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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>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X87A1E2E279597B8D">15.1-3 Bogomology</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X7C374F188523A659">15.1-4 Coclass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X819E8AEC835F8CD1">15.1-5 EpiCentre</a></span>
<span class="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">15.1-7 NonabelianSymmetricKernel</a></span>
<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="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X854F2DB382723504">15.1-12 TensorCentre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X79D037908746C65C">15.1-13 ThirdHomotopyGroupOfSuspensionB</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15_mj.html#X7967760B7E0F1E5F">15.1-14 UpperEpicentralSeries</a></span>
</div></div>
</div>
<h3>15 <span class="Heading"> Commutator and nonabelian tensor computations</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>15.1 <span class="Heading"> </span></h4>
<p><a id="X7DFB0FC4834DF183" name="X7DFB0FC4834DF183"></a></p>
<h5>15.1-1 BaerInvariant</h5>
<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>, <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 integer <span class="SimpleMath">\(c\)</span>><span class="SimpleMath">\(0\)</span>. It returns the Baer invariant <span class="SimpleMath">\(M^(c)(G)\)</span> defined as follows. For an arbitrary group <span class="SimpleMath">\(G\)</span> let <span class="SimpleMath">\(L^*_{c+1}(G)\)</span> be the <span class="SimpleMath">\((c+1)\)</span>-st term of the upper central series of the group <span class="SimpleMath">\(U=F/[[[R,F],F]...]\)</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>. This is an invariant of <span class="SimpleMath">\(G\)</span> and we define <span class="SimpleMath">\(M^{(c)}(G)\)</span> to be the kernel of the canonical homomorphism <span class="SimpleMath">\(M^{(c)}(G) \longrightarrow G\)</span>. For <span class="SimpleMath">\(c=1\)</span> the Baer invariant <span class="SimpleMath">\(M^(1)(G)\)</span> is isomorphic to the second integral homology <span class="SimpleMath">\(H_2(G,Z)\)</span>.</p>
< NQ.p
java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 139
<p><a id="X7D1614F87AAF5B97" name="< "java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 152
<h5>15.1-2 BogomolovMultiplier</h5>
<class"unc"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 226
<ivf"java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 254
<p>nputsa 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 subgroup of <span class="SimpleMath">\(H_2(G,Z)\)</span> generated by the images of all homomorphisms <span class="SimpleMath">\(H_2(A,Z) \rightarrow H_2(G,Z)\)</span> induced from abelian subgroups of <span class="SimpleMath">\(G\)</span>.</p>
<java.lang.StringIndexOutOfBoundsException: Range [143, 138) out of bounds for length 315
<p><strong class="button"span class""< ><class="ocss"&&<spana=html>.- java.lang.StringIndexOutOfBoundsException: Range [141, 138) out of bounds for length 149
<p><a id
<h5>15.1-3 Bogomology</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Bogomologyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Inputs a finite
<p><strong class="button">Examples:</strong> <span class="URL"><a href="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X7C374F188523A659" name="X7C374F188523A659"></a></p>
<h5>15.1-4 Coclass</h5>
<div class="func"><table class="func" width="100%"><tr><td classpInputsa group < =SimpleMath"\G\<span> and integer <span class="SimpleMath">\(c\)</span>><span class="SimpleMath">\(0\)</span>. It returns the Baer invariant <span class="SimpleMath">\(M^(c)(G)\)</span> defined as follows. For an arbitrary group <span
<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 251
<p><strong class="button">Examples:</strong></p>
<p><a id="X819E8AEC835F8CD1" name="X819E8AEC835F8CD1"></a></p>
<h5>15.1-5 EpiCentre</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EpiCentre</code>( <var class="Arg">G</var>, <
<class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#style='color: green'>8227; EpiCentre</code>( <var class="Arg">G</var> )</td><td div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BogomolovMultiplier</code>( <var class="Arg">G</var>, <vartd<td class"tdright">( function&)/td>/tr>table></div>
< crossed module <span class="SimpleMath">\(d:E \longrightarrow G\)</span> with <span class="SimpleMath">\(N=Image(d))</span> with <pan class="SimpleMath">\(Ker(d)\)</span> equal to the subgroup of <span class="SimpleMath">\(E\)</span> consisting of those elements on which <span class="SimpleMath">\(G\)</span> acts trivially.</p>
<p>If no value for <span class="SimpleMath">\(N\)</span> is entered then it is assumed that <span class="SimpleMath">\(N=G\)</span>. In this case the group <span class="SimpleMath">\(Z^\ast(G,G)\)</span> is trivial if and only if <span class="SimpleMath">\(G\)</span> is isomorphic to a quotient <span class="SimpleMath">\(G=E/Z(E
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.htmljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X7EAF8A2A79C86181" name="X7EAF8A2A79C86181"></a></p>
<h5>15.1-6 NonabelianExteriorProduct</h5>
<java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
<p>Inputs a
<ul>
<>java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 370
</li>
<li><p><span class="SimpleMath">\(E.pairing(x,y)\)</span> is a function which inputs an element <span class="p>Inputs a finite group <span class="SimpleMath">\(G\)</span> and positive integer <span class="SimpleMath">\(n\) span classSimpleMath"(GZ/(GG\java.lang.StringIndexOutOfBoundsException: Range [197, 196) out of bounds for length 607
</li>
</ul>
<p>This function should work for reasonably small nilpotent groups or extremely small non-nilpotent groups.</p>
<p><
<p><a id="X8155A3747A66FE81" name
<h5>15.
<="java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 232
< "unc"<table= "00"<tr>td""<codejava.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 258
<>a nilpotentinfinitegroup span =SimpleMath>\)/ returns abelianinvariantsof group span class="SimpleMath">\(SG\)</span> of the double suspension <span class="SimpleMath">\(SSK(G,1)\)</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="java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 0
<p>Alternatively, the optional second varible <span class="SimpleMath">\java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><strongdiv =">table class""width=100%>td class=tdleft"code ">#;EpiCentre</java.lang.StringIndexOutOfBoundsException: Range [117, 116) out of bounds for length 216
<p><a id="X7856BC54856452DE" name="p>Inputs a finite group <span class="SimpleMathG\)/span>and subgroup <span class="SimpleMath">\(N\)</span> and returns a subgroup <span class="SimpleMath">\(Z^\ast(G,N)\)</span> of the centre ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>5.1-8 NonabelianSymmetricSquare<h5>
<div class="func"><table class="func" width="100%java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="<h5>15.1-6 NonabelianExteriorProduct</h5>
<p>Inputs a finite or nilpotent infinite group <span class="SimpleMath">\(G\)</span> and
<ul>
<li><p><span <p> finite <panclass="impleMath">(G)/span>and subgroup <span class="SimpleMath">\(N\)</span>. It returns a record <span class="SimpleMath">\(E\)</span> with the following components.</p>
</li>
<li><p><span class="SimpleMath">\(T.pairing(x,y)\)</span> is a function which inputs two elements <span class="SimpleMath">java.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 to a multiple of the order of the symmetric square <span class="SimpleMath">\((G \tilde\otimes G)\)</span>. This might help java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>The optional second variable <span java.lang.StringIndexOutOfBoundsException: Range [0, 43) out of bounds for length 0
<
<p><strong class="button<div java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 258
<p><a id="X829476FA82D5759B" name="X829476FA82D5759B"></a></p>
<h5>15.1-9 NonabelianTensorProduct</h5>
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<p>ajava.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 214
<ul>
<li><p><span class="java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
</li>
<li><p><span class="SimpleMath">\(E.pairing(x,y)\)</span> is a function which inputs an element <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
</ul>
<p>This function should work for reasonably small nilpotent java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutNonabelian.html">1</a></span> </p>
<p><a id="X7C0DF7C97F78C666" name="X7C0DF7C97F78C666"></adiv class="func"><table class="func" width="100%"><tr><td "impleMath">(\<span a <java.lang.StringIndexOutOfBoundsException: Range [112, 111) out of bounds for length 178
<h5>15.1-10 NonabelianTensorSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 5
<p;function )</td></tr<<div>
<p>Inputs a finite or nilpotent infinite group <span class="SimpleMath">\(G\)</span> and returns a record <span class="SimpleMath">\(T\)</span> with the following components.</p>
<ul
<li><p><spanpAn optional <spanclass="">(m)/>be equaltoamultiple order symmetricsquare span>(G tildeotimes )\<span> might helpwhen <pan S>\G\)</> butnot nilpotent(especially <class"">\(m\)<span> is reasonable accurate) boundjava.lang.StringIndexOutOfBoundsException: Range [401, 400) out of bounds for length 445
</li>
<java.lang.StringIndexOutOfBoundsException: Range [5, 3) out of bounds for length 329
</li>
</ul>
<p>An optional second varible <span class="SimpleMath">\(m\)</span> can be set equal to a
<p>The java.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 0
<p>function work for small groups extremely non-groups./java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
<p><strong class="java.lang.StringIndexOutOfBoundsException: Index 23 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 class="java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 0
<p>Inputs a finite group <span class
<p><span class="SimpleMath">\(\ldots\longrightarrow H_3(G,Z) \longrightarrow H_3(G/N,Z) \longrightarrow H_2(G,N
<p>This function should work for reasonably small nilpotent groups <span class="SimpleMath">\(G\)</span> or java.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 0
<>strong ""></ span class"<java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 139
<p><a id="X854F2DB382723504p>Inputsa java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 178
<h5>15.1-12 TensorCentre</h5>
<div class="java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 0
<> a group <spanclass=SimpleMath>(\</>andreturns largestcentral span =SimpleMath"(\)<span> such thattheinducedhomomorphism nonabeliantensor < ">((G otimes )\(/N\GN\)span>is isomorphism.Equivalently <spanclass=SimpleMath"\N\<span>is largest central that span class">\(\pi_3((,))\java.lang.StringIndexOutOfBoundsException: Range [454, 453) out of bounds for length 497
<p><strong class="button">Examples:</strong></p>
<p><a id="X79D037908746C65C" name="X79D037908746C65C"></a></p>
</li>
<div class="func"><table class="func" width="100%"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div=func>< class=func =100%><><td ""<class=func">#227;ThirdHomotopyGroupOfSuspensionBcode(< ="Arg><var,<class""m<var <td<td ">(&function />/>/<div>
<p>Inputs a finite or nilpotent infinite group <span class="java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 0
<p>For non-nilpotent groups the implementation of the function <span class="SimpleMath">\(ThirdHomotopyGroupOfSuspensionB(G)\)</span> is far from optimal and will soon be java.lang.StringIndexOutOfBoundsException: Index 179 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [51, 16) out of bounds for length 505
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X7967760B7E0F1E5F" name="X7967760B7E0F1E5F"></a></p>
<h5
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Inputs a nilpotent group span class=SimpleMath">(\<span and an integer<spanclass=SimpleMath">()/. Itreturns spanclassSimpleMath"\c\<span-thtermof theupper epicentralseries span java.lang.StringIndexOutOfBoundsException: Range [224, 223) out of bounds for length 411
<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_cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>15.-12TensorCentre<h5>
<p>strong "">xamples:<strong><java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 139
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