Documentation

Permutation Representations of Linear Groups

Each of the functions in this family returns two values:

(a): A permutation group G corresponding to the action of a designated matrix group M on a vector space V; and
(b): An indexed set of affine or projective points on which M acts, such that the indexing gives the correspondence between this set and the G-set of M.

Furthermore, most of the function in this family are parameterized by two objects: the degree and the coefficient field of the matrix group. These can be supplied in one of the following three forms:

(i): Integers n and q corresponding to the degree and the field GF(q) of M (GF(q²) in the case of the unitary groups).
(ii): An integer n and a finite field K corresponding to the degree and the coefficient field of M.
(iii): A vector space V = Kⁿ on which M naturally acts.

The Suzuki group, however, is only parametrised by the field, as the degree is always four. As such, it can be described by the integer q, the field K = GF(q), or the vector space K⁴.

AffineGeneralLinearGroup(arguments)

AGL(arguments)

AffineGeneralLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

AffineGeneralLinearGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

AffineGeneralLinearGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

AGL(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

AGL(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

AGL(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the affine general linear group G = AGL(n, q), i.e., the group corresponding to the action of GL(n, q) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the affine points and the G-set of G.

AffineSpecialLinearGroup(arguments)

ASL(arguments)

AffineSpecialLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

AffineSpecialLinearGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

AffineSpecialLinearGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

ASL(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ASL(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ASL(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the affine special linear group G = ASL(n, q), i.e., the group corresponding to the action of SL(n, q) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the affine points and the G-set of G.

AffineGammaLinearGroup(arguments)

AGammaL(arguments)

AffineGammaLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

AffineGammaLinearGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

AffineGammaLinearGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

AGammaL(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

AGammaL(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

AGammaL(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the affine gamma linear group G = AGammaL(n, q), i.e., the group corresponding to the action of GammaL(n, q) (the automorphism group of GL(n, q)) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the points and the G-set of G.

AffineSigmaLinearGroup(arguments)

ASigmaL(arguments)

AffineSigmaLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

AffineSigmaLinearGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

AffineSigmaLinearGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

ASigmaL(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ASigmaL(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ASigmaL(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the affine sigma linear group G = ASigmaL(n, q), i.e., the group corresponding to the action of SigmaL(n, q) (the automorphism group of SL(n, q)) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the points and the G-set of G.

AffineSymplecticGroup(arguments)

ASp(arguments)

AffineSymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

AffineSymplecticGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

AffineSymplecticGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

ASp(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ASp(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ASp(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the affine symplectic linear group G, i.e., the group corresponding to the action of Sp(n, q) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the affine points and the G-set of G.

AffineSigmaSymplecticGroup(arguments)

ASigmaSp(arguments)

AffineSigmaSymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

AffineSigmaSymplecticGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

AffineSigmaSymplecticGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

ASigmaSp(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ASigmaSp(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ASigmaSp(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the affine sigma symplectic linear group G, i.e., the group corresponding to the action of Sp(n, q) on the affine points of the n-dimensional vector space V over K = GF(q), plus the action of a field automorphism. The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the affine points and the G-set of G.

ProjectiveGeneralLinearGroup(arguments)

PGL(arguments)

ProjectiveGeneralLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralLinearGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralLinearGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PGL(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PGL(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PGL(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective general linear group G = PGL(n, q), i.e., the group corresponding to the action of GL(n, q) on the projective points of the n-dimensional vector space V over K = GF(q), where n ≥2 and q is a prime power. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

ProjectiveSpecialLinearGroup(arguments)

PSL(arguments)

ProjectiveSpecialLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialLinearGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialLinearGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSL(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSL(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSL(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective special linear group G = PSL(n, q), i.e., the group corresponding to the action of SL(n, q) on the projective points of the n-dimensional vector space V over K = GF(q), where n ≥2 and q is a prime power. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

ProjectiveGammaLinearGroup(arguments)

PGammaL(arguments)

ProjectiveGammaLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGammaLinearGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGammaLinearGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PGammaL(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PGammaL(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PGammaL(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct an automorphism group G = PGammaL(n, q) of the projective general linear group B = PGL(n, q), by adding the field automorphisms of GF(q) to B. The permutation action corresponds to the natural action on 1-dimensional subspaces of the n-dimensional vector space V over the field K = GF(q), where n ≥2 and q is a prime power. The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the points and the G-set of G.

ProjectiveSigmaLinearGroup(arguments)

PSigmaL(arguments)

ProjectiveSigmaLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSigmaLinearGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSigmaLinearGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSigmaL(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSigmaL(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSigmaL(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct an automorphism group G = PSigmaL(n, q) of the projective special linear group B = PSL(n, q), by adding the field automorphisms of GF(q) to B. The permutation action corresponds to the natural action on 1-dimensional subspaces of the n-dimensional vector space V over the field K = GF(q), where n ≥2 and q is a prime power. The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the points and the G-set of G.

ProjectiveGeneralUnitaryGroup(arguments)

PGU(arguments)

ProjectiveGeneralUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralUnitaryGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PGU(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PGU(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PGU(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective general unitary group G = PGU(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q²), where n ≥2 and q is a prime power. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

ProjectiveSpecialUnitaryGroup(arguments)

PSU(arguments)

ProjectiveSpecialUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialUnitaryGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSU(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSU(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSU(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective special unitary group G = PSU(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q²), where n ≥2 and q is a prime power. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of V, giving the correspondence between these vectors and the G-set of G.

ProjectiveGammaUnitaryGroup(arguments)

PGammaU(arguments)

ProjectiveGammaUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGammaUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGammaUnitaryGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PGammaU(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PGammaU(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PGammaU(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct an automorphism group G = PGammaU(n, q) of the projective general unitary group B = PGU(n, q), by adding the field automorphisms of GF(q²) to B. The permutation action corresponds to the natural action on 1-dimensional subspaces of the n-dimensional vector space V over the field K = GF(q²), where n ≥2 and q is a prime power. The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the points and the G-set of G.

ProjectiveSigmaUnitaryGroup(arguments)

PSigmaU(arguments)

ProjectiveSigmaUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSigmaUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSigmaUnitaryGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSigmaU(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSigmaU(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSigmaU(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the automorphism group G = PSigmaU(n, q) of the projective special unitary group B = PSU(n, q), by adding the field automorphisms of GF(q²) to B. The permutation action corresponds to the natural action on 1-dimensional subspaces of the n-dimensional vector space V over the field K = GF(q²), where n ≥2 and q is a prime power. The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the points and the G-set of G.

ProjectiveSymplecticGroup(arguments)

PSp(arguments)

ProjectiveSymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSymplecticGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSymplecticGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSp(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSp(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSp(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective symplectic group G = PSp(n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 4. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

ProjectiveSigmaSymplecticGroup(arguments)

PSigmaSp(arguments)

ProjectiveSigmaSymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSigmaSymplecticGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSigmaSymplecticGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSigmaSp(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSigmaSp(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSigmaSp(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the group G = PSigmaSp(n, q) of the projective symplectic group PSp(n, q) extended by field automorphisms of K = GF(q), where V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 4. The function returns:

(a)
The group G;
(b)
An indexed set giving the correspondence between the points and the G-set of G.

PGO(arguments)

ProjectiveGeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralOrthogonalGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PGO(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PGO(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PGO(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective general orthogonal group G = PGO(n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an odd integer greater than or equal to 3. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

PGOPlus(arguments)

ProjectiveGeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralOrthogonalGroupPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PGOPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PGOPlus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PGOPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective general orthogonal group G = PGO^ + (n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 2. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

PGOMinus(arguments)

ProjectiveGeneralOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveGeneralOrthogonalGroupMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PGOMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PGOMinus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PGOMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective general orthogonal group G = PGO^ - (n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 2. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

PSO(arguments)

ProjectiveSpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialOrthogonalGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSO(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSO(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSO(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective special orthogonal group G = PSO(n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an odd integer greater than or equal to 3. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

PSOPlus(arguments)

ProjectiveSpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialOrthogonalGroupPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSOPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSOPlus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSOPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective special orthogonal group G = PSO^ + (n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 2. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

PSOMinus(arguments)

ProjectiveSpecialOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSpecialOrthogonalGroupMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSOMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSOMinus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

PSOMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective general orthogonal group G = PSO^ - (n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 2. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

ProjectiveOmega(arguments)

POmega(arguments)

ProjectiveOmega(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveOmega(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveOmega(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

POmega(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

POmega(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

POmega(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective orthogonal group G = POmega(n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an odd integer greater than or equal to 3. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

ProjectiveOmegaPlus(arguments)

POmegaPlus(arguments)

ProjectiveOmegaPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveOmegaPlus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveOmegaPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

POmegaPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

POmegaPlus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

POmegaPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective orthogonal group G = POmega(n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 2. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

ProjectiveOmegaMinus(arguments)

POmegaMinus(arguments)

ProjectiveOmegaMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveOmegaMinus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveOmegaMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

POmegaMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}

POmegaMinus(n, K) : RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt @}

POmegaMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the projective orthogonal group G = POmega(n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 2. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

ProjectiveSuzukiGroup(arguments)

PSz(arguments)

ProjectiveSuzukiGroup(q) : RngIntElt -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSuzukiGroup(K) : FldFin -> GrpPerm, {@ ModTupFldElt @}

ProjectiveSuzukiGroup(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

PSz(q): RngIntElt -> GrpPerm, {@ ModTupFldElt @}

PSz(K) : FldFin -> GrpPerm, {@ ModTupFldElt @}

PSz(V) : ModTupRng -> GrpPerm, {@ ModTupFldElt @}

Construct the permutation representation G = PSz(q) of the Suzuki simple group Sz(q), given by its action on projective points, where q is of the form 2^{2n + 1}. If K is given, its cardinality is q. If V is given, it must be 4-dimensional, and over K. The function returns:

(a)
The group G;
(b)
An indexed set of the generators of the 1-dimensional subspaces of K⁽ⁿ⁾, giving the correspondence between these vectors and the G-set of G.

AffineGroup(M) : GrpMat[FldFin] -> GrpPerm, {@ ModTupFldElt @}

Given a matrix group of degree d over a finite field F, construct the semidirect product V:M, where V=F^d is the natural M-module. The result G is a standard permutation group of degree |V| = |F|^d, where the second return value gives the correspondence between the elements of V and the standard G-set.

V2.28, 28 February 2025