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 have been included 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) F
Degree(
G) O
DimensionOfMatrixGroup(
G) A
Dimension(
G) O
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.
CharacteristicOfField(
G) A
Characteristic(
G) O
This is the characteristic of FieldOfMatrixGroup
(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 evalueated easily.
It is usually advantageous to carry out computations in H, and to translate
them to G via RepresentationIsomorphism
(G). (Note that the inverse
mapping of RepresentationIsomorphism
is not required to be efficient.)
IsIrreducibleMatrixGroup(
G) A
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, F = FieldOfMatrixGroup
(G) is assumed.
IsAbsolutelyIrreducibleMatrixGroup(
G) A
If present, this attribute is true if G is absolutely irreducible, i. e., irreducible over any
extension field of FieldOfMatrixGroup
(G).
IsMaximalAbsolutelyIrreducibleSolvableMatrixGroup(
G) A
This attribute, if present, is true if, and only if, G is absolutely irreducible and maximal among
the soluble subgroups of GL(d, F), where
d = DegreeOfMatrixGroup
(G) and
F = 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, F =
FieldOfMatrixGroup
(G) is assumed.
IsPrimitiveMatrixGroup(
G) A
IsPrimitiveMatrixGroup(
G,
F) O
IsLinearlyPrimitive(
G [,
F]) F
IsPrimitive(
G [,
F]) O
The 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, F = FieldOfMatrixGroup
(G) is assumed.
TraceField(
G) A
This is the field generated by the traces of the elements of G.
If G is a matrix group over a finite field, then By Brauer's
theorem, G has a conjugate which is a matrix group over TraceField
(G).
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 absolutely irreducible groups G (described in
GH) and certain trivial cases.
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