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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% matgroups.tex IRREDSOL documentation Burkhard Höfling %% %% Copyright © 2003–2016 Burkhard Höfling %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Additional functionality for matrix groups} 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Basic attributes for matrix groups}\null \index{attributes!basic, for matrix groups} \index{matrix group!basic attributes} \>DegreeOfMatrixGroup(<G>) A \>Degree(<G>)!{for matrix groups} O \>DimensionOfMatrixGroup(<G>) A \>Dimension(<G>)!{for matrix groups} A This is the degree of the matrix group or, equivalently, the dimension of the natural underlying vector space. See also "ref:DimensionOfMatrixGroup". \>FieldOfMatrixGroup(<G>) A This is the field generated by the matrix entries of the elements of~<G>. See also "ref:FieldOfMatrixGroup". \>DefaultFieldOfMatrixGroup(<G>) A This is a field containing all matrix entries of the elements of~<G>. See also "ref:DefaultFieldOfMatrixGroup". \>SplittingField(<G>) A Let <G> be an irreducible subgroup of $GL(n, F)$, where $F = `FieldOfMatrixGroup'( 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 $E^n$ 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>)!{for matrix groups} O This is the characteristic of `FieldOfMatrixGroup'( \>RepresentationIsomorphism(<G>) A This attribute stores an isomorphism $<H> \to <G>$, 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 conjuga The concept of a representation isomorphism is related to nice monomorphisms; see Section "ref:Nice Monomorphisms". However, unlike nice monomorph `RepresentationIsomorphism' need not be efficient for computing preimages (and, %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Irreducibility and maximality of matrix groups}\null \index{matrix group!irreducibility} \index{matrix group!maximality} \index{irreducibility!of matrix group} \index{maximality!of matrix group} \>IsIrreducibleMatrixGroup(<G>) P \>IsIrreducibleMatrixGroup(<G>, <F>) O \>IsIrreducible(<G> [, <F>])!{for matrix groups} O 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'( \beginexample 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 \endexample \>IsAbsolutelyIrreducibleMatrixGroup(<G>) P \>IsAbsolutelyIrreducible(<G>) O If present, this operation returns true if <G> is absolutely irreducible, i.~e., irreducible ov extension field of `FieldOfMatrixGroup'( \beginexample gap> G := IrreducibleSolubleMatrixGroup(4, 2, 2, 3); <matrix group of size 10 with 2 generators> gap> IsAbsolutelyIrreducibleMatrixGroup(G); false \endexample \>`IsMaximalAbsolutelyIrreducibleSolubleMatrixGroup(<G>)'% {IsMaximalAbsolutelyIrreducibleSolubleMatrixGroup}% @{`IsMaximalAbsolutelyIrreducibleSoluble\\MatrixGroup'} P \>`IsMaximalAbsolutelyIrreducibleSolvableMatrixGroup(<G>)'% {IsMaximalAbsolutelyIrreducibleSolvableMatrixGroup}% @{`IsMaximalAbsolutelyIrreducibleSolvable\\MatrixGroup'} P This property, if present, is `true' if, and only if, the soluble subgroups of $GL(d, F)$, where $d$ is `DegreeOfMatrixGroup'( $F$ equals `FieldOfMatrixGroup'( %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Primitivity of matrix groups}\null \index{matrix group!primitivity} \index{primitivity!of matrix group} \>MinimalBlockDimensionOfMatrixGroup(<G>) A \>MinimalBlockDimensionOfMatrixGroup(<G>, <F>) O \>MinimalBlockDimension(<G> [, <F>])!{for matrix groups} O Let <G> be a matrix group of degree <d> over the field <F>. A decomposition $V_1 \oplus \cdots \oplus V_k$ 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 $V_1 \oplus \cdots \oplus V_k$. The function `MinimalBlockDimensionOfMatrixGroup' returns the minimum of the dimensions of all block systems of <G>. If <F> is not specified, `FieldOfMatrixGroup'( is used as <F>. At present, only methods for groups which are irreducible over <F> are available. \beginexample gap> G := IrreducibleSolubleMatrixGroup(2,3,1,4);; gap> MinimalBlockDimension(G, GF(3)); 2 gap> MinimalBlockDimension(G, GF(9)); 1 \endexample \>IsPrimitiveMatrixGroup(<G>) P \>IsPrimitiveMatrixGroup(<G>, <F>) O \>IsPrimitive(<G> [, <F>])!{for matrix groups} 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 $<F>^d$ or, equivalently, if $`MinimalBlockDimensionOfMatrixGroup'( specified, it is assumed that <F> is `FieldOfMatrixGroup'( \beginexample 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 \endexample \>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'( Each imprimitivity system is given by a record with the following entries: \beginitems `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 "ref: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 `min' & is true if the imprimitivity system is minimal, that is, if `stab1' acts primitively on <W>, and false otherwise \enditems \beginexample 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(()) ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Conjugating matrix groups into smaller fields}\null \index{matrix group!conjugating into smaller fieldss} \>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'( \beginexample gap> repeat > G := IrreducibleSolubleMatrixGroup(8, 2, 2, 7)^RandomInvertibleMat(8, GF(8)); > until FieldOfMatrixGroup(G) = GF(8); gap> TraceField(G); GF(2) \endexample \>ConjugatingMatTraceField(<G>) A If bound, this is a matrix <x> over `FieldOfMatrixGroup'( $<G>^<x>$ is a matrix group over `TraceField'( only methods available for irreducible matrix groups <G> over finite fields and certain trivial cases. The method for absolutely irreducible groups is described in \cite{GH}. Note that, for matrix groups over infinite fields, such a matrix <x> need not exist. \beginexample gap> repeat > G := IrreducibleSolubleMatrixGroup(8, 2, 2, 7) ^ > RandomInvertibleMat(8, GF(8)); > until FieldOfMatrixGroup(G) = GF(8); gap> FieldOfMatrixGroup(G^ConjugatingMatTraceField(G)); GF(2) \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E %% ¤ Dauer der Verarbeitung: 0.13 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28