This chapter explains some attributes, properties, and operations which may be useful for working with matrix groups. Some of these are part of the GAP library and are listed for the sake of completeness, and some are provided by the package IRREDSOL. Note that groups constructed by functions in IRREDSOL already have the appropriate properties and attributes.
DegreeOfMatrixGroup(
G) A
Degree(
G) O
DimensionOfMatrixGroup(
G) A
Dimension(
G) A
This is the degree of the matrix group or, equivalently, the dimension of the natural underlying vector space. See also DimensionOfMatrixGroup.
FieldOfMatrixGroup(
G) A
This is the field generated by the matrix entries of the elements of G. See also FieldOfMatrixGroup.
DefaultFieldOfMatrixGroup(
G) A
This is a field containing all matrix entries of the elements of G. See also DefaultFieldOfMatrixGroup.
SplittingField(
G) A
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.
CharacteristicOfField(
G) A
Characteristic(
G) O
This is the characteristic of FieldOfMatrixGroup
(G) (see FieldOfMatrixGroup).
RepresentationIsomorphism(
G) A
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).
IsIrreducibleMatrixGroup(
G) P
IsIrreducibleMatrixGroup(
G,
F) O
IsIrreducible(
G [,
F]) O
The matrix group G of degree d is irreducible over the field F if no subspace of Fd 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
IsAbsolutelyIrreducibleMatrixGroup(
G) P
IsAbsolutelyIrreducible(
G) O
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
IsMaximalAbsolutelyIrreducibleSolubleMatrixGroup(
G) P
IsMaximalAbsolutelyIrreducibleSolvableMatrixGroup(
G) P
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).
MinimalBlockDimensionOfMatrixGroup(
G) A
MinimalBlockDimensionOfMatrixGroup(
G,
F) O
MinimalBlockDimension(
G [,
F]) O
Let G be a matrix group of degree d over the field F. A
decomposition V1 opluscdotsoplusVk of Fd into F-subspaces
Vi is a block system of G if the Vi are permuted by the natural
action of G. Obviously, all Vi have the same dimension; this is the
dimension of the block system
V1 opluscdotsoplusVk. 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
IsPrimitiveMatrixGroup(
G) P
IsPrimitiveMatrixGroup(
G,
F) O
IsPrimitive(
G [,
F]) O
IsLinearlyPrimitive(
G [,
F]) O
An irreducible matrix group G of degree d is primitive over the field F if it
only has the trivial block system Fd 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
ImprimitivitySystems(
G [,
F]) O
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
OnSubspacesByCanonicalBasis
stab1
bases[1]
min
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(()) ]
TraceField(
G) A
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)
ConjugatingMatTraceField(
G) A
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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