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<title>GAP (HAP commands) - Chapter 15:  Commutator and nonabelian tensor computations</title>
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<pp>aid""<<p
div=>="chap15.html#X86DE968B7B20BD48">15 <spanjava.lang.StringIndexOutOfBoundsException: Range [79, 77) out of bounds for length 145
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<spanclass=ContSS< >span class"">nbsp&;/>ahrefchap15htmlX7DFB0FC4834DF183>.- BaerInvariant</a<span
<span
< =
< =ContSS< >span="&;/>ahref".>. java.lang.StringIndexOutOfBoundsException: Range [128, 126) out of bounds for length 137
< =C">br >span =nocss>&bsp;&;/span><ahref"html">15.12 BogomolovMultiplier</></span
<span "> /< class""&;nbsp</><a href=chap15.#819"1515EpiCentre<a<span
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< =ContSS< >span="  /> href=chap15html#>1515EpiCentre/>/span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X7856BC54856452DE">15.1-8 NonabelianSymmetricSquare</a></span>
<span classspan =>/< =nocss&;nbsp;/><ahref="chap15html#X7C0DF7C97F78C666">15.1-10 NonabelianTensorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X7C0DF7C97F78C666">15.1-10 NonabelianTensorSquare< class"ContSS">br/< "nocss">&;&bsp<span< =chap15htmlX7EF8F3FF7FB900F8"15.111RelativeSchurMultiplier</a></span>
< ="">br/< ""&;nbsp/>ahrefchap15java.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 148
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<class"">br><&&;<spanahrefchap15#151- UpperEpicentralSeries<a<span
</div></div>
</p><aid""name"">/></p

<>15<span class=Heading Commutator and  tensorcomputations<span<h3

<>aidX7CFDEEC07F15CF82name""<a<p

<h4p<=X7DFB0FC4834DF183 =<a<p

<h5>1- <h5

<

<p>    spanclassSimpleMath><>and span">/>gt< =">/. the  spanSMc)/defined follows   arbitrary   ="/let span="L*c1/>bethe span="1<>st    the  "">U=/[R,].]/>( spanclass""c ofclass""<>in < =SimpleMath>//span  anyof< =>/>    java.lang.StringIndexOutOfBoundsException: Range [716, 715) out of bounds for length 1092
<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=java.lang.StringIndexOutOfBoundsException: Index 447 out of bounds for length 0

<p>This function requires the NQ package.</p>

<p><strong class="button">Examples:</strong> <span class="URLdiv class="func>="unc width="100%<>tdclass"">code =func>#;BogomolovMultiplier<code varclass""><var varclass""></ar <tdtd =tdright(nbspfunctionnbsp;<td<tr/></>

<p><a id="X7D1614F87AAF5B97" name="java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0

<h5>15.1-2 

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func<><strong class="buttonExamples:/trong<class"">ahref".tutorial/."1<a<span <class="">href.wwwSideLinksAboutaboutBogomolov."2/a>/> </>
<class""< =func =100">tr>td class="dleft< =func&8227 /> < =""Gvar< Arg>/><td="tdright">&function<td/>tablediv
<p>Inputs a finite group

<pInputs    spanclass=G/>andinteger spanclassn/,    spanclass=""H_n,)K)     =SimpleMath>span homology  span=SimpleMath>/>span=<  the subgroupof<spanclass="impleMath>H_n(G,)</span>generatedbytheimages of all homomorphisms span class=SimpleMath>(A,Z)→H_nG,Z</> induced from abelian subgroupsof <pan class="SimpleMath"G</pan>.<p>

<h515.4/h5>

<>aid"" name"X87A1E2E279597B8D"></a></p>

<h5>15.1-3 Bogomology</h5>

< class=func>tableclass=width"%">td=tdleft"<codeclass">;/> ""G/>,< =Argn/>)/>td=tdright(nbspnbsp;<td<>/>/>
<p>Inputs

<>strongclassbutton<strong>< =URL< =".www/ideLinks//."1/>/><p

<p><

<

< =func>< "func" =100"<>tdclass""classfunc"&#; />/><class=tdright>&global  )td<tr<table<div
div=>table"" 100"<>< class""<code "unc&8227;/> <var =ArgG/ar,var=>N<var <tdtd=tdright(nbspfunction;<td<tr<table/

<>class"">:/>/>

<pp>    span="G/> normalsubgroup< =SimpleMathN/>and  a subgroup <span class="SimpleMath"^(,)/  centre  spanclass=SimpleMath"NspanZG)/>is    if  is acrossed  span=SimpleMath">:E  G</pan>with<span class=SimpleMath>N=Image(d)<span and with <span 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>

<h5>15.1-5 EpiCentre<

<div class="func"><table class="func" width="100%"><tr
< ="func">tableclass=func width=100">tr><td class="tdleft"<code java.lang.StringIndexOutOfBoundsException: Range [92, 85) out of bounds for length 216
<> =SimpleMath><span  normal subgroup <span class="SimpleMath"><span  returns  < =SimpleMathZ^(,)/>ofthecentreof< =SimpleMathN/>  < ="SimpleMath>^(,)/>istrivial   only  there isacrossedmodule<span class="SimpleMath">d:E ⟶ G</span> with <span class="SimpleMath">N=Image(d)</span> and with <span 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>

<>If novalue for span class=SimpleMath"N</span is entered thenitisassumedthat<class"SimpleMath">NG/span.Inthiscase  group<class=SimpleMath>Z∗G,)</span istrivialifandonly spanclass"SimpleMath"G<span is  to a  <span class=SimpleMath>GE/(E)</span>  group< ="E/pan     <spanclass"impleMath></>. (ee also the  <span =SimpleMath>UpperEpicentralSeries(Gc</>.)<p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1<<p><a ="7EAF8A2A79C86181" "X7EAF8A2A79C86181">/a/p

<p><a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>15.1-6 NonabelianExteriorProduct<pInputsafinitegroup span G/>andnormalsubgroup spanclass""><span> It returnsarecord< ="impleMath">E</span> with the following components.</p>

<div class="func"><table class
<<ul>


<ul
</i>

<<li><p< class"SimpleMath"Epairing(,)/ afunction  anelement< =SimpleMathx<> in =java.lang.StringIndexOutOfBoundsException: Range [154, 152) out of bounds for length 370
<<< =>xy)span functioninputsanelementspan=>/>inspanclass""G/>andan java.lang.StringIndexOutOfBoundsException: Range [178, 177) out of bounds for length 370

</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> <span class="URL"><a href="../www/SideLinks/About/aboutNonabelian.html">1</a></span> </p>

<p><a id="X8155A3747A66FE81" name="X8155A3747A66FE81"></a></p>

<>17/>

< =func< class"" =100%>tr< =" =func"&8227 <code( varclass""><var> )/>td class"tdright">&functionnbsp)/></>/
<
p afiniteorinfinite  spanclass=>/>and theabelianinvariantsofthe  < =SimpleMathSG/> ofthedouble  <span class=SimpleMath"(G1<span of  Eilenberg- space< =SimpleMath>KG1<span./>

<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div =func>tableclass"" ="100%>tr< class=">codeclass""&8227 <>( <var class""G/>)td ="(nbspfunction&;)td<tr<table<div

<p><strong class="button">Examples:</strong> <span class="URL"><ap afiniteornilpotentinfinite  spanclass"impleMath>G/> and theabelianinvariants  Fourthhomotopy <span class/>ofthe   spanclass"impleMath>(,)span   -   spanclass=SimpleMathKG1<span><p>

<p><a id="X7856BC54856452DE" name="X7856BC54856452DE"></a></

<h5p>strong =>:<strong < "">ahref=.aboutIntrohtml><a<><p

<
< =func> =funcwidth"%><>tdclass=tdleft"><code class="func">&#8227; NonabelianSymmetricSquare</code>( <var class="Arg">G<
<>afiniteornilpotent  spanclass""><span  arecord< =SimpleMath>/>withthefollowing components</>


<ul>
<li><p><span class="SimpleMath">T.homomorphism</span> is a group homomorphism <span class="SimpleMath">µ : (G tilde⊗ G) ⟶ G</span> from the nonabelian symmetric square of <span class="SimpleMath">G</span> to <span class="SimpleMath">G</span>. The kernel of <span class="SimpleMath">µ</span> is isomorphic to the fourth homotopy group of<pafinite nilpotentinfinite  spanclass=><span  arecordspan =SimpleMathT/>withthefollowing ./>

</li>
<li><p><span class="SimpleMath">T.pairing(x,y)</span> is a function which inputs two elements <span<li><>span =SimpleMath>.homomorphism<span> isagrouphomomorphismspan=SimpleMathµ:( ⊗G  G/> from the symmetricsquare  <span class"SimpleMath">G</span> to <span class="SimpleMath">G</span>. The kernel of <span class="SimpleMath">µ</span> is isomorphic

</li>
</ul>
<p>An optional second varible <span class="SimpleMath">m</span> can be set equal to a multiple of

<>Theoptional variable < class"">/>canalsobeset equal  spanclass=>/>.In casethe Coxeter  willbe  enumeratethe squareeven  spanclass=SimpleMath><span  solvable./>

<p>This function should workpAn optional second  spanclass=><span   setequal  multipleoftheorder   square< class=SimpleMath>Gtilde⊗)/>.Thismight when< "SimpleMath"G/>issolvablenot ( theupperbound< =SimpleMathm/>isreasonable  thebound     solvablequotient./>

<p><strong class=

<java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

<h5>15.1-9 NonabelianTensorProduct</h5>

<div class="func"><table class="func" width="100%"><tr>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Inputs a 


<
<divclass"unc"< ="unc" =100%<tr<tdclass=tdleft"<codeclass"func"ã </code>(varclassArg><var>,var class=""N/>)/td>tdclass="tdright"> function )/td>/tr>/><div>

</li>
<li><p><span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
</ul>
<p>This<li>span=SimpleMathE.pairing(x,y)</span> is a function which inputs an element<spanclass""><span  spanclass"">/>   spanclass=>/>  < class"">N<span  classSimpleMathx y<spanin product span="G />./>

<>strongclass=button><strong spanclass" =.///About/.html">/><span />

<p<>a =X7C0DF7C97F78C666name"">/>/>

<>.110 />

< =func< class=""width100%>tdclass="tdleft"<code ""&8227 java.lang.StringIndexOutOfBoundsException: Range [124, 123) out of bounds for length 229
<div class="func"><table class="func" divclass""> class"" =%> =tdleft"< class=func>#; NonabelianTensorSquare/> <var class="ArgG/> varclass""m << =>&; )/<tr<>
<p>Inputs a finite or nilpotent infinite<>afiniteornilpotentgroupspanclass="SimpleMath"G/  a  class"/ withfollowing./>


<l>
< Thomomorphism/ is  homomorphism spanclassSimpleMathµ G) <spanfromthenonabelian  of< ""><span spanclass="/.The  span=>/>isomorphictothirdjava.lang.StringIndexOutOfBoundsException: Range [319, 318) out of bounds for length 418

</li>
<li<>span=>.xy<span>is      span ="x,<span<span>andthe  span=">(x ⊗ y)<spanin   <class"">G  )/><p

</li>
</ul>
<p>An optional second varible <span class="SimpleMath">m</span> can be set equal to a multiple of the order of the tensor square <span class

<p>The optional second java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0

<p>This java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p><strong class="button">Examples:</

<

<

p    span ="<andnormal""><span  thehomology  span =SimpleMathH_2G,Z/  fitsinto   sequence<>
<<p< =SimpleMath.. (GZ⟶H_3(NZ  GNZ  G,)⟶(/,)⟶../><>

<p shouldworkreasonablynilpotent< =SimpleMathG/>orextremelysmall - groups<p

<p< =button<strong>spanclass""< =.Aboutjava.lang.StringIndexOutOfBoundsException: Range [116, 115) out of bounds for length 139

<ph5151- />

<< =X854F2DB382723504name""<a/>

<112TensorCentre/>

<iv"< class"unc =%>tr< => =>8227 <  =>var<>class>;unction)td/>/div
<p>Inputs 

<

<pdivclassfunc< =funcwidth""tr< classtdleft> =func&8227/>( varclassArgG<var<td classtdright(; )/>;/>/>/>

<h5>15.1-< =func< =funcwidth""<>tdclasstdleft< =func&8227 />< class="Arg">G</var>, < "">/>)/>td=tdright>nbspnbsp)/d<>/>/

<div class=<Inputsafinite   group< =java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 312
div ="< =funcwidth"100"<tr>tdclass"tdleft> ""&#8227 ThirdHomotopyGroupOfSuspensionB> varclass"Arg"><var var =Argm/var> <td<dclass""(java.lang.StringIndexOutOfBoundsException: Range [234, 233) out of bounds for length 264
<p>Inputs a finite or nilpotent infinite group <span class=pFor -  implementationof functionspanclass=SimpleMathThirdHomotopyGroupOfSuspensionB(<span    optimalandsoon .   solution   ,an second  span">span      span =>0/>     java.lang.StringIndexOutOfBoundsException: Range [369, 368) out of bounds for length 574

<For- implementationof function< =SimpleMathThirdHomotopyGroupOfSuspensionB(<spanisfar optimalandwill   . atemporary     an    spanclassSm/>canset  spanclass"0/>      returnstheabelianof groups< =SimpleMath><span> span =><span        span =SimpleMath>⟶B ⟶ JG    <span><p

<p>Alternatively, the 

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a

<

<>Inputs   span =Gand   spanclass""c/> Itreturns  spanclassc/-      seriesspanclass""1/ lt < =SimpleMathZ_1∗)/ &;<class=Z_2∗G)/>&;< ""../><p
<p>Inputs a nilpotent group <span class="

<p>The upper epicentral series is defined for

<

<


<


<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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