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Standard Groups and Extensions


Standard Groups and Extensions






Subsections



Construction of a Classical Matrix Group






A number of functions are provided which construct various classical groups. 
The effect of these functions is to define the group in terms of a set of 
generating matrices.



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



Construct the general linear group GL(n, K), where 
K = GF(q) and V is an n-dimensional vector space over K.



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



Construct the special linear group SL(n, K), where 
K = GF(q) and V is an n-dimensional vector space over K.



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



Construct the general unitary group GU(n, K)
corresponding to the n-dimensional vector space V over the field 
K = GF(q^2), where n >= 2 and q is a prime power.



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



Construct the special unitary group SU(n, K)
corresponding to the n-dimensional vector space V over the field 
K = GF(q^2), where n >= 2 and q is a prime power.



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



Construct the symplectic group Sp(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.



SuzukiGroup(q) : RngIntElt -> GrpMat


SuzukiGroup(K) : FldFin -> GrpMat


SuzukiGroup(V) : ModTupRng -> GrpMat


Sz(q) : RngIntElt -> GrpMat


Sz(K) : FldFin -> GrpMat


Sz(V) : ModTupRng -> GrpMat



Construct the Suzuki simple group Sz(q), 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.





Example GrpMat_Symplectic (H15E9)

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







> F<u> := FiniteField(8);
> G := SymplecticGroup(10, F);
> print 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 GrpMat_Suzuki (H15E10)

We create the Suzuki group over GF(128):







> F<w> := FiniteField(128);
> V := VectorSpace(F, 4);
> S := SuzukiGroup(V);
> print 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]
> print Order(S);
34093383680
> print FactoredOrder(S);
[ <2, 14>, <5, 1>, <29, 1>, <113, 1>, <127, 1> ]





Construction of Extensions





DirectProduct(G, H) : GrpMat, GrpMat -> GrpMat



Given two matrix groups G and H of degrees m and n respectively, 
construct the direct product of G and H as a matrix group of degree
m + n.




DirectProduct(Q) : [ GrpMat ] -> GrpMat



Given a sequence Q of n matrix groups, construct the direct product
Q[1] x Q[2] x ... x Q[n] as a matrix group of degree equal
to the sum of the degrees of the groups Q[i], (i = 1, ..., n).



TensorProduct(G, H) : GrpMat, GrpPerm -> GrpMat



Given a matrix group G and a permutation group H, construct the tensor 
product G tensor H of G and H.



WreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat



Given a matrix group G and a permutation group H, construct 
the wreath product G wreath H of G and H.





Example GrpMat_Constructions (H15E11)

We define G to be SU(3, 4) and H to be the symmetric group 
of order 6. We then proceed to form the direct product of G with itself and the 
tensor and wreath products of G and H.







> K<w> := FiniteField(4);
> G := SpecialUnitaryGroup(3, K);
> D := DirectProduct(G, G);
> print D;
MatrixGroup(6, GF(2, 2))
Generators:
[  1   w   w   0   0   0]
[  0   1 w^2   0   0   0]
[  0   0   1   0   0   0]
[  0   0   0   1   0   0]
[  0   0   0   0   1   0]
[  0   0   0   0   0   1]


[w 1 1 0 0 0]
[1 1 0 0 0 0]
[1 0 0 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]


[  1   0   0   0   0   0]
[  0   1   0   0   0   0]
[  0   0   1   0   0   0]
[  0   0   0   1   w   w]
[  0   0   0   0   1 w^2]
[  0   0   0   0   0   1]


[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 w 1 1]
[0 0 0 1 1 0]
[0 0 0 1 0 0]
> print Order(D);
46656
> H := SymmetricGroup(3);
> E := WreathProduct(G, H);
> print Degree(E);
9
> print Order(E);
60466176
> F := TensorProduct(G, H);
> print Degree(F);
27
> print Order(F);
6718464





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