4 Additional functionality for matrix groups

This is the degree of the matrix group or, equivalently, the dimension of the natural underlying vector space. See also DimensionOfMatrixGroup.

This is the field generated by the matrix entries of the elements of G. See also FieldOfMatrixGroup.

This is a field containing all matrix entries of the elements of G. See also DefaultFieldOfMatrixGroup.

Let G be an irreducible subgroup of GL(n, F), where F = FieldOfMatrixGroup(G) is a finite field. This attribute stores the splitting field E for G, that is, the (unique) smallest field E containing F such that the natural E G-module Eⁿ is the direct sum of absolutely irreducible E G- submodules. The number of these absolutely irreducible summands equals the dimension of E as an F-vector space.

This is the characteristic of FieldOfMatrixGroup(G) (see FieldOfMatrixGroup).

This attribute stores an isomorphism H toG, where H is a group in which computations can be carried out more efficiently than in G, and the isomorphism can be evaluated easily. Every group in the IRREDSOL library has such a representation isomorphism from a pc group H to G.

In this way, computations which only depend on the isomorphism type of G can be carried out in the group H and translated back to the group G via the representation isomorphism. Possible applications are the conjugacy classes of G, Sylow subgroups, composition and chief series, normal subgroups, group theoretical properties of G, and many more.

The concept of a representation isomorphism is related to nice monomorphisms; see Section Nice Monomorphisms. However, unlike nice monomorphisms, RepresentationIsomorphism need not be efficient for computing preimages (and, indeed, will not usually be, in the case of the groups in the IRREDSOL library).

4.2 Irreducibility and maximality of matrix groups

The matrix group G of degree d is irreducible over the field F if no subspace of F^d is invariant under the action of G. If F is not specified, FieldOfMatrixGroup(G) is used as F.

gap> G := IrreducibleSolubleMatrixGroup(4, 2, 2, 3);
<matrix group of size 10 with 2 generators>
gap> IsIrreducibleMatrixGroup(G);
true
gap> IsIrreducibleMatrixGroup(G, GF(2));
true
gap> IsIrreducibleMatrixGroup(G, GF(4));
false

If present, this operation returns true if G is absolutely irreducible, i. e., irreducible over any extension field of FieldOfMatrixGroup(G).

gap> G := IrreducibleSolubleMatrixGroup(4, 2, 2, 3);
<matrix group of size 10 with 2 generators>
gap> IsAbsolutelyIrreducibleMatrixGroup(G);
false

This property, if present, is true if, and only if, G is absolutely irreducible and maximal among the soluble subgroups of GL(d, F), where d is DegreeOfMatrixGroup(G) and F equals FieldOfMatrixGroup(G).

4.3 Primitivity of matrix groups

Let G be a matrix group of degree d over the field F. A decomposition V₁ opluscdotsoplusV_k of F^d into F-subspaces V_i is a block system of G if the V_i are permuted by the natural action of G. Obviously, all V_i have the same dimension; this is the dimension of the block system V₁ opluscdotsoplusV_k. The function MinimalBlockDimensionOfMatrixGroup returns the minimum of the dimensions of all block systems of G. If F is not specified, FieldOfMatrixGroup(G) is used as F. At present, only methods for groups which are irreducible over F are available.

gap> G := IrreducibleSolubleMatrixGroup(2,3,1,4);;
gap> MinimalBlockDimension(G, GF(3));
2
gap> MinimalBlockDimension(G, GF(9));
1

An irreducible matrix group G of degree d is primitive over the field F if it only has the trivial block system F^d or, equivalently, if MinimalBlockDimensionOfMatrixGroup(G, F) = d. If F is not specified, it is assumed that F is FieldOfMatrixGroup(G).

gap> G := IrreducibleSolubleMatrixGroup(2,2,1,1);;
gap> IsPrimitiveMatrixGroup(G, GF(2));
true
gap> IsIrreducibleMatrixGroup(G, GF(4));
true
gap> IsPrimitiveMatrixGroup(G, GF(4));
false

This function returns the list of all imprimitivity systems of the irreducible matrix group G over the field F. If F is not given, FieldOfMatrixGroup(G) is used. Each imprimitivity system is given by a record with the following entries:

bases

a list of the bases of the subspaces which form the imprimitivity system. Note that a basis here is just a list of vectors, not a basis in the sense of GAP (see Basis). Each basis is in Hermite normal form so that the action of G on the imprimitivity system can be determined by OnSubspacesByCanonicalBasis

stab1

the subgroup of G stabilizing the subspace W spanned by bases[1]

min

is true if the imprimitivity system is minimal, that is, if stab1 acts primitively on W, and false otherwise

gap> G := IrreducibleSolubleMatrixGroup(6, 2, 1, 9);
<matrix group of size 54 with 4 generators>
gap> impr := ImprimitivitySystems(G, GF(2));;
gap> List(ImprimitivitySystems(G, GF(2)), r -> Length(r.bases));
[ 3, 3, 1 ]
gap> List(ImprimitivitySystems(G, GF(4)), 
>        r -> Action(G, r.bases, OnSubspacesByCanonicalBasis));
[ Group([ (), (1,2)(3,6)(4,5), (1,3,4)(2,5,6), (1,4,3)(2,6,5) ]), 
  Group([ (1,2,4)(3,5,6), (1,3)(2,5)(4,6), (), () ]), 
  Group([ (1,2,4)(3,5,6), (1,3)(2,5)(4,6), (1,2,4)(3,6,5), (1,4,2)(3,5,6) ]), 
  Group([ (1,2,4)(3,5,6), (1,3)(2,5)(4,6), (1,4,2)(3,5,6), (1,2,4)(3,6,5) ]), 
  Group([ (), (1,2), (), () ]), Group([ (1,2,3), (), (), () ]), 
  Group([ (), (2,3), (1,2,3), (1,3,2) ]), 
  Group([ (), (2,3), (1,2,3), (1,3,2) ]), 
  Group([ (), (2,3), (1,2,3), (1,3,2) ]), Group(()) ]

4.4 Conjugating matrix groups into smaller fields

This is the field generated by the traces of the elements of the matrix group G. If G is an irreducible matrix group over a finite field then, by a theorem of Brauer, G has a conjugate which is a matrix group over TraceField(G).

gap> repeat
>        G := IrreducibleSolubleMatrixGroup(8, 2, 2, 7)^RandomInvertibleMat(8, GF(8));
>    until FieldOfMatrixGroup(G) = GF(8);
gap> TraceField(G);
GF(2)

If bound, this is a matrix x over FieldOfMatrixGroup(G) such that G^<x> is a matrix group over TraceField(G). Currently, there are only methods available for irreducible matrix groups G over finite fields and certain trivial cases. The method for absolutely irreducible groups is described in GH. Note that, for matrix groups over infinite fields, such a matrix x need not exist.

gap> repeat
>       G := IrreducibleSolubleMatrixGroup(8, 2, 2, 7) ^ 
>                RandomInvertibleMat(8, GF(8));
>    until FieldOfMatrixGroup(G) = GF(8);
gap> FieldOfMatrixGroup(G^ConjugatingMatTraceField(G));
GF(2)

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