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<div class="ChapSects"><a href="chap1.html#X81D076B27FE6C1FE">1 <span class="Heading">The GAP Table of Marks Library</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X84DBFB8287C3F1B4">1.1 <span class="Heading">Tables Of Marks</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A96960785BCFEA6">1.2 <span class="Heading">Installing The Table of Marks Library</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X86BA319E87365B5C">1.3 <span class="Heading">Contents</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8177A790811FBADC">1.4 <span class="Heading">Administrative Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7E56B36A794A80C0">1.4-1 LIBTOMKNOWN</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7BFF23287D8DF0F3">1.4-2 IsLibTomRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X78F117087B49838D">1.4-3 TableOfMarksFromLibrary</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X854A6959795FCE91">1.4-4 ConvertToLibTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8799978F7F338677">1.4-5 SetActualLibFileName</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C86CE2B7C0F4842">1.4-6 LIBTOM</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7E9199A0852A94E8">1.4-7 AllLibTomNames</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7912D7D487CF4461">1.4-8 NamesLibTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X789A0FF487E231E4">1.4-9 NotifiedFusionsOfLibTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83DE997E7A3698FB">1.4-10 NotifiedFusionsToLibTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X838400737B446706">1.4-11 UnloadTableOfMarksData</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7F8107C282BB991C">1.5 <span class="Heading">Standard Generators of Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7984E27078B20557">1.5-1 StandardGeneratorsInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C6A2D3A8762F11B">1.5-2 HumanReadableDefinition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X867C9F52847B7A5E">1.5-3 StandardGeneratorsFunctions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7E5546217E418DE3">1.5-4 IsStandardGeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X830DDDC287DFBA0A">1.5-5 StandardGeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79316D2487927FF5">1.5-6 StandardGeneratorsInfo</a></span>
</div></div>
</div>
<h3>1 <span class="Heading">The GAP Table of Marks Library</span></h3>
<p><a id="X84DBFB8287C3F1B4" name="X84DBFB8287C3F1B4"></a></p>
<h4>1.1 <span class="Heading">Tables Of Marks</span></h4>
<p>The concept of a <em>Table of Marks</em> was introduced by W.Burnside in his book ``Theory of Groups of Finite Order'' <a href="chapBib.html#biBBur55">[Bur55]</a>. Therefore a table of marks is sometimes called a <em>Burnside matrix</em>. The table of marks of a finite group <span class="SimpleMath">G</span> is a matrix whose rows and columns are labelled by the conjugacy classes of subgroups of <span class="SimpleMath">G</span> and where for two subgroups <span class="SimpleMath">H</span> and <span class="SimpleMath">K</span> the <span class="SimpleMath">(H, K)</span>–entry is the number of fixed points of <span class="SimpleMath">K</span> in the transitive action of <span class="SimpleMath">G</span> on the cosets of <span class="SimpleMath">H</span> in <span class="SimpleMath">G</span>. So the table of marks characterizes the set of all permutation representations of <span class="SimpleMath">G</span>. Moreover, the table of marks gives a compact description of the subgroup lattice of <span class="SimpleMath">G</span>, since from the numbers of fixed points the numbers of conjugates of a subgroup <span class="SimpleMath">K</span> contained in a subgroup <span class="SimpleMath">H</span> can be derived. For small groups the table of marks of <span class="SimpleMath">G</span> can be constructed directly in GAP by first computing the entire subgroup lattice of <span class="SimpleMath">G</span>. However, for larger groups this method is unfeasible. The GAP Table of Marks library provides access to several hundred table of marks and their maximal subgroups.</p>
<p><a id="X7A96960785BCFEA6" name="X7A96960785BCFEA6"></a></p>
<h4>1.2 <span class="Heading">Installing The Table of Marks Library</span></h4>
<p>Download the archives in your preferred format. Unpack the archives inside the pkg dirctory of your GAP installation. Load the package</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("tomlib");</span>
true</pre></div>
<p><a id="X86BA319E87365B5C" name="X86BA319E87365B5C"></a></p>
<h4>1.3 <span class="Heading">Contents</span></h4>
<p>TomLib contains several hundred tables of marks. For a complete list of the contents of the library do the following.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">names:=AllLibTomNames();;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">"A5" in names;</span>
true
</pre></div>
<p>The current version of the tomlib contains the tables of marks of the groups listed below as well as the tables of many of their maximal subgroups and automorphism groups. The Alternating groups <span class="SimpleMath">A_n</span></p>
<ul>
<li><p>for <span class="SimpleMath">n = 5, 6, 7, 8, 9, 10, 11, 12, 13</span>.</p>
</li>
</ul>
<p>The Symmetric groups <span class="SimpleMath">S_n</span></p>
<ul>
<li><p>for <span class="SimpleMath">n = 4, 5, 6, 7, 8, 9, 10, 11, 12, 13</span>.</p>
</li>
</ul>
<p>The Linear groups <span class="SimpleMath">L_2(n)</span> for</p>
<ul>
<li><p><span class="SimpleMath">n = 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53</span></p>
</li>
<li><p><span class="SimpleMath">n = 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125 .</span></p>
</li>
</ul>
<p>along with</p>
<ul>
<li><p><span class="SimpleMath">L_3(4), L_3(3), L_3(5), L_3(7), L_3(9)</span></p>
</li>
<li><p><span class="SimpleMath">L_4(3), L_3(8), L_3(11)</span>.</p>
</li>
</ul>
<p>The Unitary groups</p>
<ul>
<li><p><span class="SimpleMath">U_3(3), U_4(3), U_3(5), U_3(4), U_3(11), U_3(7), U_3(8)</span></p>
</li>
<li><p><span class="SimpleMath">U_3(9), U_4(2), U_5(2)</span></p>
</li>
</ul>
<p>The Sporadic Groups</p>
<ul>
<li><p><span class="SimpleMath">Co_3, HS, McL, He, J_1, J_2, J_3, M_11, M_12, M_22, M_23, M_24</span></p>
</li>
</ul>
<p>The names given to each subgroup are consistent with those used in Robert Wilson's atlas <a href="chapBib.html#biBAGR">[WWT+]</a> For example if you wish to access the table of marks of the maximal subgroup <span class="SimpleMath">"5:4 × A5"</span> of the Higman-Sims group do the following:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TableOfMarks("5:4xA5");</span>
TableOfMarks( "5:4xA5" )
</pre></div>
<p><a id="X8177A790811FBADC" name="X8177A790811FBADC"></a></p>
<h4>1.4 <span class="Heading">Administrative Functions</span></h4>
<p>Here we document some of the administrative facilities for the the <strong class="pkg">GAP</strong> library of tables of marks.</p>
<p><a id="X7E56B36A794A80C0" name="X7E56B36A794A80C0"></a></p>
<h5>1.4-1 LIBTOMKNOWN</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LIBTOMKNOWN</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p><code class="code">LIBTOMKNOWN</code> is a record that controls the loading of data files of the <strong class="pkg">GAP</strong> library of tables of marks. It has the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">ACTUAL</code> </strong></dt>
<dd><p>the name of the file to be read at the moment (set by <code class="code">SetActualLibFileName</code>),</p>
</dd>
<dt><strong class="Mark"><code class="code">LOADSTATUS</code> </strong></dt>
<dd><p>a record whose components are names of files in the library of tables of marks, with value a list whose first entry is one of <code class="code">"loaded"</code>, <code class="code">"unloaded"</code>, <code class="code">"userloaded"</code> and whose second entry is an integer that controls when the corresponding tables of marks can be removed from <strong class="pkg">GAP</strong>,</p>
</dd>
<dt><strong class="Mark"><code class="code">MAX</code> </strong></dt>
<dd><p><strong class="pkg">GAP</strong> remembers at most <code class="code">MAX</code> of the previously loaded library files (the default value is <span class="SimpleMath">10</span>),</p>
</dd>
<dt><strong class="Mark"><code class="code">UNLOAD</code> </strong></dt>
<dd><p>is it allowed to remove previously loaded library files (is set to <code class="keyw">true</code> by default),</p>
</dd>
<dt><strong class="Mark"><code class="code">STDGEN</code> </strong></dt>
<dd><p>a list describing standard generators of almost simple groups in the table of marks library.</p>
</dd>
</dl>
<p>Additionally the names of the files already loaded occur as components of <code class="code">LIBTOMKNOWN</code>; the corresponding values are given by the data of the files.</p>
<p><a id="X7BFF23287D8DF0F3" name="X7BFF23287D8DF0F3"></a></p>
<h5>1.4-2 IsLibTomRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLibTomRep</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Library tables of marks have their own representation. <code class="code">IsLibTomRep</code> tests if <var class="Arg">obj</var> is a library representation.</p>
<p><a id="X78F117087B49838D" name="X78F117087B49838D"></a></p>
<h5>1.4-3 TableOfMarksFromLibrary</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarksFromLibrary</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the table of marks with name <var class="Arg">string</var>.</p>
<p><a id="X854A6959795FCE91" name="X854A6959795FCE91"></a></p>
<h5>1.4-4 ConvertToLibTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConvertToLibTom</code>( <var class="Arg">record</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">ConvertToLibTom</code> converts a record with components from <code class="code">TableOfMarksComponents</code> into a library table of marks object with the corresponding attribute values set.</p>
<p><a id="X8799978F7F338677" name="X8799978F7F338677"></a></p>
<h5>1.4-5 SetActualLibFileName</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SetActualLibFileName</code>( <var class="Arg">filename</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Sets the file name for <var class="Arg">filename</var>.</p>
<p><a id="X7C86CE2B7C0F4842" name="X7C86CE2B7C0F4842"></a></p>
<h5>1.4-6 LIBTOM</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LIBTOM</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AFLT</code>( <var class="Arg">source</var>, <var class="Arg">destination</var>, <var class="Arg">fusion</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ACLT</code>( <var class="Arg">identifier</var>, <var class="Arg">component</var>, <var class="Arg">value</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The library format of a library table of marks is a call to the function <code class="code">LIBTOM</code>, with the arguments sorted as in <code class="code">TableOfMarksComponents</code> .</p>
<p><code class="code">AFLT</code> adds a fusion map <var class="Arg">value</var> from the table of marks with name <var class="Arg">source</var> to the table of marks with name <var class="Arg">destination</var>. The fusion map is a list of positive integers, storing at position <span class="SimpleMath">i</span> the position of the class in <var class="Arg">destination</var> that contains the subgroups in the <span class="SimpleMath">i</span>-th class of <var class="Arg">source</var>.</p>
<p><code class="code">ACLT</code> adds the value <var class="Arg">value</var> of a component with name <var class="Arg">component</var> to the table of marks with identifier value <var class="Arg">identifier</var> in the library of tables of marks.</p>
<p><a id="X7E9199A0852A94E8" name="X7E9199A0852A94E8"></a></p>
<h5>1.4-7 AllLibTomNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllLibTomNames</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list containing all names of available library tables of marks.</p>
<p><a id="X7912D7D487CF4461" name="X7912D7D487CF4461"></a></p>
<h5>1.4-8 NamesLibTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NamesLibTom</code>( <var class="Arg">string</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NamesLibTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: all names of the library table <var class="Arg">tom</var> or of the library table with name <var class="Arg">string</var></p>
<p><a id="X789A0FF487E231E4" name="X789A0FF487E231E4"></a></p>
<h5>1.4-9 NotifiedFusionsOfLibTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NotifiedFusionsOfLibTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NotifiedFusionsOfLibTom</code>( <var class="Arg">string</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionsOfLibTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionsOfLibTom</code>( <var class="Arg">string</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Are there any fusions from the library table of marks <var class="Arg">tom</var> or the table of marks with name <var class="Arg">string</var> into other library tables marks?</p>
<p><code class="code">NotifiedFusionsOfLibTom</code> returns the names of all such tables of marks. <code class="code">FusionsOfLibTom</code> returns the complete fusion maps. For that the corresponding library file has to be loaded.</p>
<p><a id="X83DE997E7A3698FB" name="X83DE997E7A3698FB"></a></p>
<h5>1.4-10 NotifiedFusionsToLibTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NotifiedFusionsToLibTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NotifiedFusionsToLibTom</code>( <var class="Arg">string</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionsToLibTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionsToLibTom</code>( <var class="Arg">string</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Are there any fusions from other library table of marks to <var class="Arg">tom</var> or the table of marks with name <var class="Arg">string</var>.</p>
<p><code class="code">NotifiedFusionsToLibTom</code> returns the names of all such tables of marks. <code class="code">FusionsToLibTom</code> returns the complete fusion maps. For that, the correponding library files have to be loaded.</p>
<p><a id="X838400737B446706" name="X838400737B446706"></a></p>
<h5>1.4-11 UnloadTableOfMarksData</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnloadTableOfMarksData</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">UnloadTableOfMarksData</code> clears the cache of tables of marks. This can be used to free up to several hundred megabytes of space in the current <strong class="pkg">GAP</strong> session.</p>
<p><a id="X7F8107C282BB991C" name="X7F8107C282BB991C"></a></p>
<h4>1.5 <span class="Heading">Standard Generators of Groups</span></h4>
<p>An <span class="SimpleMath">s</span>-tuple of <em>standard generators</em> of a given group <span class="SimpleMath">G</span> is a vector <span class="SimpleMath">(g_1, g_2, ..., g_s)</span> of elements <span class="SimpleMath">g_i ∈ G</span> satisfying certain conditions (depending on the isomorphism type of <span class="SimpleMath">G</span>) such that</p>
<ol>
<li><p><span class="SimpleMath">⟨ g_1, g_2, ..., g_s ⟩ = G</span> and</p>
</li>
<li><p>the vector is unique up to automorphisms of <span class="SimpleMath">G</span>, i.e., for two vectors <span class="SimpleMath">(g_1, g_2, ..., g_s)</span> and <span class="SimpleMath">(h_1, h_2, ..., h_s)</span> of standard generators, the map <span class="SimpleMath">g_i ↦ h_i</span> extends to an automorphism of <span class="SimpleMath">G</span>.</p>
</li>
</ol>
<p>For details about standard generators, see <a href="chapBib.html#biBWil96">[Wil96]</a>.</p>
<p><a id="X7984E27078B20557" name="X7984E27078B20557"></a></p>
<h5>1.5-1 StandardGeneratorsInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardGeneratorsInfo</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>When called with the group <var class="Arg">G</var>, <code class="func">StandardGeneratorsInfo</code> returns a list of records with at least one of the components <code class="code">script</code> and <code class="code">description</code>. Each such record defines <em>standard generators</em> of groups isomorphic to <var class="Arg">G</var>, the <span class="SimpleMath">i</span>-th record is referred to as the <span class="SimpleMath">i</span>-th set of standard generators for such groups. The value of <code class="code">script</code> is a dense list of lists, each encoding a command that has one of the following forms.</p>
<dl>
<dt><strong class="Mark">A <em>definition</em> <span class="SimpleMath">[ i, n, k ]</span> or <span class="SimpleMath">[ i, n ]</span></strong></dt>
<dd><p>means to search for an element of order <span class="SimpleMath">n</span>, and to take its <span class="SimpleMath">k</span>-th power as candidate for the <span class="SimpleMath">i</span>-th standard generator (the default for <span class="SimpleMath">k</span> is <span class="SimpleMath">1</span>),</p>
</dd>
<dt><strong class="Mark">a <em>relation</em> <span class="SimpleMath">[ i_1, k_1, i_2, k_2, ..., i_m, k_m, n ]</span> with <span class="SimpleMath">m > 1</span></strong></dt>
<dd><p>means a check whether the element <span class="SimpleMath">g_{i_1}^{k_1} g_{i_2}^{k_2} ⋯ g_{i_m}^{k_m}</span> has order <span class="SimpleMath">n</span>; if <span class="SimpleMath">g_j</span> occurs then of course the <span class="SimpleMath">j</span>-th generator must have been defined before,</p>
</dd>
<dt><strong class="Mark">a <em>relation</em> <span class="SimpleMath">[ [ i_1, i_2, ..., i_m ], <var class="Arg">slp</var>, n ]</span></strong></dt>
<dd><p>means a check whether the result of the straight line program <var class="Arg">slp</var> (see <span class="RefLink">Reference: Straight Line Programs</span>) applied to the candidates <span class="SimpleMath">g_{i_1}, g_{i_2}, ..., g_{i_m}</span> has order <span class="SimpleMath">n</span>, where the candidates <span class="SimpleMath">g_j</span> for the <span class="SimpleMath">j</span>-th standard generators must have been defined before,</p>
</dd>
<dt><strong class="Mark">a <em>condition</em> <span class="SimpleMath">[ [ i_1, k_1, i_2, k_2, ..., i_m, k_m ], f, v ]</span></strong></dt>
<dd><p>means a check whether the <strong class="pkg">GAP</strong> function in the global list <code class="func">StandardGeneratorsFunctions</code> (<a href="chap1.html#X867C9F52847B7A5E"><span class="RefLink">1.5-3</span></a>) that is followed by the list <span class="SimpleMath">f</span> of strings returns the value <span class="SimpleMath">v</span> when it is called with <span class="SimpleMath">G</span> and <span class="SimpleMath">g_{i_1}^{k_1} g_{i_2}^{k_2} ⋯ g_{i_m}^{k_m}</span>.</p>
</dd>
</dl>
<p>Optional components of the returned records are</p>
<dl>
<dt><strong class="Mark"><code class="code">generators</code></strong></dt>
<dd><p>a string of names of the standard generators,</p>
</dd>
<dt><strong class="Mark"><code class="code">description</code></strong></dt>
<dd><p>a string describing the <code class="code">script</code> information in human readable form, in terms of the <code class="code">generators</code> value,</p>
</dd>
<dt><strong class="Mark"><code class="code">classnames</code></strong></dt>
<dd><p>a list of strings, the <span class="SimpleMath">i</span>-th entry being the name of the conjugacy class containing the <span class="SimpleMath">i</span>-th standard generator, according to the <strong class="pkg">Atlas</strong> character table of the group (see <code class="func">ClassNames</code> (<span class="RefLink">Reference: ClassNames</span>)), and</p>
</dd>
<dt><strong class="Mark"><code class="code">ATLAS</code></strong></dt>
<dd><p>a boolean; <code class="keyw">true</code> means that the standard generators coincide with those defined in Rob Wilson's <strong class="pkg">Atlas</strong> of Group Representations (see <a href="chapBib.html#biBAGR">[WWT+]</a>), and <code class="keyw">false</code> means that this property is not guaranteed.</p>
</dd>
<dt><strong class="Mark"><code class="code">standardization</code></strong></dt>
<dd><p>a positive integer <span class="SimpleMath">i</span>; Whenever <code class="code">ATLAS</code> returns <code class="keyw">true</code> the value of <span class="SimpleMath">i</span> means that the generators stored in the group are standard generators w.r.t. standardization <span class="SimpleMath">i</span>, in the sense of Rob Wilson's <strong class="pkg">Atlas</strong> of Group Representations.</p>
</dd>
</dl>
<p>There is no default method for an arbitrary isomorphism type, since in general the definition of standard generators is not obvious.</p>
<p>The function <code class="func">StandardGeneratorsOfGroup</code> (<span class="RefLink">???</span>) can be used to find standard generators of a given group isomorphic to <var class="Arg">G</var>.</p>
<p>The <code class="code">generators</code> and <code class="code">description</code> values, if not known, can be computed by <code class="func">HumanReadableDefinition</code> (<a href="chap1.html#X7C6A2D3A8762F11B"><span class="RefLink">1.5-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">StandardGeneratorsInfo( TableOfMarks( "L3(3)" ) );</span>
[ rec( ATLAS := true,
description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4",
generators := "a, b",
script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
[ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ],
standardization := 1 ) ]
</pre></div>
<p><a id="X7C6A2D3A8762F11B" name="X7C6A2D3A8762F11B"></a></p>
<h5>1.5-2 HumanReadableDefinition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HumanReadableDefinition</code>( <var class="Arg">info</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ScriptFromString</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">info</var> be a record that is valid as value of <code class="func">StandardGeneratorsInfo</code> (<a href="chap1.html#X7984E27078B20557"><span class="RefLink">1.5-1</span></a>). <code class="func">HumanReadableDefinition</code> returns a string that describes the definition of standard generators given by the <code class="code">script</code> component of <var class="Arg">info</var> in human readable form. The names of the generators are taken from the <code class="code">generators</code> component (default names <code class="code">"a"</code>, <code class="code">"b"</code> etc. are computed if necessary), and the result is stored in the <code class="code">description</code> component.</p>
<p><code class="func">ScriptFromString</code> does the converse of <code class="func">HumanReadableDefinition</code>, i.e., it takes a string <var class="Arg">string</var> as returned by <code class="func">HumanReadableDefinition</code>, and returns a corresponding <code class="code">script</code> list.</p>
<p>If "condition" lines occur in the script (see <code class="func">StandardGeneratorsInfo</code> (<a href="chap1.html#X7984E27078B20557"><span class="RefLink">1.5-1</span></a>)) then the functions that occur must be contained in <code class="func">StandardGeneratorsFunctions</code> (<a href="chap1.html#X867C9F52847B7A5E"><span class="RefLink">1.5-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">scr:= ScriptFromString( "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4" );</span>
[ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], [ 1, 1, 2, 1, 13 ],
[ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">info:= rec( script:= scr );</span>
rec( script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
[ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">HumanReadableDefinition( info );</span>
"|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4"
<span class="GAPprompt">gap></span> <span class="GAPinput">info;</span>
rec( description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4",
generators := "a, b",
script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
[ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )
</pre></div>
<p><a id="X867C9F52847B7A5E" name="X867C9F52847B7A5E"></a></p>
<h5>1.5-3 StandardGeneratorsFunctions</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardGeneratorsFunctions</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p><code class="func">StandardGeneratorsFunctions</code> is a list of even length. At position <span class="SimpleMath">2i-1</span>, a function of two arguments is stored, which are expected to be a group and a group element. At position <span class="SimpleMath">2i</span> a list of strings is stored such that first inserting a generator name in all holes and then forming the concatenation yields a string that describes the function at the previous position; this string must contain the generator enclosed in round brackets <code class="code">(</code> and <code class="code">)</code>.</p>
<p>This list is used by the functions <code class="func">StandardGeneratorsInfo</code> (<a href="chap1.html#X7984E27078B20557"><span class="RefLink">1.5-1</span></a>)), <code class="func">HumanReadableDefinition</code> (<a href="chap1.html#X7C6A2D3A8762F11B"><span class="RefLink">1.5-2</span></a>), and <code class="func">ScriptFromString</code> (<a href="chap1.html#X7C6A2D3A8762F11B"><span class="RefLink">1.5-2</span></a>). Note that the lists at even positions must be pairwise different.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">StandardGeneratorsFunctions{ [ 1, 2 ] };</span>
[ function( G, g ) ... end, [ "|C(",, ")|" ] ]
</pre></div>
<p><a id="X7E5546217E418DE3" name="X7E5546217E418DE3"></a></p>
<h5>1.5-4 IsStandardGeneratorsOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsStandardGeneratorsOfGroup</code>( <var class="Arg">info</var>, <var class="Arg">G</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">info</var> be a record that is valid as value of <code class="func">StandardGeneratorsInfo</code> (<a href="chap1.html#X7984E27078B20557"><span class="RefLink">1.5-1</span></a>), <var class="Arg">G</var> a group, and <var class="Arg">gens</var> a list of generators for <var class="Arg">G</var>. In this case, <code class="func">IsStandardGeneratorsOfGroup</code> returns <code class="keyw">true</code> if <var class="Arg">gens</var> satisfies the conditions of the <code class="code">script</code> component of <var class="Arg">info</var>, and <code class="keyw">false</code> otherwise.</p>
<p>Note that the result <code class="keyw">true</code> means that <var class="Arg">gens</var> is a list of standard generators for <var class="Arg">G</var> only if <var class="Arg">G</var> has the isomorphism type for which <var class="Arg">info</var> describes standard generators.</p>
<p><a id="X830DDDC287DFBA0A" name="X830DDDC287DFBA0A"></a></p>
<h5>1.5-5 StandardGeneratorsOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardGeneratorsOfGroup</code>( <var class="Arg">info</var>, <var class="Arg">G</var>[, <var class="Arg">randfunc</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">info</var> be a record that is valid as value of <code class="func">StandardGeneratorsInfo</code> (<a href="chap1.html#X7984E27078B20557"><span class="RefLink">1.5-1</span></a>), and <var class="Arg">G</var> a group of the isomorphism type for which <var class="Arg">info</var> describes standard generators. In this case, <code class="func">StandardGeneratorsOfGroup</code> returns a list of standard generators of <var class="Arg">G</var>, see Section <a href="chap1.html#X7F8107C282BB991C"><span class="RefLink">1.5</span></a>.</p>
<p>The optional argument <var class="Arg">randfunc</var> must be a function that returns an element of <var class="Arg">G</var> when called with <var class="Arg">G</var>; the default is <code class="func">PseudoRandom</code> (<span class="RefLink">Reference: PseudoRandom</span>).</p>
<p>In each call to <code class="func">StandardGeneratorsOfGroup</code> (<span class="RefLink">???</span>), the <code class="code">script</code> component of <var class="Arg">info</var> is scanned line by line. <var class="Arg">randfunc</var> is used to find an element of the prescribed order whenever a definition line is met, and for the relation and condition lines in the <code class="code">script</code> list, the current generator candidates are checked; if a condition is not fulfilled, all candidates are thrown away, and the procedure starts again with the first line. When the conditions are fulfilled after processing the last line of the <code class="code">script</code> list, the standard generators are returned.</p>
<p>Note that if <var class="Arg">G</var> has the wrong isomorphism type then <code class="func">StandardGeneratorsOfGroup</code> (<span class="RefLink">???</span>) returns a list of elements in <var class="Arg">G</var> that satisfy the conditions of the <code class="code">script</code> component of <var class="Arg">info</var> if such elements exist, and does not terminate otherwise. In the former case, obviously the returned elements need not be standard generators of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a5:= AlternatingGroup( 5 );</span>
Alt( [ 1 .. 5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">info:= StandardGeneratorsInfo( TableOfMarks( "A5" ) )[1];</span>
rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5",
generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ],
standardization := 1 )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (3,4,5) ] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (1,2,3) ] );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">s5:= SymmetricGroup( 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction( s5, [ (1,3)(2,4), (3,4,5) ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> StandardGeneratorsOfGroup( info, a5 ), OnPairs ) <> fail;</span>
true
</pre></div>
<p><a id="X79316D2487927FF5" name="X79316D2487927FF5"></a></p>
<h5>1.5-6 StandardGeneratorsInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardGeneratorsInfo</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var>, a stored value of <code class="func">StandardGeneratorsInfo</code> (<span class="RefLink">???</span>) equals the value of this attribute for the underlying group (see <code class="func">UnderlyingGroup</code> (<span class="RefLink">Reference: UnderlyingGroup for tables of marks</span>)) of <var class="Arg">tom</var>, cf. Section <span class="RefLink">???</span>.</p>
<p>In this case, the <code class="func">GeneratorsOfGroup</code> (<span class="RefLink">Reference: GeneratorsOfGroup</span>) value of the underlying group <span class="SimpleMath">G</span> of <var class="Arg">tom</var> is assumed to be in fact a list of standard generators for <span class="SimpleMath">G</span>; So one should be careful when setting the <code class="func">StandardGeneratorsInfo</code> value by hand.</p>
<p>There is no default method to compute the <code class="func">StandardGeneratorsInfo</code> value of a table of marks if it is not yet stored.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a5:=TableOfMarks("A5");</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">std:= StandardGeneratorsInfo( a5 );</span>
[ rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5",
generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ]
], standardization := 1 ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># Now find standard generators of an isomorphic group.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= SL(2,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repeat</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x:= PseudoRandom( g );</span>
<span class="GAPprompt">></span> <span class="GAPinput">until Order( x ) = 2;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repeat </span>
<span class="GAPprompt">></span> <span class="GAPinput"> y:= PseudoRandom( g );</span>
<span class="GAPprompt">></span> <span class="GAPinput">until Order( y ) = 3 and Order( x*y ) = 5;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># Compute a representative w.r.t. these generators.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeTomByGenerators( a5, 4, [ x, y ] );</span>
Group([ [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ],
[ [ Z(2^2), Z(2^2)^2 ], [ Z(2^2)^2, Z(2^2) ] ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput"># Show that the new generators are really good.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">grp:= UnderlyingGroup( a5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso:= GroupGeneralMappingByImages( grp, g,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GeneratorsOfGroup( grp ), [ x, y ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGroupHomomorphism( iso );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBijective( iso );</span>
true
</pre></div>
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