Documentation

Creating Finite Groups of Lie Type

Several functions are provided which construct various classical groups and other groups of Lie type. The effect of these functions is to define the group in terms of a set of generating matrices.

As shown by Chevalley, for each simple Lie algebra L over the complex field and for each finite field GF(q) there is an associated matrix group L(q). In general, these groups are perfect but not simple. To obtain the simple group, it is necessary to form the quotient by the centre. Similarly, as Steinberg, Ree and others have shown, if the associated Coxeter graph has an automorphism, of order t say, then there will be a `twisted' version sL(q) of L(q).

Generators for the series A, C, ()²A and ()²B are described in [Tay87]. Generators for the series B, D and ()²D are as given by Rylands and Taylor [RT98]. Generators for the exceptional groups of Lie type are described by Howlett, Rylands and Taylor in [HRT01].

Generic Creation Function

ChevalleyGroup(X, n, K: parameters) : MonStgElt, RngIntElt, FldFin -> GrpMat

ChevalleyGroup(X, n, q: parameters) : MonStgElt, RngIntElt, RngIntElt -> GrpMat

    Irreducible: BoolElt                Default: false

Construct a matrix group over the field K (or over GF(q)) which has the adjoint Chevalley group of Lie series X and Lie rank n as the quotient modulo scalar matrices. In most cases the group returned is the universal Chevalley group X_n(q); however, for series B, D and 2D the universal group is the spin group and the matrix group returned by ChevalleyGroup is Ω(2n + 1, q), Ω^ + (2n, q) or Ω^ - (2n, q).
For the twisted groups the meaning of the parameter q is consistent with the (abbreviated) notation in the `Atlas of Finite Groups' and in the monograph series `The Classification of the Finite Simple Groups' by Gorenstein, Lyons and Solomon. For a Chevalley group of rank n and type X with an automorphism of order t the Atlas defines the twisted Chevalley group ()^tX_n(q, q^t) to be the set of elements of X_n(q^t) fixed by the quotient of the twisting automorphism and the field automorphism induced by x |-> x^q of GF(q^t). In the Atlas the abbreviated notation for the twisted group is ()^tX_n(q) but in Carter [Car72] it is ()^tX_n(q^t). The first signature of the intrinsic expects the field GF(q^t) but the second signature expects the parameter q.
For example, for the series "2A", the group 2A_n(q) is SU(n + 1, q) but, in the first form of the signature, K must be the field GF(q²). Similarly the first form of the signature for the groups ()³D₄(q) and ()²E₆(q) requires the fields GF(q³) and GF(q²), respectively.
The possible series and the groups returned are:

"A": n≥0, A_n(q), the special linear group SL(n + 1, q).
"B": n≥1, B_n(q), the orthogonal group Ω(2n + 1, q).
"C": n≥1, C_n(q), the symplectic group Sp(2n, q).
"D": n≥1, D_n(q), the orthogonal group D_n(q) = Omega+(2n, q).

"E": n∈{ 6, 7, 8 }, the exceptional groups E_n(q). E₆(q) is represented as a matrix group of degree 27. It is simple unless q ≡ 1 mod 3, in which case its centre has order 3. E₇(q) is represented as a matrix group of degree 56. It is simple unless q ≡ 1 mod 2, in which case its centre has order 2. E₈(q) is represented as a matrix group of degree 248.

"F": n = 4, the exceptional group F₄(q) represented as a matrix group of degree 26. If q = 3^k then this representation is reducible. An irreducible representation is not yet available.
"G": n = 2, the exceptional group G₂(q) represented as a matrix group of degree 7. If q = 2^k then this representation is reducible. An irreducible representation of degree 6 can be obtained by setting the parameter Irreducible := true.
"2A": n≥1, K = GF(q²), the special unitary group 2A_n(q) = SU(n + 1, q).
"2B": n = 2, K = GF(q), q = 2^{2k + 1}, the Suzuki group 2B_2(q) = Sz(q).
"2D": n≥1, K =GF(q), ()²D_n(q), the orthogonal group 2D_n(q) = Omega-(2n, q).
"3D": n = 4, K = GF(q³), the exceptional group 3D_4(q).
"2E": n = 6, K = GF(q²), the exceptional group 2E_6(q).
"2F": n = 4, K=GF(q), q = 2^{2k + 1}, the Ree group 2F_4(q), simple except when q = 2 when the derived group is simple and is returned by the function TitsGroup.
"2G": n = 2, K = GF(q), q = 3^{2k + 1}, the Ree group 2G_2(q), simple except when q = 3.

The Orders of the Chevalley Groups

ChevalleyOrderPolynomial(type, n: parameters) : MonStgElt, RngIntElt -> RngUPolElt

The orders of the universal Chevalley groups X_n(q) and ()^tX_n(q) are polynomials in q. For the twisted groups of types ()²A_n, ()³D₄ and ()²E₆ the parameter q is the order of the fixed field of the Frobenius automorphism.
Other versions of Chevalley groups are quotients of universal Chevalley groups modulo a subgroup of the centre.

FactoredChevalleyGroupOrder(type, n, F: parameters) : MonStgElt, RngIntElt, FldFin -> RngIntEltFact

FactoredChevalleyGroupOrder(type, n, q: parameters) : MonStgElt, RngIntElt, RngIntElt -> RngIntEltFact

    Proof: BoolElt                      Default: true

    Version: MonStgElt                  Default: em "Default"

ChevalleyGroupOrder(type, n, F: parameters) : MonStgElt, RngIntElt, FldFin -> RngIntEltFact

ChevalleyGroupOrder(type, n, q: parameters) : MonStgElt, RngIntElt, RngIntElt -> RngIntEltFact

    Version: MonStgElt                  Default: em "Default"

The (factored) order of the Chevalley group of a given type and rank over the field F (or GF(q)). The default is the order of the group returned by ChevalleyGroup, which except for types B_n, D_n and ()²D_n is the universal group. The orders of the universal and adjoint Chevalley group can be obtained by setting the parameter Version to Universal or Adjoint. In the factored version the value of Proof is passed to the Magma's factorisation function (q.v.).

Classical Groups

Magma offers several functions to construct the classical groups. For most of these functions, it is possible to specify the particular group by giving one of the following combinations of arguments:

(i): The degree n and the coefficient field K of the desired matrix group;
(ii): The degree n of the desired matrix group and a prime power q which relates the group to the appropriate Lie algebra. With the exception of the unitary groups (which will be defined over GF(q²)), the resulting group will be defined over GF(q); or,
(iii): A full vector space V = Kⁿ on which the desired matrix group should act naturally.

Linear Groups

GeneralLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

GeneralLinearGroup(n, K) : RngIntElt, FldFin -> GrpMat

GeneralLinearGroup(V) : ModTupRng -> GrpMat

GL(n, q) : RngIntElt, RngIntElt -> GrpMat

GL(n, K) : RngIntElt, FldFin -> GrpMat

GL(V) : ModTupRng -> GrpMat

Here n is a positive integer, q is the power of a prime, K is a finite field GF(q), and V is an n-dimensional vector space over K. This function constructs the general linear group GL(n, q) (resp. GL(n, K), GL(V)) in terms of generating matrices. The intrinsic name may be abbreviated to GL.

SpecialLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

SpecialLinearGroup(n, K) : RngIntElt, FldFin -> GrpMat

SpecialLinearGroup(V) : ModTupRng -> GrpMat

SL(n, q) : RngIntElt, RngIntElt -> GrpMat

SL(n, K) : RngIntElt, FldFin -> GrpMat

SL(V) : ModTupRng -> GrpMat

Here n is a positive integer, q is the power of a prime, K is a finite field GF(q), and V is an n-dimensional vector space over K. This function constructs the special linear group SL(n, q) (resp. SL(n, K), SL(V)), namely the group of n x n matrices of determinant 1, in terms of generating matrices. The intrinsic name may be abbreviated to SL.

AffineGeneralLinearGroup(GrpMat, n, q) : Cat, RngIntElt, RngIntElt -> GrpMat

AffineGeneralLinearGroup(GrpMat, n, K) : Cat, RngIntElt, FldFin -> GrpMat

AffineGeneralLinearGroup(GrpMat, V) : Cat, ModTupRng -> GrpMat

AffineGeneralLinearGroup(E) : GrpPerm -> GrpPerm

AGL(GrpMat, n, q) : Cat, RngIntElt, RngIntElt -> GrpMat

AGL(GrpMat, n, K) : Cat, RngIntElt, FldFin -> GrpMat

AGL(GrpMat, V) : Cat, ModTupRng -> GrpMat

AGL(E) : GrpPerm -> GrpPerm

Here n is a positive integer greater than or equal to 2, q is the power of a prime, K is a finite field GF(q), and V is an n-dimensional vector space over K. This function constructs the affine general linear group AGL(n, q) (resp. AGL(n, K), AGL(V)) as a subgroup of GL(n + 1, K). If the category name GrpMat is omitted the affine group will be returned as a permutation group. The intrinsic name may be abbreviated to AGL.
If E is a regular elementary abelian permutation group the function constructs the affine general linear group of E, regarded as a vector space over the field GF(p), where p is the exponent of E.

AffineSpecialLinearGroup(GrpMat, n, q) : Cat, RngIntElt, RngIntElt -> GrpMat

AffineSpecialLinearGroup(GrpMat, n, K) : Cat, RngIntElt, FldFin -> GrpMat

AffineSpecialLinearGroup(GrpMat, V) : Cat, ModTupRng -> GrpMat

ASL(GrpMat, n, q) : Cat, RngIntElt, RngIntElt -> GrpMat

ASL(GrpMat, n, K) : Cat, RngIntElt, FldFin -> GrpMat

ASL(GrpMat, V) : Cat, ModTupRng -> GrpMat

Here n is a positive integer greater than or equal to 2, q is the power of a prime, K is a finite field GF(q), and V is an n-dimensional vector space over K. This function constructs the affine special linear group ASL(n, q) (resp. ASL(n, K), ASL(V)) as a subgroup of SL(n + 1, K). If the category name GrpMat is omitted, the affine group will be returned as a permutation group. The intrinsic name may be abbreviated to ASL.

Unitary Groups

ConformalUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

ConformalUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpMat

ConformalUnitaryGroup(V): ModTupRng -> GrpMat

CU(n, q) : RngIntElt, RngIntElt -> GrpMat

CU(n, K) : RngIntElt, FldFin -> GrpMat

CU(V): ModTupRng -> GrpMat

Here n ≥2 is a positive integer, q is the power of a prime, K is the finite field GF(q²), and V is the n-dimensional vector space over K. This function constructs the conformal unitary group CU(n, q) (resp. CU(n, K), CU(V)) in terms of generating matrices. It preserves the standard hermitian form up to a non-zero scalar multiple. The intrinsic name may be abbreviated to CU.

GeneralUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

GeneralUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpMat

GeneralUnitaryGroup(V): ModTupRng -> GrpMat

GU(n, q) : RngIntElt, RngIntElt -> GrpMat

GU(n, K) : RngIntElt, FldFin -> GrpMat

GU(V): ModTupRng -> GrpMat

Here n ≥2 is a positive integer, q is the power of a prime, K is the finite field GF(q²), and V is the n-dimensional vector space over K. This function constructs the general unitary group GU(n, q) (resp. GU(n, K), GU(V)) in terms of generating matrices. It preserves the standard hermitian form. The intrinsic name may be abbreviated to GU.

SpecialUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

SpecialUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpMat

SpecialUnitaryGroup(V): ModTupRng -> GrpMat

SU(n, q) : RngIntElt, RngIntElt -> GrpMat

SU(n, K) : RngIntElt, FldFin -> GrpMat

SU(V): ModTupRng -> GrpMat

Here n is an integer greater than or equal to 2, q is the power of a prime, K is the finite field GF(q²), and V is the n-dimensional vector space over K. This function constructs the special unitary group SU(n, q) (resp. SU(n, K), SU(V)) in terms of generating matrices. The intrinsic name may be abbreviated to SU.

ConformalSpecialUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

CSU(n, q) : RngIntElt, RngIntElt -> GrpMat

For an integer n≥2 and a prime power q, this function constructs the normaliser of the special unitary group SU(n, q) in SL(n, q²). The intrinsic name may be abbreviated to CSU.

Symplectic Groups

ConformalSymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

ConformalSymplecticGroup(n, K) : RngIntElt, FldFin -> GrpMat

ConformalSymplecticGroup(V) : ModTupRng -> GrpMat

CSp(n, q) : RngIntElt, RngIntElt -> GrpMat

CSp(n, K) : RngIntElt, FldFin -> GrpMat

CSp(V) : ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 4, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the conformal symplectic group CSp(n, q) (resp. CSp(n, K), CSp(V)) in terms of generating matrices. It preserves the standard alternating form up to a non-zero scalar multiple. The intrinsic name may be abbreviated to CSp.

SymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

SymplecticGroup(n, K) : RngIntElt, FldFin -> GrpMat

SymplecticGroup(V) : ModTupRng -> GrpMat

Sp(n, q) : RngIntElt, RngIntElt -> GrpMat

Sp(n, K) : RngIntElt, FldFin -> GrpMat

Sp(V) : ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 4, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the symplectic group Sp(n, q) (resp. Sp(n, K), Sp(V)) in terms of generating matrices. It preserves the standard alternating form up. The intrinsic name may be abbreviated to Sp.

Orthogonal and Spin Groups

ConformalOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

ConformalOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpMat

ConformalOrthogonalGroup(V): ModTupRng -> GrpMat

CO(n, q) : RngIntElt, RngIntElt -> GrpMat

CO(n, K) : RngIntElt, FldFin -> GrpMat

CO(V): ModTupRng -> GrpMat

Here n is an odd integer greater than or equal to 3, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the conformal orthogonal group CO(n, q) (resp. CO(n, K), CO(V)) in terms of generating matrices. It preserves the standard quadratic form up to a non-zero scalar multiple. The intrinsic name may be abbreviated to CO.

GeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

GeneralOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpMat

GeneralOrthogonalGroup(V): ModTupRng -> GrpMat

GO(n, q) : RngIntElt, RngIntElt -> GrpMat

GO(n, K) : RngIntElt, FldFin -> GrpMat

GO(V): ModTupRng -> GrpMat

Here n is an odd integer greater than or equal to 3, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the general orthogonal group GO(n, q) (resp. GO(n, K), GO(V)) in terms of generating matrices. It preserves the standard quadratic form. The intrinsic name may be abbreviated to GO.

SpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

SpecialOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpMat

SpecialOrthogonalGroup(V): ModTupRng -> GrpMat

SO(n, q) : RngIntElt, RngIntElt -> GrpMat

SO(n, K) : RngIntElt, FldFin -> GrpMat

SO(V): ModTupRng -> GrpMat

Here n is an odd integer greater than or equal to 3, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the special orthogonal group SO(n, q) (resp. SO(n, K), SO(V)) in terms of generating matrices. The intrinsic name may be abbreviated to SO. In characteristic 2 the group coincides with the general orthogonal group.

ConformalSpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

CSO(n, q) : RngIntElt, RngIntElt -> GrpMat

For an odd integer n≥3 and a prime power q, this function constructs the normaliser of the special orthogonal group SO(n, q) in SL(n, q). The intrinsic name may be abbreviated to CSO.

ConformalOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

ConformalOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpMat

ConformalOrthogonalGroupPlus(V): ModTupRng -> GrpMat

COPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

COPlus(n, K) : RngIntElt, FldFin -> GrpMat

COPlus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the conformal orthogonal group CO^ + (n, q) (resp. CO^ + (n, K), CO^ + (V)) in terms of generating matrices. It preserves the standard quadratic form up to a non-zero scalar multiple. The intrinsic name may be abbreviated to COPlus.

GeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

GeneralOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpMat

GeneralOrthogonalGroupPlus(V): ModTupRng -> GrpMat

GOPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

GOPlus(n, K) : RngIntElt, FldFin -> GrpMat

GOPlus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the general orthogonal group GO^ + (n, q) (resp. GO^ + (n, K), GO^ + (V)) in terms of generating matrices. It preserves the standard quadratic form. The intrinsic name may be abbreviated to GOPlus.

SpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

SpecialOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpMat

SpecialOrthogonalGroupPlus(V): ModTupRng -> GrpMat

SOPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

SOPlus(n, K) : RngIntElt, FldFin -> GrpMat

SOPlus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the special orthogonal group SO^ + (n, q) (resp. SO^ + (n, K), SO^ + (V)) in terms of generating matrices. The intrinsic name may be abbreviated to SOPlus. In characteristic 2 the group coincides with GOPlus and is not the kernel of the Dickson invariant.

ConformalSpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

CSOPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

For an odd integer n≥3 and a prime power q, this function constructs the normaliser of the special orthogonal group SO^ + (n, q) in SL(n, q). The intrinsic name may be abbreviated to CSOPlus.

ConformalOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

ConformalOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpMat

ConformalOrthogonalGroupMinus(V): ModTupRng -> GrpMat

COMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

COMinus(n, K) : RngIntElt, FldFin -> GrpMat

COMinus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the conformal orthogonal group CO^ - (n, q) (resp. CO^ - (n, K), CO^ - (V)) in terms of generating matrices. It preserves the standard quadratic form (of Minus type) up to a non-zero scalar multiple. The intrinsic name may be abbreviated to COMinus.

GeneralOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

GeneralOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpMat

GeneralOrthogonalGroupMinus(V): ModTupRng -> GrpMat

GOMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

GOMinus(n, K) : RngIntElt, FldFin -> GrpMat

GOMinus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the general orthogonal group GO^ - (n, q) (resp. GO^ - (n, K), GO^ - (V)) in terms of generating matrices. It preserves the standard quadratic form (of Minus type). The intrinsic name may be abbreviated to GOMinus.

SpecialOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

SpecialOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpMat

SpecialOrthogonalGroupMinus(V): ModTupRng -> GrpMat

SOMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

SOMinus(n, K) : RngIntElt, FldFin -> GrpMat

SOMinus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the special orthogonal group SO^ - (n, q) (resp. SO^ - (n, K), SO^ - (V)) in terms of generating matrices. The intrinsic name may be abbreviated to SOMinus. In characteristic 2 the group coincides with GOMinus and is not the kernel of the Dickson invariant.

ConformalSpecialOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

CSOMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

For an odd integer n≥3 and a prime power q, this function constructs the normaliser of the special orthogonal group SO^ - (n, q) in SL(n, q). The intrinsic name may be abbreviated to CSOMinus.

Omega(n, q) : RngIntElt, RngIntElt -> GrpMat

Omega(n, K) : RngIntElt, FldFin -> GrpMat

Omega(V): ModTupRng -> GrpMat

Here n is an odd integer greater than or equal to 3, q is a power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the orthogonal group Ω(n, K) (resp. Ω(n, K), Ω(V)) in terms of two generating matrices. If q is odd, the group Ω(n, K) is the kernel of the spinor norm map on SO(n, K)); if q is even, it is the kernel of the Dickson invariant.

OmegaPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

OmegaPlus(n, K) : RngIntElt, FldFin -> GrpMat

OmegaPlus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is a power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the orthogonal group Ω^ + (n, q) (resp. Ω^ + (n, K), Ω^ + (V)) in terms of two generating matrices. If q is odd, the group Ω^ + (n, K) is the kernel of the spinor norm map on SO^ + (n, K). If q is even, it is the kernel of the Dickson invariant.

OmegaMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

OmegaMinus(n, K) : RngIntElt, FldFin -> GrpMat

OmegaMinus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is a power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the orthogonal group Ω^ - (n, q) (resp. Ω^ - (n, K), Ω^ - (V)) in terms of two generating matrices. If q is odd, the group Ω^ - (n, K) is the kernel of the spinor norm map on SO^ - (n, K). If q is even, it is the kernel of the Dickson invariant.

Spin(n, q) : RngIntElt, RngIntElt -> GrpMat

Spin(n, K) : RngIntElt, FldFin -> GrpMat

Spin(V): ModTupRng -> GrpMat

Here n is an odd integer greater than or equal to 1, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the spin group Spin(n, K) (resp. Spin(n, K), Spin(V)).

SpinPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

SpinPlus(n, K) : RngIntElt, FldFin -> GrpMat

SpinPlus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 2, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the spin group Spin^ + (n, K) (resp. Spin^ + (n, K), Spin^ + (V)).

SpinMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

SpinMinus(n, K) : RngIntElt, FldFin -> GrpMat

SpinMinus(V): ModTupRng -> GrpMat

Here n is an even integer greater than or equal to 4, q is the power of a prime, K is the finite field GF(q), and V is the n-dimensional vector space over K. This function constructs the spin group Spin^ - (n, K) (resp. Spin^ - (n, K), Spin^ - (V)).

Exceptional Groups

Suzuki Groups

The Suzuki groups are specified slightly differently, as the degree of the group is always four. Thus for this family of groups, the possible combinations of arguments are:

(i): A finite field K = GF(2^{2m + 1}), over which the resulting matrix group is defined;
(ii): An integer q = 2^{2m + 1}, corresponding to the field K = GF(q) over which the resulting matrix group is defined; or,
(iii): A vector space V = K⁴ where K = GF(2^{2m + 1}) on which the resulting matrix group acts naturally. which the resulting

SuzukiGroup(q) : RngIntElt -> GrpMat

SuzukiGroup(K) : FldFin -> GrpMat

SuzukiGroup(V) : ModTupRng -> GrpMat

Here q is a prime power of the form 2^{2n + 1}, K is the finite field GF(q), and V is the 4-dimensional vector space over K. This function constructs the Suzuki simple group Sz(q) (resp. Sz(K), Sz(V)) in terms of two generating matrices. The intrinsic name may be abbreviated to Sz.

Example `GrpASim_Symplectic (H71E1)`

We create the 10-dimensional symplectic group over GF(8):

> F<u> := FiniteField(8);
> G := SymplecticGroup(10, F);
> G;
MatrixGroup(10, GF(2, 3))
Generators:
[  u   0   0   0   0   0   0   0   0   0]
[  0   1   0   0   0   0   0   0   0   0]
[  0   0   1   0   0   0   0   0   0   0]
[  0   0   0   1   0   0   0   0   0   0]
[  0   0   0   0   u   0   0   0   0   0]
[  0   0   0   0   0   u   0   0   0   0]
[  0   0   0   0   0   0   1   0   0   0]
[  0   0   0   0   0   0   0   1   0   0]
[  0   0   0   0   0   0   0   0   1   0]
[  0   0   0   0   0   0   0   0   0 u^6]
[0 0 0 1 1 1 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0]

Example `GrpASim_Suzuki (H71E2)`

We create the Suzuki group over GF(128):

> F<w> := FiniteField(128);
> V := VectorSpace(F, 4);
> S := SuzukiGroup(V);
> S;
MatrixGroup(4, GF(2, 7))
Generators:
[0 0 0 1]
[0 0 1 0]
[0 1 0 0]
[1 0 0 0]
[  w^8     0     0     0]
[    0 w^120     0     0]
[    0     0   w^7     0]
[    0     0     0 w^119]
[   1    0    0    0]
[ w^8    1    0    0]
[   0    w    1    0]
[w^17  w^9  w^8    1]
> Order(S);
34093383680
> FactoredOrder(S);
[ <2, 14>, <5, 1>, <29, 1>, <113, 1>, <127, 1> ]

Small Ree Groups

The Ree groups ((²)G₂(q)) are given in an irreducible matrix representation of degree seven. The possible combinations of arguments are:

(i)

A finite field K = GF(3^{2m + 1}) with m > 0, over which the matrix group is defined.

(ii)

An integer q = 3^{2m + 1} with m > 0, corresponding to the field K = GF(q) over which the group is defined; or,

(iii)

A vector space V = K⁷ where K = GF(3^{2m + 1}) with m > 0, on which the matrix group acts naturally.

ReeGroup(q) : RngIntElt -> GrpMat

ReeGroup(K) : FldFin -> GrpMat

ReeGroup(V) : ModTupRng -> GrpMat

Here q is a prime power of the form q = 3^{2m + 1} with m > 0, K is the finite field GF(q), and V is the 7-dimensional vector space over K. This function constructs the Ree group (²)G₂(q) (resp. (²)G₂(K), (²)G₂(V)) in terms of standard generating matrices. The intrinsic name may be abbreviated to Ree.

Large Ree Groups

The Ree groups ((²)F₄(q)) are given in an irreducible matrix representation of degree twenty-six. The possible combinations of arguments are: