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 \({\bf F}_{q}\) 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 \({}^tL(q)\) of \(L(q)\).

Generators for the series A, C, \({}^2\)A and \({}^2\)B are described in [Taylor, 1987]. Generators for the series B, D and \({}^2\)D are as given by Rylands and Taylor [Rylands and Taylor, 1998]. Generators for the exceptional groups of Lie type are described by Howlett, Rylands and Taylor in [Howlett et al., 2001].

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 \({\bf F}_{q}\)) 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 \(\Omega(2n+1,q)\), \(\Omega^+(2n,q)\) or \(\Omega^-(2n,q)\).

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 \(x \mapsto x^q\) of \({\bf F}_{q^t}\). In the Atlas the abbreviated notation for the twisted group is \({}^tX_n(q)\) but in Carter [Carter, 1972] it is \({}^tX_n(q^t)\). The first signature of the intrinsic expects the field \({\bf F}_{q^t}\) but the second signature expects the parameter \(q\).

For example, for the series "2A", the group \({}^2A_n(q)\) is \({\operatorname{SU}}(n + 1, q)\) but, in the first form of the signature, \(K\) must be the field \({\bf F}_{q^2}\). Similarly the first form of the signature for the groups \({}^3D_4(q)\) and \({}^2E_6(q)\) requires the fields \({\bf F}_{q^3}\) and \({\bf F}_{q^2}\), respectively.

The possible series and the groups returned are:

"A"

\(n\geq 0\), \(A_n(q)\), the special linear group \({\operatorname{SL}}(n + 1, q)\).

"B"

\(n\geq 1\), \(B_n(q)\), the orthogonal group \(\Omega(2n + 1, q)\).

"C"

\(n\geq 1\), \(C_n(q)\), the symplectic group \({\operatorname{Sp}}(2n, q)\).

"D"

\(n\geq 1\), \(D_n(q)\), the orthogonal group \(\Omega^+(2n, q)\).

"E"

\(n\in\lbrace6,7,8\rbrace\), the exceptional groups \(E_n(q)\). \(E_6(q)\) is represented as a matrix group of degree 27. It is simple unless \(q\equiv 1\bmod 3\), in which case its centre has order 3. \(E_7(q)\) is represented as a matrix group of degree 56. It is simple unless \(q\equiv 1\bmod 2\), in which case its centre has order 2. \(E_8(q)\) is represented as a matrix group of degree \(248\).

"F"

\(n = 4\), the exceptional group \(F_4(q)\) 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 \(G_2(q)\) 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"

\(n\geq 1\), \(K = {\bf F}_{q^2}\), the special unitary group \({}^2A_n(q) = {\operatorname{SU}}(n + 1, q)\).

"2B"

\(n = 2\), \(K = {\bf F}_{q}\), \(q = 2^{2k+1}\), the Suzuki group \({}^2B_2(q) = {\operatorname{Sz}}(q)\).

"2D"

\(n\geq 1\), \(K ={\bf F}_{q}\), \({}^2D_n(q)\), the orthogonal group \(\Omega^-(2n, q)\).

"3D"

\(n = 4\), \(K = {\bf F}_{q^3}\), the exceptional group \({}^3D_4(q)\).

"2E"

\(n = 6\), \(K = {\bf F}_{q^2}\), the exceptional group \({}^2E_6(q)\).

"2F"

\(n = 4\), \(K={\bf F}_{q}\), \(q = 2^{2k+1}\), 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"

\(n = 2\), \(K = {\bf F}_{q}\), \(q = 3^{2k+1}\), 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 \({}^2A_n\), \({}^3D_4\) and \({}^2E_6\) 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

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

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

Proof  : BoolElt                      Default: true
Version: MonStgElt                    Default: "Default"

Version: MonStgElt                    Default: "Default"

The (factored) order of the Chevalley group of a given type and rank over the field \(F\) (or \({\bf F}_{q}\)). 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 \({\bf F}_{q^2}\)), the resulting group will be defined over \({\bf F}_{q}\); or,

(iii)

A full vector space \(V = K^n\) 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 \({\bf F}_{q}\), and \(V\) is an \(n\)-dimensional vector space over \(K\). This function constructs the general linear group \({\operatorname{GL}}(n, q)\) (resp. \({\operatorname{GL}}(n, K)\), \({\operatorname{GL}}(V)\)) 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 \({\bf F}_{q}\), and \(V\) is an \(n\)-dimensional vector space over \(K\). This function constructs the special linear group \({\operatorname{SL}}(n, q)\) (resp. \({\operatorname{SL}}(n, K)\), \({\operatorname{SL}}(V)\)), namely the group of \(n\times n\) 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 \({\bf F}_{q}\), and \(V\) is an \(n\)-dimensional vector space over \(K\). This function constructs the affine general linear group \({\operatorname{AGL}}(n, q)\) (resp. \({\operatorname{AGL}}(n, K)\), \({\operatorname{AGL}}(V)\)) as a subgroup of \({\operatorname{GL}}(n+1, K)\). 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 \({\bf F}_{p}\), 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 \({\bf F}_{q}\), and \(V\) is an \(n\)-dimensional vector space over \(K\). This function constructs the affine special linear group \({\operatorname{ASL}}(n, q)\) (resp. \({\operatorname{ASL}}(n, K)\), \({\operatorname{ASL}}(V)\)) as a subgroup of \({\operatorname{SL}}(n+1, K)\). 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 \(n \geq 2\) is a positive integer, \(q\) is the power of a prime, \(K\) is the finite field \({\bf F}_{q^2}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the conformal unitary group \({\operatorname{CU}}(n, q)\) (resp. \({\operatorname{CU}}(n, K)\), \({\operatorname{CU}}(V)\)) 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 \(n \geq 2\) is a positive integer, \(q\) is the power of a prime, \(K\) is the finite field \({\bf F}_{q^2}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the general unitary group \({\operatorname{GU}}(n, q)\) (resp. \({\operatorname{GU}}(n, K)\), \({\operatorname{GU}}(V)\)) 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 \({\bf F}_{q^2}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the special unitary group \({\operatorname{SU}}(n, q)\) (resp. \({\operatorname{SU}}(n, K)\), \({\operatorname{SU}}(V)\)) 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 \(n\ge 2\) and a prime power \(q\), this function constructs the normaliser of the special unitary group \({\operatorname{SU}}(n,q)\) in \({\operatorname{SL}}(n,q^2)\). 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the conformal symplectic group \({\operatorname{CSp}}(n, q)\) (resp. \({\operatorname{CSp}}(n, K)\), \({\operatorname{CSp}}(V)\)) 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the symplectic group \({\operatorname{Sp}}(n, q)\) (resp. \({\operatorname{Sp}}(n, K)\), \({\operatorname{Sp}}(V)\)) in terms of generating matrices. It preserves the standard alternating form. 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the conformal orthogonal group \({\operatorname{CO}}(n, q)\) (resp. \({\operatorname{CO}}(n, K)\), \({\operatorname{CO}}(V)\)) 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the general orthogonal group \({\operatorname{GO}}(n, q)\) (resp. \({\operatorname{GO}}(n, K)\), \({\operatorname{GO}}(V)\)) in terms of generating matrices. It preserves the standard quadratic form. The intrinsic name may be abbreviated to GO.

Example: Gen Orthog Odd (ex-681a40)

We create the \(5\)-dimensional orthogonal group \(G\) over \({\bf F}_{4}\). Note that \(G\) is defined by a standard non-degenerate quadratic form \(Q\) and it preserves the polar form of \(Q\). Since the characteristic is 2 this is a symplectic form. In addition, the dimension is odd and so the symplectic form has a non-trivial radical, which is invariant under the action of \(G\). We check that \(Q\) is non-degenerate; that is, there are no singular elements in the radical.

> G := GO(5,4);
> qq := InvariantQuadraticForms(G);
> qq;
[
    [    0     0     0     0     1]
    [    0     0     0     1     0]
    [    0     0     1     0     0]
    [    0     0     0     0     0]
    [    0     0     0     0     0]
]
> Q := qq[1];
> Q eq StandardQuadraticForm(5,4);
true
> V := QuadraticSpace(Q);
> R := Radical(V);
> Dimension(R);
1
> S := SingularRadical(V);
> Dimension(S);
0;
> QuadraticNorm(R.1);
1
> R.1;
(    0     0     1     0     0)
> forall{ g : g in Generators(G) | R.1 * g eq R.1 };
true

Run in calculatorCopy input

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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the special orthogonal group \({\operatorname{SO}}(n, q)\) (resp. \({\operatorname{SO}}(n, K)\), \({\operatorname{SO}}(V)\)) 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 \(n\ge 3\) and a prime power \(q\), this function constructs the normaliser of the special orthogonal group \({\operatorname{SO}}(n,q)\) in \({\operatorname{SL}}(n,q)\). 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the conformal orthogonal group \({\operatorname{CO}}^+(n, q)\) (resp. \({\operatorname{CO}}^+(n, K)\), \({\operatorname{CO}}^+(V)\)) 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the general orthogonal group \({\operatorname{GO}}^+(n, q)\) (resp. \({\operatorname{GO}}^+(n, K)\), \({\operatorname{GO}}^+(V)\)) 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the special orthogonal group \({\operatorname{SO}}^+(n, q)\) (resp. \({\operatorname{SO}}^+(n, K)\), \({\operatorname{SO}}^+(V)\)) 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 \(n\ge 3\) and a prime power \(q\), this function constructs the normaliser of the special orthogonal group \({\operatorname{SO}}^+(n,q)\) in \({\operatorname{SL}}(n,q)\). 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the conformal orthogonal group \({\operatorname{CO}}^-(n, q)\) (resp. \({\operatorname{CO}}^-(n, K)\), \({\operatorname{CO}}^-(V)\)) 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the general orthogonal group \({\operatorname{GO}}^-(n, q)\) (resp. \({\operatorname{GO}}^-(n, K)\), \({\operatorname{GO}}^-(V)\)) 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the special orthogonal group \({\operatorname{SO}}^-(n, q)\) (resp. \({\operatorname{SO}}^-(n, K)\), \({\operatorname{SO}}^-(V)\)) 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 \(n\ge 3\) and a prime power \(q\), this function constructs the normaliser of the special orthogonal group \({\operatorname{SO}}^-(n,q)\) in \({\operatorname{SL}}(n,q)\). 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the orthogonal group \(\Omega(n, K)\) (resp. \(\Omega(n, K)\), \(\Omega(V)\)) in terms of two generating matrices. If \(q\) is odd, the group \(\Omega(n, K)\) is the kernel of the spinor norm map on \({\operatorname{SO}}(n, K)\)); 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the orthogonal group \(\Omega^+(n, q)\) (resp. \(\Omega^+(n, K)\), \(\Omega^+(V)\)) in terms of two generating matrices. If \(q\) is odd, the group \(\Omega^+(n, K)\) is the kernel of the spinor norm map on \({\operatorname{SO}}^+(n, K)\). 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the orthogonal group \(\Omega^-(n, q)\) (resp. \(\Omega^-(n, K)\), \(\Omega^-(V)\)) in terms of two generating matrices. If \(q\) is odd, the group \(\Omega^-(n, K)\) is the kernel of the spinor norm map on \({\operatorname{SO}}^-(n, K)\). 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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the spin group \({\operatorname{Spin}}(n, K)\) (resp. \({\operatorname{Spin}}(n, K)\), \({\operatorname{Spin}}(V)\)).

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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the spin group \({\operatorname{Spin}}^+(n, K)\) (resp. \({\operatorname{Spin}}^+(n, K)\), \({\operatorname{Spin}}^+(V)\)).

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 \({\bf F}_{q}\), and \(V\) is the \(n\)-dimensional vector space over \(K\). This function constructs the spin group \({\operatorname{Spin}}^-(n, K)\) (resp. \({\operatorname{Spin}}^-(n, K)\), \({\operatorname{Spin}}^-(V)\)).

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 \(K = {\bf F}_{2^{2m+1}}\), over which the resulting matrix group is defined;

(ii)

An integer \(q = 2^{2m+1}\), corresponding to the field \(K = {\bf F}_{q}\) over which the resulting matrix group is defined; or,

(iii)

A vector space \(V = K^4\) where \(K = {\bf F}_{2^{2m+1}}\) 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 \({\bf F}_{q}\), and \(V\) is the \(4\)-dimensional vector space over \(K\). This function constructs the Suzuki simple group \({\operatorname{Sz}}(q)\) (resp. \({\operatorname{Sz}}(K)\), \({\operatorname{Sz}}(V)\)) in terms of two generating matrices. The intrinsic name may be abbreviated to Sz.

Example: Symplectic (ex-df72cd)

We create the \(10\)-dimensional symplectic group over \({\bf F}_{8}\):

> 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]

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Example: Suzuki (ex-58f4ed)

We create the Suzuki group over \({\bf F}_{128}\):

> 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> ]

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Small Ree Groups

The Ree groups (\({^2}G_2(q)\)) are given in an irreducible matrix representation of degree seven. The possible combinations of arguments are:

(i)

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

(ii)

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

(iii)

A vector space \(V = K^7\) where \(K = {\bf F}_{3^{2m+1}}\) 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 \({\bf F}_{q}\), and \(V\) is the \(7\)-dimensional vector space over \(K\). This function constructs the Ree group \({^2}G_2(q)\) (resp. \({^2}G_2(K)\), \({^2}G_2(V)\)) in terms of standard generating matrices. The intrinsic name may be abbreviated to Ree.

Large Ree Groups

The Ree groups (\({^2}F_4(q)\)) are given in an irreducible matrix representation of degree twenty-six. The possible combinations of arguments are:

(i)

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

(ii)

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

(iii)

A vector space \(V = K^{26}\) where \(K = {\bf F}_{2^{2m+1}}\) with \(m > 0\), on which the matrix group acts naturally.

LargeReeGroup(q): RngIntElt -> GrpMat

LargeReeGroup(K): FldFin -> GrpMat

LargeReeGroup(V): ModTupRng -> GrpMat

Here \(q\) is a prime power of the form \(q = 2^{2m+1}\) with \(m > 0\), \(K\) is the finite field \({\bf F}_{q}\), and \(V\) is the \(26\)-dimensional vector space over \(K\). This function constructs the Ree group \({^2}F_4(q)\) (resp. \({^2}F_4(K)\), \({^2}F_4(V)\)) in terms of standard generating matrices. The intrinsic name may be abbreviated to LargeRee.