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<div class="ChapSects"><a href="chap39_mj.html#X7C5563A37D566DA5">39 <span class="Heading"> Miscellaneous</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.html#X7CFDEEC07F15CF82">39.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7EBCEA9A7AD002DE">39.1-1 SL2Z</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X85B2390887156117">39.1-2 BigStepLCS</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7F121EAB85DE75A3">39.1-3 Classify</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87DEBDDA81F4A5E7">39.1-4 RefineClassification</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X798C13F97F955B72">39.1-5 Compose</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7C6A67F179AEB20A">39.1-6 HAPcopyright</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E722F7F7A7FD8B7">39.1-7 IsLieAlgebraHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8189AFE9871A8329">39.1-8 IsSuperperfect</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X780C131986E57C4D">39.1-9 MakeHAPManual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7FE5376D7F49D028">39.1-10 PermToMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87896E307D758299">39.1-11 SolutionsMatDestructive</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7B503C638795B001">39.1-12 LinearHomomorphismsPersistenceMat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7922CD92832949E5">39.1-13 NormalSeriesToQuotientHomomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7DBF782B7D06B0FA">39.1-14 TestHap</a></span>
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<h3>39 <span class="Heading"> Miscellaneous</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>39.1 <span class="Heading"> </span></h4>
<p><a id="X7EBCEA9A7AD002DE" name="X7EBCEA9A7AD002DE"></a></p>
<h5>39.1-1 SL2Z</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SL2Z</code>( <var class="Arg">p</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">‣ SL2Z</code>( <var class="Arg">1/m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a prime <span class="SimpleMath">\(p\)</span> or the reciprocal <span class="SimpleMath">\(1/m\)</span> of a square free integer <span class="SimpleMath">\(m\)</span>. In the first case the function returns the conjugate <span class="SimpleMath">\(SL(2,Z)^P\)</span> of the special linear group <span class="SimpleMath">\(SL(2,Z)\)</span> by the matrix <span class="SimpleMath">\(P=[[1,0],[0,p]]\)</span>. In the second case it returns the group <span class="SimpleMath">\(SL(2,Z[1/m])\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../tutorial/chap13.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">4</a></span> </p>
<p><a id="X85B2390887156117" name="X85B2390887156117"></a></p>
<h5>39.1-2 BigStepLCS</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BigStepLCS</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">\(G\)</span> and a positive integer <span class="SimpleMath">\(n\)</span>. It returns a subseries <span class="SimpleMath">\(G=L_1\)</span>><span class="SimpleMath">\(L_2\)</span>><span class="SimpleMath">\( \ldots L_k=1\)</span> of the lower central series of <span class="SimpleMath">\(G\)</span> such that <span class="SimpleMath">\(L_i/L_{i+1}\)</span> has order greater than <span class="SimpleMath">\(n\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutPerformance.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">2</a></span> </p>
<p><a id="X7F121EAB85DE75A3" name="X7F121EAB85DE75A3"></a></p>
<h5>39.1-3 Classify</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Classify</code>( <var class="Arg">L</var>, <var class="Arg">Inv</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list of objects <span class="SimpleMath">\(L\)</span> and a function <span class="SimpleMath">\(Inv\)</span> which computes an invariant of each object. It returns a list of lists which classifies the objects of <span class="SimpleMath">\(L\)</span> according to the invariant..</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap4.html">2</a></span> , <span class="URL"><a href="../tutorial/chap5.html">3</a></span> , <span class="URL"><a href="../tutorial/chap12.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutquasi.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTensorSquare.html">8</a></span> </p>
<p><a id="X87DEBDDA81F4A5E7" name="X87DEBDDA81F4A5E7"></a></p>
<h5>39.1-4 RefineClassification</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RefineClassification</code>( <var class="Arg">C</var>, <var class="Arg">Inv</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(C:=Classify(L,OldInv)\)</span> and returns a refined classification according to the invariant <span class="SimpleMath">\(Inv\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap4.html">1</a></span> , <span class="URL"><a href="../tutorial/chap12.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">4</a></span> </p>
<p><a id="X798C13F97F955B72" name="X798C13F97F955B72"></a></p>
<h5>39.1-5 Compose</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Compose</code>( <var class="Arg">f</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two <span class="SimpleMath">\(FpG\)</span>-module homomorphisms <span class="SimpleMath">\( f:M \longrightarrow N\)</span> and <span class="SimpleMath">\(g:L \longrightarrow M\)</span> with <span class="SimpleMath">\(Source(f)=Target(g)\)</span> . It returns the composite homomorphism <span class="SimpleMath">\(fg:L \longrightarrow N\)</span> .</p>
<p>This also applies to group homomorphisms <span class="SimpleMath">\(f,g\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutFunctorial.html">1</a></span> </p>
<p><a id="X7C6A67F179AEB20A" name="X7C6A67F179AEB20A"></a></p>
<h5>39.1-6 HAPcopyright</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HAPcopyright</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function provides details of HAP'S GNU public copyright licence.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7E722F7F7A7FD8B7" name="X7E722F7F7A7FD8B7"></a></p>
<h5>39.1-7 IsLieAlgebraHomomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLieAlgebraHomomorphism</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an object <span class="SimpleMath">\(f\)</span> and returns true if <span class="SimpleMath">\(f\)</span> is a homomorphism <span class="SimpleMath">\(f:A \longrightarrow B\)</span> of Lie algebras (preserving the Lie bracket).</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8189AFE9871A8329" name="X8189AFE9871A8329"></a></p>
<h5>39.1-8 IsSuperperfect</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSuperperfect</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">\(G\)</span> and returns "true" if both the first and second integral homology of <span class="SimpleMath">\(G\)</span> is trivial. Otherwise, it returns "false".</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X780C131986E57C4D" name="X780C131986E57C4D"></a></p>
<h5>39.1-9 MakeHAPManual</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MakeHAPManual</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function creates the manual for HAP from an XML file.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7FE5376D7F49D028" name="X7FE5376D7F49D028"></a></p>
<h5>39.1-10 PermToMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermToMatrixGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group <span class="SimpleMath">\(G\)</span> and its degree <span class="SimpleMath">\(n\)</span>. Returns a bijective homomorphism <span class="SimpleMath">\(f:G \longrightarrow M\)</span> where <span class="SimpleMath">\(M\)</span> is a group of permutation matrices.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">2</a></span> </p>
<p><a id="X87896E307D758299" name="X87896E307D758299"></a></p>
<h5>39.1-11 SolutionsMatDestructive</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SolutionsMatDestructive</code>( <var class="Arg">M</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(m×n\)</span> matrix <span class="SimpleMath">\(M\)</span> and a <span class="SimpleMath">\(k×n\)</span> matrix <span class="SimpleMath">\(B\)</span> over a field. It returns a k×m matrix <span class="SimpleMath">\(S\)</span> satisfying <span class="SimpleMath">\(SM=B\)</span>.</p>
<p>The function will leave matrix <span class="SimpleMath">\(M\)</span> unchanged but will probably change matrix <span class="SimpleMath">\(B\)</span>.</p>
<p>(This is a trivial rewrite of the standard GAP function <span class="SimpleMath">\(SolutionMatDestructive(\)</span><<span class="SimpleMath">\(mat\)</span>>,<<span class="SimpleMath">\(vec\)</span>>) .)</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7B503C638795B001" name="X7B503C638795B001"></a></p>
<h5>39.1-12 LinearHomomorphismsPersistenceMat</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinearHomomorphismsPersistenceMat</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a composable sequence <span class="SimpleMath">\(L\)</span> of vector space homomorphisms. It returns an integer matrix containing the dimensions of the images of the various composites. The sequence <span class="SimpleMath">\(L\)</span> is determined up to isomorphism by this matrix.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7922CD92832949E5" name="X7922CD92832949E5"></a></p>
<h5>39.1-13 NormalSeriesToQuotientHomomorphisms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalSeriesToQuotientHomomorphisms</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an (increasing or decreasing) chain <span class="SimpleMath">\(L\)</span> of normal subgroups in some group <span class="SimpleMath">\(G\)</span>. This <span class="SimpleMath">\(G\)</span> is the largest group in the chain. It returns the sequence of composable group homomorphisms <span class="SimpleMath">\(G/L[i] \rightarrow G/L[i+/-1]\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7DBF782B7D06B0FA" name="X7DBF782B7D06B0FA"></a></p>
<h5>39.1-14 TestHap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TestHap</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>This runs a representative sample of HAP functions and checks to see that they produce the correct output.</p>
<p><strong class="button">Examples:</strong></p>
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