type even orthogonal finite simple groups

Parameter domain

Reference properties

Exceptions and identifications

Names and aliases

Claims and sources

$\mathrm P\Omega^{-}(2n,q)$ Minus-type even orthogonal finite simple groups