<html><head><title>[IRREDSOL] 3 Recognition of matrix groups</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>3 Recognition of matrix groups</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP003.htm#SECT001">Identification of irreducible groups</a> <li> <A HREF="CHAP003.htm#SECT002">Compatibility with other data libraries</a> <li> <A HREF="CHAP003.htm#SECT003">Loading and unloading recognition data manually</a> </ol><p> <p> This chapter describes some functions which, given an irreducible matrix group, identify a group in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library which is conjugate to that group, see Section <a href="CHAP003.htm#SECT001">Identification of irreducible groups</a>. Moreover, Section <a href="CHAP003.htm#SECT002">Compatibility with other data libraries</a> describes how to translate between groups in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library and the <font face="Gill Sans,Helvetica,Arial">GAP</font> library of irreducible solvable groups. Section <a href="CHAP003.htm#SECT003">Loading and unloading recognition data manually</a> describes some functions which allow to load and unload the recognition data in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> package manually. <p> <p> <h2><a name="SECT001">3.1 Identification of irreducible groups</a></h2> <p><p> <a name = "SSEC001.1"></a> <li><code>IsAvailableIdIrreducibleSolvableMatrixGroup(</code><var>G</var><code>) F</code> <p> This function returns <code>true</code> if <code>IdIrreducibleSolvableMatrixGroup</code> (see <a href="CHAP003.htm#SSEC001.3">IdIrreducibleSolvableMatrixGroup</a>) will work for the irreducible matrix group <var>G</var>, and <code>false</code> otherwise. <p> <a name = "SSEC001.2"></a> <li><code>IsAvailableIdAbsolutelyIrreducibleSolvableMatrixGroup(</code><var>G</var><code>) F</code> <p> This function returns <code>true</code> if <code>IdIrreducibleSolvableMatrixGroup</code> (see <a href="CHAP003.htm#SSEC001.3">IdIrreducibleSolvableMatrixGroup</a>) will work for the absolutely irreducible matrix group <var>G</var>, and <code>false</code> otherwise. <p> <a name = "SSEC001.3"></a> <li><code>IdIrreducibleSolvableMatrixGroup(</code><var>G</var><code>) A</code> <p> If the matrix group <var>G</var> is solvable and irreducible over <var>F = <code>FieldOfMatrixGroup</code>(<var>G</var>)</var>, (see <a href="../../../doc/htm/ref/CHAP042.htm#SSEC001.3">FieldOfMatrixGroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual), and a conjugate in <var>GL(<var>n</var>, <var>F</var>)</var> of <var><var>G</var></var> belongs to the data base of irreducible solvable groups in <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font>, this function returns a list <code>[</code><var>n</var><code>, </code><var>q</var><code>, </code><var>d</var><code>, </code><var>k</var><code>]</code> such that <var>G</var> is conjugate to <code>IrreducibleSolvableMatrixGroup</code>(<var>n</var>, <var>q</var>, <var>d</var>, <var>k</var>) (see <a href="CHAP002.htm#SSEC002.3">IrreducibleSolvableMatrixGroup</a>). <p> <pre> gap> G := IrreducibleSolvableMatrixGroup (12, 2, 3, 52)^RandomInvertibleMat (12, GF(8)); <matrix group of size 2340 with 6 generators> gap> IdIrreducibleSolvableMatrixGroup (G); [ 12, 2, 3, 52 ] </pre> <p> <a name = "SSEC001.4"></a> <li><code>RecognitionIrreducibleSolvableMatrixGroup(</code><var>G</var><code>, </code><var>wantmat</var><code>, </code><var>wantgroup</var><code>) F</code> <a name = "SSEC001.4"></a> <li><code>RecognitionIrreducibleSolvableMatrixGroupNC(</code><var>G</var><code>, </code><var>wantmat</var><code>,</code><var>wantgroup</var><code>) F</code> <p> Let <var>G</var> be an irreducible solvable matrix group over a finite field, and let <var>wantmat</var> and <var>wantmat</var> be <code>true</code> or <code>false</code>. These functions identify a conjugate <var>H</var> of <var>G</var> group in the library. They return a record which has the following entries: <p> <dl compact> <dt><code>id</code> <dd> contains the id of <var>H</var> (and thus of <var>G</var>); cf. <code>IdIrreducibleSolvableMatrixGroup</code> (<a href="CHAP003.htm#SSEC001.3">IdIrreducibleSolvableMatrixGroup</a>) <p> <dt><code>mat</code> (present if <code>wantmat</code> is <code>true</code>) <dd> a matrix <var>x</var> such that <var>G<sup>x</sup> = H</var> <p> <dt><code>group</code> (present if <code>wantmat</code> is <code>true</code>) <dd> the group <var>H</var> <p> </dl> Note that in most cases, <code>RecognitionIrreducibleSolvableMatrixGroup</code> and <code>RecognitionIrreducibleSolvableMatrixGroupNC</code> are much slower if <var>wantmat</var> is set to true. <p> <code>RecognitionIrreducibleSolvableMatrixGroupNC</code> does not check its arguments. If the group <var>G</var> is beyond the scope of the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library (see <a href="CHAP003.htm#SSEC001.1">IsAvailableIdIrreducibleSolvableMatrixGroup</a>), <code>RecognitionIrreducibleSolvableMatrixGroupNC</code> returns <code>fail</code>, while <code>RecognitionIrreducibleSolvableMatrixGroup</code> raises an error. <p> <pre> gap> G := IrreducibleSolvableMatrixGroup (6, 2, 3, 5) ^ > RandomInvertibleMat (6, GF(4)); <matrix group of size 42 with 3 generators> gap> r := RecognitionIrreducibleSolvableMatrixGroup (G, true, false);; gap> r.id; [ 6, 2, 3, 5 ] gap> G^r.mat = CallFuncList (IrreducibleSolvableMatrixGroup, r.id); true </pre> <p> <a name = "SSEC001.5"></a> <li><code>IdAbsolutelyIrreducibleSolvableMatrixGroup(</code><var>G</var><code>) A</code> <a name = "SSEC001.5"></a> <li><code>RecognitionAbsolutelyIrreducibleSolvableMatrixGroup(</code><var>G</var><code>, </code><var>wantmat</var><code>, </code><var>wantgroup</var><code>) F</code> <a name = "SSEC001.5"></a> <li><code>RecognitionAbsolutelyIrreducibleSolvableMatrixGroupNC(</code><var>G</var><code>, </code><var>wantmat</var><code>,</code><var>wantgroup</var><code>) F</code> <p> These functions are no longer available. These functions have been replaced by the functions <code>IdIrreducibleSolvableMatrixGroup</code> (<a href="CHAP003.htm#SSEC001.3">IdIrreducibleSolvableMatrixGroup</a>), <code>RecognitionIrreducibleSolvableMatrixGroup</code> (<a href="CHAP003.htm#SSEC001.4">RecognitionIrreducibleSolvableMatrixGroup</a>), or <code>RecognitionIrreducibleSolvableMatrixGroupNC</code> (<a href="CHAP003.htm#SSEC001.4">RecognitionIrreducibleSolvableMatrixGroupNC</a>). <p> Note that the ids returned by the functions for absolutely irreducible groups was a triple <code>[</code><var>n</var><code>, </code><var>d</var><code>, </code><var>k</var><code>]</code>, while the replacement functions use ids of the form <code>[</code><var>n</var><code>, </code><var>d</var><code>, </code><var>d</var><code>, </code><var>k</var><code>]</code>, where <var><var>d</var> = 1</var> in the absolutely irreducible case. <p> <p> <h2><a name="SECT002">3.2 Compatibility with other data libraries</a></h2> <p><p> A library of irreducible solvable subgroups of <var>GL(n, p)</var>, where <var>p</var> is a prime and <var>p<sup>n</sup> leq255</var> already exists in <font face="Gill Sans,Helvetica,Arial">GAP</font>, see Section <a href="../../../doc/htm/ref/CHAP048.htm#SECT011">Irreducible Solvable Matrix Groups</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual. The following functions allow one to translate between between that library and the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library. <p> <a name = "SSEC002.1"></a> <li><code>IdIrreducibleSolvableMatrixGroupIndexMS(</code><var>n</var><code>, </code><var>p</var><code>, </code><var>k</var><code>) F</code> <p> This function returns the id (see <a href="CHAP003.htm#SSEC001.3">IdIrreducibleSolvableMatrixGroup</a>) of <var>G</var>, where <var>G</var> is <code>IrreducibleSolvableGroupMS</code>(<var>n</var>, <var>p</var>, <var>k</var>) (see <a href="../../../doc/htm/ref/CHAP048.htm#SSEC011.1">IrreducibleSolvableGroupMS</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual). <p> <pre> gap> IdIrreducibleSolvableMatrixGroupIndexMS (6, 2, 5); [ 6, 2, 2, 4 ] gap> G := IrreducibleSolvableGroupMS (6,2,5); <matrix group of size 27 with 2 generators> gap> H := IrreducibleSolvableMatrixGroup (6, 2, 2, 4); <matrix group of size 27 with 3 generators> gap> G = H; false # groups in the libraries need not be the same gap> r := RecognitionIrreducibleSolvableMatrixGroup (G, true, false);; gap> G^r.mat = H; true </pre> <p> <a name = "SSEC002.2"></a> <li><code>IndexMSIdIrreducibleSolvableMatrixGroup(</code><var>n</var><code>, </code><var>q</var><code>, </code><var>d</var><code>, </code><var>k</var><code>) F</code> <p> This function returns a triple [<var>n</var>, <var>p</var>, <var>l</var>] such that <code>IrreducibleSolvableGroupMS</code>(<var>n</var>, <var>p</var>, <var>l</var>) (see <a href="../../../doc/htm/ref/CHAP048.htm#SSEC011.1">IrreducibleSolvableGroupMS</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual) is conjugate to <code>IrreducibleSolvableMatrixGroup</code>(<var>n</var>, <var>q</var>, <var>d</var>, <var>k</var>) (see <a href="CHAP002.htm#SSEC002.3">IrreducibleSolvableMatrixGroup</a>). <p> <pre> gap> IndexMSIdIrreducibleSolvableMatrixGroup (6, 2, 2, 7); [ 6, 2, 13 ] gap> G := IrreducibleSolvableGroupMS (6,2,13); <matrix group of size 27 with 2 generators> gap> H := IrreducibleSolvableMatrixGroup (6, 2, 2, 7); <matrix group of size 27 with 3 generators> gap> G = H; false # groups in the libraries need not be the same gap> r := RecognitionIrreducibleSolvableMatrixGroup (G, true, false);; gap> G^r.mat = H; true </pre> <p> <p> <h2><a name="SECT003">3.3 Loading and unloading recognition data manually</a></h2> <p><p> The data required by the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library is loaded into <font face="Gill Sans,Helvetica,Arial">GAP</font>'s workspace automatically whenever required, but is never unloaded automatically. The functions described in this and the previous section describe how to load and unload this data manually. They are only relevant if timing or conservation of memory is an issue. <a name = "I0"></a> <a name = "SSEC003.1"></a> <li><code>LoadAbsolutelyIrreducibleSolvableGroupFingerprints(</code><var>n</var><code>, </code><var>q</var><code>) F</code> This function loads the fingerprint data required for the recognition of absolutely irreducible solvable subgroups of <var>GL(<var>n</var>, <var>q</var>)</var>. <p> <a name = "SSEC003.2"></a> <li><code>LoadedAbsolutelyIrreducibleSolvableGroupFingerprints() F</code> <p> This function returns a list. Each entry consists of an integer <var>n</var> and a set <var>l</var>. The set <var>l</var> contains all prime powers <var>q</var> such that the recognition data for <var>GL(<var>n</var>, <var>q</var>)</var> is currently in memory. <p> <a name = "SSEC003.3"></a> <li><code>UnloadAbsolutelyIrreducibleSolvableGroupFingerprints([n [,q]]) F</code> <p> This function can be used to delete recognition data for irreducible groups from the <font face="Gill Sans,Helvetica,Arial">GAP</font> workspace. If no argument is given, all data will be deleted. If only <var>n</var> is given, all data for degree <var>n</var> (and any <var>q</var>) will be deleted. If <var>n</var> and <var>q</var> are given, only the data for <var>GL(n, q)</var> will be deleted from the <font face="Gill Sans,Helvetica,Arial">GAP</font> workspace. Use this function if you run out of <font face="Gill Sans,Helvetica,Arial">GAP</font> workspace. The data is automatically re-loaded when required. <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>IRREDSOL manual<br>February 2007 </address></body></html>
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