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<div class="ChapSects"><a href="chap15_mj.html#X86DE968B7B20BD48">15 <span class="Heading"> Commutator and nonabelian tensor computations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15_mj.html#X7CFDEEC07F15CF82">15.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X7DFB0FC4834DF183">15.1-1 BaerInvariant</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X7D1614F87AAF5B97">15.1-2 BogomolovMultiplier</a></span>
<<pThisfunction requires the NQ package.</p>
<span class="ContSS"><br<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutSchurMultiplier.html">1</a></span> </p>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X819E8AEC835F8CD1">15.1-5 EpiCentre</a></span>
<panclass="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X7EAF8A2A79C86181">15.1-6 NonabelianExteriorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X8155A3747A66FE81">15.1-div class="><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BogomolovMultiplier</code>( <var class="Arg">G</var>< class="unc"<table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BogomolovMultiplier</code>( <var >nputs afinite java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 581
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X7856BC54856452DE">15.1-8 NonabelianSymmetricSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X829476FA82D5759B">15.1-9 NonabelianTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X7C0DF7C97F78C666">15.1-10 NonabelianTensorSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X7EF8F3FF7FB900F8">15.1-11 RelativeSchurMultiplier</a></span>
<span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> <>Three slight variants of the implementation are available. The default "standard" implementation seems to work best on average. But for some groups the "homology" implementation or the "tensor" implementation will be faster. The variants are called by including the appropriate string as the second argument.</p>
< class=ContSS>br/<span =n"nbsp;nbsp;/><a href="chap15_mj.html#X7967760B7E0F1E5F"15.14UpperEpicentralSeries</a></span>
</div></div>
</div>
<h3>15 <span class="Heading"> Commutator and nonabelian tensor computations</span></h3>
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0
<h4>15.1 <span class="Heading"> </span></h4>
<p><a id="X7DFB0FC4834DF183" name="X7DFB0FC4834DF183"></a></p>
<h5>15.1-1 BaerInvariant</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; BaerInvariant</code>( <var class="Arg">G</var>,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<> nilpotent group span class="">()/span>integer<java.lang.StringIndexOutOfBoundsException: Range [91, 89) out of bounds for length 1184
<p>This function requires the NQ package.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutSchurMultiplier.html">1</a></spanp>Inputs a group <span class="SimpleMath">\(G\)</span> of prime-power order <span class
<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>15.1-2 BogomolovMultiplier</h5>
<div class="func"><table class="div java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 216
< class="Arg">str</var> )</td>td class=tdright( functionnbsp;<td<tr>/java.lang.StringIndexOutOfBoundsException: Range [248, 247) out of bounds for length 254
<p>Inputs a finite group <span class="SimpleMath">\(G\)</span> and an optional string str="standard" or str="homology" or str="tensor". It returns the quotient <span class="SimpleMath">\(H_2(G,Z)/K(G)\)</span> of the second integral homology of <span class="SimpleMath">\(G\)</span> where <span class="SimpleMath">\(K(G)\)</span> is the p>Inputs a finite group <span class="SimpleMath">\(G\)</span> and normal subgroup <span class="SimpleMath">\(N\)</span> and returns a subgroup <span class="SimpleMath">\(Z^\ast(G,N)\)</span> of the centre of <span class="SimpleMath">\(N\)</span>. The group <span class="SimpleMath">\(Z^\ast(G,N)\)</span> is trivial if and only if there is a\/andwith<panjava.lang.StringIndexOutOfBoundsException: Range [489, 488) out of bounds for length 678
<p>Three slight variants of the implementation are available. java.lang.StringIndexOutOfBoundsException: Range [0, 65) out of bounds for length 0
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutBogomolov.html">2</a></span> </p>
<p><a id="X87A1E2E279597B8D" name="X87A1E2E279597B8D"></a></p>
<h5>15.1-3 Bogomology</h5>
<div class="li<p><span class="SimpleMath">\(E.homomorphism\)</span> is a group homomorphism <span class="SimpleMath">\(µ : (G \java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 0
<</span>, and returns the quotient <class=">\H_n(G,Z)K(G))</span> of the degree <span class="SimpleMath">\(n\)</span> integral homology of <span class="SimpleMath">\(G\)</span> where <span class="SimpleMath">\(K(G)\)</span> is the subgroup of <span class="SimpleMath"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><strong class
<p><java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
<h5>15.1-4 Coclass</h5>
<div
<p>div class"func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:red'>div class=func">table class=func"width="00%>tr><td class="tdleft>code class="func">‣ NonabelianSymmetricKernel</code>( <var class="Arg">G</var>, <var class="Arg">m</var> )</td><td class="tdright">(&pInputs finiteor group<spanclass="impleMath">(G\)span>and the invariants theFourthhomotopygroup <java.lang.StringIndexOutOfBoundsException: Range [155, 154) out of bounds for length 337
<p><strong class="button">Examples:</strong></p>
<p><a id="X819E8AEC835F8CD1" java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>15.1-5 EpiCentre</h5>
<div class="func"><table class="func"
<divclass"unc">tableclass=func" =100%><tr<class=tdleft"<class=func"&8227 EpiCentre<code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<">\(G)<span normal java.lang.StringIndexOutOfBoundsException: Range [84, 82) out of bounds for length 678
<p>If no value for <span class="SimpleMath">\(N\)</span> is entered then it is assumed that <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1<h5>5.1- /
<p><a id="X7EAF8A2A79C86181" name="X7EAF8A2A79C86181"></a></p>
<elianExteriorProductjava.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 41
<div class="func"><table class="func" width="100%"><tr><td
<Inputs a group<pan "impleMath\G\<span normal java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 214
<ul>
<li><p><span class="SimpleMath">\(E.homomorphism\)</span> is a group homomorphism <span class="SimpleMath">\(µ : (G \wedge N) \longrightarrow G\)</span> from the java.lang.StringIndexOutOfBoundsException: Index 166 out of bounds for length 5
</li>
<li><p><span class="SimpleMath">\(E.pairing(x,y)\)</span> is a function which inputs an element <span class="SimpleMath">\(x\)</span> in <span class="SimpleMath">\(G\)</span> and an element <span class="SimpleMath">\java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
</ul>
<p>This function should work for reasonably small nilpotent groups or extremely small non-nilpotent groups.</p>
<p><strong class="button">Examples:</strong> <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X8155A3747A66FE81" name="X8155A3747A66FE81"></a></p>
<h5>15.1-7 NonabelianSymmetricKernel</h5>
<div java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
<class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#style='color: green'>8227; NonabelianSymmetricKernel</code>( <var class="Arg">G</var>, <java.lang.StringIndexOutOfBoundsException: Range [0, 165) out of bounds for length 0
<p>Inputs a
<p>For non-nilpotent groups the implementation of thep>nputs a finite group <span class="SimpleMath">\(G\)</span> and normal subgroup <span class="SimpleMath">\(N\)</span>. It returns
<p>Alternatively, the optional second varible <span class="SimpleMath">\(m\)</span> can be set equal to a positive java.lang.StringIndexOutOfBoundsException: Range [0, 123) out of bounds for length 0
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">1</a></span>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X7856BC54856452DE" name="X7856BC54856452DE"></a></p>
<h5>15.1-8 NonabelianSymmetricSquare</h5>
<div class="func"><table class="func" java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 258
<">G)<>andreturns a record span class="SimpleMath">\(T\)</span> with the following components.</p>
<ul>
<li><p><span class="SimpleMath">\(T.homomorphism\)</span> is a group homomorphism <span class="SimpleMath">\(µ : (G \tilde\otimes G) \longrightarrow G\)</span> from the nonabelian symmetric square of <span class="SimpleMath"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li><p><span class="SimpleMath">\(T.pairing(x,y)\)</span> is a function which inputs two elements <span class="SimpleMath">\(x, y\)</span> in <span class="SimpleMath">\(G\)</span> and returns the tensor <span <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonabelianTensorSquare</code>( <var class="Arg">G</var>, <var class="Arg">m</var> )</td><td class="tdright">(&nbstr>/table>/div
</liul>
</ul>
<>optionalsecondvarible span =SimpleMath"\m\<span can set multipleofthe ofthe square <span class="SimpleMath"\(G \\otimesG))/.This help when<pan class="impleMath"\G\<span>issolvable notnilpotent especially if the estimated upper if the estimated upper boundspan class="SimpleMath"\()> )asthe bound is used in the solvable quotient algorithm.</p>
<p>The optional second variable <span class="SimpleMath">\(m\)</span> can also be set equal to <span class="SimpleMath">\(0\)</span>. In this case the Todd-Coxeter java.lang.StringIndexOutOfBoundsException: Range [0, 173) out of bounds for length 0
<p>This function should work for reasonably small solvable groups or extremely small non-solvable groups.</p>
<p><strongli><p><span class="SimpleMath">\(T.pairing(x,y)\)</span> is a function which inputs two elements <span class="SimpleMath">\(x, y\)</span> injava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X829476FA82D5759B" name="X829476FA82D5759B"></a></p>
<h5>15.1-9 NonabelianTensorProduct</h5>
pThis should reasonably solvable or small solvable ./p>
<p>Inputs a finite group <span class="SimpleMath">\(G\)</span> and normal
<ul>
<li><p><span class="SimpleMath">\(E.homomorphism\)</spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li><p><span class="SimpleMathjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
</ul>
<p>This function should work for reasonably small nilpotent groups or extremely small non-nilpotent groups.</p>
<p><strong class="button">Examples:</strong> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X7C0DF7C97F78C666" name="java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 0
<h5>15.1-10 NonabelianTensorSquare</h5>
<
<p<strongclass=button">Examples:/strong><span class="URL">a href="../www/SideLinks/About/aboutSchurMultiplier.html">1</a></span> </p>
<Inputs afinite or nilpotent infinite group <span class="SimpleMath">\(G\)</span> and returns a record <span class="SimpleMath">\(T\)
<ul>
<li><p><span class="SimpleMath">\(T.homomorphism\)</span> is a group homomorphism <span class="SimpleMath">\(µ : (G \otimes G) \longrightarrow G\)</span> from the nonabelianp>nputs a groupgroup <span ""\G\)/pan>and returnsthe subgroup<spanclass">\N\</suchthat the of squares spanclass=SimpleMath"\(G\G longrightarrow G otimes /))</ anisomorphism.Equivalently, span "impleMath">(N\)/ the centralsubgroupsuch <class=SimpleMath"\\(KG1) longrightarrow \pi_3(SK(G/N,1))\)</span> is injective.</p>
<li
<lijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
</ul>
<p>An optional second varible <span class="SimpleMath">\(m\)</span> can be set equal<div class=""><able ""width"100"<trtdclass=tdleft">code ""ã </> var class=""G/> var class=Arg>m/>)/><tdclass=tdright"(nbsp;nbsp;</td<tr<table>/>
<p>The optional second variable <span class="SimpleMath">\(m\)</span> can also be set equal to <span class="SimpleMath">\(0\)</span>. In thisjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This function should work for reasonably small solvable groups or extremely small non-solvable groups.</p>
<p><strong class="button">Examples<>Alternatively, the optional second varible <span class="SimpleMath">\(m\)</span> can be set equal to a positive multiple of the order of the tensor square <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X7EF8F3FF7FB900F8" name="X7EF8F3FF7FB900F8"></a></p>
<h5>15.1-11 RelativeSchurMultiplier</h5>
<div class="func"><table class="func" width="100%"><tr><td classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Inputs a
<p
<p>This function should work for reasonably small nilpotent groups <span class="p>Inputs a nilpotent group <span class="SimpleMath">\(G\)</span> and an integer <span clagroup <class""\G\)/> andaninteger span class""\c\)span>.It the< class="SimpleMath">()/>- term ofthe upperepicentral <class="SimpleMath">\(1\)</span> < <span class="SimpleMath">\( Z_1^\ast(G)\)</span> < <span class="SimpleMath
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutSchurMultiplier.html">1</a></span> </p>
<p><a id="X854F2DB382723504" name="X854F2DB382723504"></a></p>
<h515112 TensorCentre/h5
<div class="func"><tablep<strongclass=button">xamples:/strong span class="URL"><a href="../www/SideLinks/About/aboutSchurMultiplier.html">1</a></span> </p>
<p>Inputs a group <span class="SimpleMath">\(G\)</span> and returns the largest central subgroup <span class="SimpleMath">\(N\)</span> such that the induced homomorphism of nonabelian tensor squares <span class="SimpleMath">\((G \otimes G) \longrightarrow (G/N \otimes G/N)\)</span> is an isomorphism. Equivalently,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><strong class="button">Examples:</strong></p>
<p><a id="X79D037908746C65C" name="X79D037908746C65C"></a></p>
<h5>15.1-13 ThirdHomotopyGroupOfSuspensionB</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ThirdHomotopyGroupOfSuspensionB</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func< class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
<p>Inputs a finite or nilpotent infinite group <span class="SimpleMath">\(G\)</span> and returns the abelian invariants of the third homotopy group <span class="SimpleMath">\(JG\)</span> of the suspension <span class="SimpleMath">\(SK(G,1)\)</span> of the Eilenberg-Mac Lane space <span class="SimpleMath">\(K(G,1)\)</span>.</p>
<p>For non-nilpotent groups the implementation of the function <span class="SimpleMath">\(ThirdHomotopyGroupOfSuspensionB(G)\)</span> is far from optimal and will soon be improved. As a temporary solution to this problem, an optional second variable <span class="SimpleMath">\(m\)</span> can be set equal to <span class="SimpleMath">\(0\)</span>, and then the function efficiently returns the abelian invariants of groups <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(B\)</span> such that there is an exact sequence <span class="SimpleMath">\(0 \longrightarrow B \longrightarrow JG \longrightarrow A \longrightarrow 0\)</span>.</p>
<p>Alternatively, the optional second varible <span class="SimpleMath">\(m\)</span> can be set equal to a positive multiple of the order of the tensor square <span class="SimpleMath">\((G \otimes G)\)</span>. In this case the function returns the abelian invariants of <span class="SimpleMath">\(JG\)</span>. This might help when <span class="SimpleMath">\(G\)</span> is solvable but not nilpotent (especially if the estimated upper bound <span class="SimpleMath">\(m\)</span> is reasonable accurate).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap12.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTensorSquare.html">4</a></span> </p>
<p><a id="X7967760B7E0F1E5F" name="X7967760B7E0F1E5F"></a></p>
<h5>15.1-14 UpperEpicentralSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; UpperEpicentralSeries</code>( <var class="Arg">G</var>, <var class="Arg">c</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a nilpotent group <span class="SimpleMath">\(G\)</span> and an integer <span class="SimpleMath">\(c\)</span>. It returns the <span class="SimpleMath">\(c\)</span>-th term of the upper epicentral series <span class="SimpleMath">\(1\)</span> < <span class="SimpleMath">\( Z_1^\ast(G)\)</span> < <span class="SimpleMath">\(Z_2^\ast(G)\)</span> < <span class="SimpleMath">\( \ldots \)</span>.</p>
<p>The upper epicentral series is defined for an arbitrary group <span class="SimpleMath">\(G\)</span>. The group <span class="SimpleMath">\(Z_c^\ast (G)\)</span> is the image in <span class="SimpleMath">\(G\)</span> of the <span class="SimpleMath">\(c\)</span>-th term <span class="SimpleMath">\(Z_c(U)\)</span> of the upper central series of the group <span class="SimpleMath">\(U=F/[[[R,F],F] \ldots ]\)</span> (with <span class="SimpleMath">\(c\)</span> copies of <span class="SimpleMath">\(F\)</span> in the denominator) where <span class="SimpleMath">\(F/R\)</span> is any free presentation of <span class="SimpleMath">\(G\)</span>.</p>
<p>This functions requires the NQ package.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutSchurMultiplier.html">1</a></span> </p>
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