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<html><head><title>[IRREDSOL] 4 Additional functionality for matrix groups</title></head>
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<h1>4 Additional functionality for matrix groups</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP004.htm#SECT001">Basic attributes for matrix groups</a>
<li> <A HREF="CHAP004.htm#SECT002">Irreducibility and maximality of matrix groups</a>
<li> <A HREF="CHAP004.htm#SECT003">Primitivity of matrix groups</a>
<li> <A HREF="CHAP004.htm#SECT004">Conjugating matrix groups into smaller fields</a>
</ol>

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.


<h2><a name="SECT001">4.1 Basic attributes for matrix groups</a></h2>

<a name = "I0"></a>

<a name = "I1"></a>

<a name = "SSEC001.1"></a>
<li><code>DegreeOfMatrixGroup(</code><var>G</var><code>) A</code>
<a name = "SSEC001.1"></a>
<li><code>Degree(</code><var>G</var><code>) O</code>
<a name = "SSEC001.1"></a>
<li><code>DimensionOfMatrixGroup(</code><var>G</var><code>) A</code>
<a name = "SSEC001.1"></a>
<li><code>Dimension(</code><var>G</var><code>) A</code>

This is the degree of the matrix group or, equivalently, the dimension of the
natural underlying vector space. See also <a href="../../../doc/ref/chap44.html#X7E55258C783C50CA">DimensionOfMatrixGroup</a>.

<a name = "SSEC001.2"></a>
<li><code>FieldOfMatrixGroup(</code><var>G</var><code>) A</code>

This is the field generated by the matrix entries of the elements of <var>G</var>. See also
<a href="../../../doc/ref/chap44.html#X78A9F0E580DA613A">FieldOfMatrixGroup</a>.

<a name = "SSEC001.3"></a>
<li><code>DefaultFieldOfMatrixGroup(</code><var>G</var><code>) A</code>

This is a field containing all matrix entries of the elements of <var>G</var>. See also
<a href="../../../doc/ref/chap44.html#X7D540083793CD496">DefaultFieldOfMatrixGroup</a>.

<a name = "SSEC001.4"></a>
<li><code>SplittingField(</code><var>G</var><code>) A</code>

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

<a name = "SSEC001.5"></a>
<li><code>CharacteristicOfField(</code><var>G</var><code>) A</code>
<a name = "SSEC001.5"></a>
<li><code>Characteristic(</code><var>G</var><code>) O</code>

This is the characteristic of <code>FieldOfMatrixGroup</code>(<var>G</var>) (see <a href="CHAP004.htm#SSEC001.2">FieldOfMatrixGroup</a>).

<a name = "SSEC001.6"></a>
<li><code>RepresentationIsomorphism(</code><var>G</var><code>) A</code>

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

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

The concept of a representation isomorphism is related to
nice monomorphisms; see Section <a href="../../../doc/ref/chap40.html#X7FFD731684606BC6">Nice Monomorphisms</a>. However, unlike nice monomorphisms,
<code>RepresentationIsomorphism</code> need not be efficient for computing preimages (and, indeed, will not usually be, in the case of the groups in the IRREDSOL library).


<h2><a name="SECT002">4.2 Irreducibility and maximality of matrix groups</a></h2>

<a name = "I2"></a>

<a name = "I3"></a>

<a name = "I4"></a>

<a name = "I5"></a>

<a name = "SSEC002.1"></a>
<li><code>IsIrreducibleMatrixGroup(</code><var>G</var><code>) P</code>
<li><code>IsIrreducibleMatrixGroup(</code><var>G</var><code>, </code><var>F</var><code>) O</code>
<a name = "SSEC002.1"></a>
<li><code>IsIrreducible(</code><var>G</var><code> [, </code><var>F</var><code>]) O</code>

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

<pre>
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
</pre>

<a name = "SSEC002.2"></a>
<li><code>IsAbsolutelyIrreducibleMatrixGroup(</code><var>G</var><code>) P</code>
<a name = "SSEC002.2"></a>
<li><code>IsAbsolutelyIrreducible(</code><var>G</var><code>) O</code>

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

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

<a name = "SSEC002.3"></a>
<li><code>IsMaximalAbsolutelyIrreducibleSolubleMatrixGroup(</code><var>G</var><code>) P</code>
<a name = "SSEC002.3"></a>
<li><code>IsMaximalAbsolutelyIrreducibleSolvableMatrixGroup(</code><var>G</var><code>) P</code>

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


<h2><a name="SECT003">4.3 Primitivity of matrix groups</a></h2>

<a name = "I6"></a>

<a name = "I7"></a>

<a name = "SSEC003.1"></a>
<li><code>MinimalBlockDimensionOfMatrixGroup(</code><var>G</var><code>) A</code>
<li><code>MinimalBlockDimensionOfMatrixGroup(</code><var>G</var><code>, </code><var>F</var><code>) O</code>
<a name = "SSEC003.1"></a>
<li><code>MinimalBlockDimension(</code><var>G</var><code> [, </code><var>F</var><code>]) O</code>

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

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

<a name = "SSEC003.2"></a>
<li><code>IsPrimitiveMatrixGroup(</code><var>G</var><code>) P</code>
<li><code>IsPrimitiveMatrixGroup(</code><var>G</var><code>, </code><var>F</var><code>) O</code>
<a name = "SSEC003.2"></a>
<li><code>IsPrimitive(</code><var>G</var><code> [, </code><var>F</var><code>]) O</code>
<a name = "SSEC003.2"></a>
<li><code>IsLinearlyPrimitive(</code><var>G</var><code> [, </code><var>F</var><code>]) O</code>

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

<pre>
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
</pre>

<a name = "SSEC003.3"></a>
<li><code>ImprimitivitySystems(</code><var>G</var><code> [, </code><var>F</var><code>]) O</code>

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


<dl compact>
<dt><code>bases</code> <dd> 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 <a href="../../../doc/ref/chap61.html#X837BE54C80DE368E">Basis</a>). Each basis is
 in Hermite normal form so that the action of <var>G</var> on the imprimitivity
 system can be determined by <code>OnSubspacesByCanonicalBasis</code>

<dt><code>stab1</code> <dd> the subgroup of <var>G</var> stabilizing the subspace <var>W</var> spanned by <code>bases[1]</code>

<dt><code>min</code> <dd> is true if the imprimitivity system is minimal, that is, if <code>stab1</code>
 acts primitively on <var>W</var>, and false otherwise
</dl>

<pre>
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(()) ]
</pre>


<h2><a name="SECT004">4.4 Conjugating matrix groups into smaller fields</a></h2>

<a name = "I8"></a>

<a name = "SSEC004.1"></a>
<li><code>TraceField(</code><var>G</var><code>) A</code>

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

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

<a name = "SSEC004.2"></a>
<li><code>ConjugatingMatTraceField(</code><var>G</var><code>) A</code>

If bound, this is a matrix <var>x</var> over <code>FieldOfMatrixGroup</code>(<var>G</var>) such that
<var><var>G</var><x></var> is a matrix group over <code>TraceField</code>(<var>G</var>). Currently, there are
only methods available for irreducible matrix groups <var>G</var> over finite fields
and certain trivial cases.
The method for absolutely irreducible groups is described in
<a href="biblio.htm#GH"><cite>GH</cite></a>.
Note that, for matrix groups over infinite fields, such a matrix <var>x</var>
need not exist.
<pre>
gap> repeat
> G := IrreducibleSolubleMatrixGroup(8, 2, 2, 7) ^
> RandomInvertibleMat(8, GF(8));
> until FieldOfMatrixGroup(G) = GF(8);
gap> FieldOfMatrixGroup(G^ConjugatingMatTraceField(G));
GF(2)
</pre>


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