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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="java.lang.StringIndexOutOfBoundsException: Index 190 out of bounds for length 6

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap14.htmljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p id="mathjaxlink" class="pcenter"><a href="chap15_mj.html">[MathJax on]</a></
<<a ="X86DE968B7B20BD48" name=X86DE968B7B20BD48>/a>/p>
<div class=ChapSects"<a href class="Heading"> Commutator and nonabelian tensor computations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7CFDEEC07F15CF82">15.1 <span class="Heading">  </span></a>
</span>
<div class="ContSSBlock">
< "">br/< =nocss&;nbsp<span< =".#"1511a>/>
<span/span>
<spandiv class"ContSSBlock"><panclass"">br/< class"nocss> nbsp<span< href=chap15.html#X7DFB0FC4834DF183""151-1BaerInvariant</a></span>
<span class="ContSS"><br /><span class="nocss">&spanclass"ontSS><br/<class""nnbsp<span =chap15.#X7D1614F87AAF5B97>15- BogomolovMultipliera/>
<spanclass="ontSS"<br>span=nocss>nbsp;;span>a href=htmlXE8AEC835F8CD1>15.- </>/>
<span =ContSS><r >span class=nocss>nbsp&;<span< =chap15htmlX7EAF8A2A79C86181"1516 NonabelianExteriorProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X8155A3747A66FE81">15.spanclass"">br/< class"nocss">;<span<a=.X819E8AEC835F8CD1".- <a</>
< =ContSS> <class"">nbsp;;/>< href"chap15.html#X7856BC54856452DE""15.18NonabelianSymmetricSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X829476FA82D5759B">15.1-9 NonabelianTensorProduct</span class=ContSS>br >spanclass"">nbsp&;<span< "chap15.htmlX8155A3747A66FE81""15.1-7 NonabelianSymmetricKernel<a></>
<class"ContSS">br >spanclass"">nbsp&<span =.java.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 147
<span=< >spanclass=>nbsp&;/>ahref".#">1- java.lang.StringIndexOutOfBoundsException: Range [138, 137) out of bounds for length 148
<span class="ContSS"><br /><span class="nocss">  spanclassContSS< >spanclass=nocss>nbsp&;<span< href=".html#X7EF8F3FF7FB900F8">15.1-11 RelativeSchurMultiplier</a></span>
<panclassContSS< <classnocss>nbsp&;/span<a ="chap15.html#X79D037908746C65C">15.1-13 ThirdHomotopyGroupOfSuspensionB</a></span>
<span class="ContSS"<panclass"">br/< class"nocss">nbspn<< =chap15html#">15.1-13 ThirdHomotopyGroupOfSuspensionB</a></span>
</div></div>
</div>

<h3>15 <span <pan class=ContSS< /><span class="nocss">nbsp;nbsp;</>< ="chap15.htmlX7967760B7E0F1E5F">.14 />/>

< =X7CFDEEC07F15CF82 =X7CFDEEC07F15CF82<ap>
h315 span"">Commutatorandnonabelian />/>
<p< ="" =X7CFDEEC07F15CF82>/>/>

<>< id"X7DFB0FC4834DF183" name"X7DFB0FC4834DF183">/>/>

<15.1 BaerInvariant/>

<
<pInputsanilpotentgroup< class="SimpleMath"G<span  integer<span class="impleMath"></pan>&;<spanclass"impleMath"></span> It returns the Baerinvariant< class="impleMath">^()(G</pan> defined as.Forangroup<spanclass"SimpleMath>G<span>  < class"SimpleMath>^_+1(G)<span  < class"SimpleMath>(c+1)/span-st term ofthe upper central series of thegroup<span class="impleMath>=F[[[R,F],F..<span with< ="SimpleMath></span> copies of <span class=SimpleMath>F/span> in the denominator) where span class""F/R</> is free presentation of <spanclass="SimpleMath"G<span>. Thisis 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) ⟶ 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>

<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></span> </p>

<p><a id="X7D1614F87AAF5B97" name="X7D1614F87AAF5B97"></a></p>

<h5>15.1-2 BogomolovMultiplier</h5>

<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 0
< func"<table classf"%>tr< =tdleft<class""&8227 />(< =ArgG/>,< =Argstr/>)/><tdclass"">&;&)/>/><table<div
<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 

<p>Three slight variants of the implementation are available. The default "standard" implementation seems to work best on average.

pstrongclass""></> span =URL< href=./chap6html>1/>/span>  span =URL"<a =".//SideLinks/About/aboutBogomolovhtml><a><span<

<p><a id="X87A1E2E279597B8D" name="X87A1E2E279597B8D"></a></p>

<h5>15.1-3 Bogomology</h5div ="func>table class"func"width"100%><< ="">codeclass"">#;BogomolovMultiplier<code(varclass"rg></var>, var class="Arg"str</ar> )/><td class(nbsp; )/><tr</table></div>

<div 
<>afinitegroup< "SimpleMath"><span>  positive < ="SimpleMath"><span> andreturnsthequotient< class="impleMath>H_n(G,Z)/K(G</span> ofthedegree<spanclass="SimpleMath">n</span> integral homologyof <span class"SimpleMath"G</span where <span class"SimpleMath">K(G)/span>is  subgroup  span S",Zspan     all<""H_nA  (,)spaninduced   < >/pan<

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

<p><a id="X7C374F188523A659" name="X7C374F188523A659"></a></p>

<>15.- Coclass</

<p< =X87A1E2E279597B8D=java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
<div""< class"func" =100%"<tr><td class"> =func"‣ Bogomology</code(<var class="Arg><var varclass""><var <td< class"">&;function&)/><tr<table<div

<<p< class="button">Examples:/ span class"">ahref./www/AboutaboutBogomolovhtml><a<span />

<

<divclass"">tableclass=funcwidth"%>tr< =tdleft><code ="func>8227Coclass<code<tdtd ""(nbsp;lobalvariable)/>/>/>/>

< class"func"< class=func"width="100%>trtd=tdleft>code class="">&#8227 EpiCentre<code>(<varclass""></> < class"Arg">N/>)/>< class"tdright">&; )/>/>/table></iv>
<<p<strong =button>Examples<strong<p
<Inputsa finitegroup<span class"SimpleMath"></span and normal subgroup spanclass"SimpleMath"></span andreturns"Z∗GN</span> ofthe of< "SimpleMath"N</>. The group <span class="SimpleMath">^∗(G,N</pan  trivial ifandonlyifthere  module< class""d⟶/  span""NImaged)/>and java.lang.StringIndexOutOfBoundsException: Range [429, 428) out of bounds for length 618

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

<p>divclass"< """%<tdleft> class="func">&#8227; EpiCentre</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )pInputsa finite group <span class""G/>andN/>andreturnsasubgroup spanclass"">^∗GN<span     spanclass""><span.Thegroup spanclass"Z^GN<span  ifandonlyifis    java.lang.StringIndexOutOfBoundsException: Range [344, 343) out of bounds for length 618
p <">/>isentered      span =SimpleMath>=<span>   casethe span ""^(G</>      if< =>/>isisomorphic  quotient<span""=/E<span>ofsome  spanclass=SimpleMath></>bythecentre ofspan class=S"Espan(commandspanclass"">UpperEpicentralSeries,)span> </>
aid"name=<a><p>

<h5>15.1-6 NonabelianExteriorProduct</h5>

<div class="func"><table class="func" width="100%"><tr
<>   <class="SimpleMath"><span   subgroup< =SimpleMathN/.It    spanclass"java.lang.StringIndexOutOfBoundsException: Range [158, 157) out of bounds for length 202


<


/i
li>span=>.x,y<span>is  function whichinputs  spanclass""><span  <span class"SimpleMath">G</span> and an element <span class="SimpleMath">y</span> in <span class="SimpleMath">N</span> and returns <span class="SimpleMath">(x ∧ y)</span> <i>p>spanclass"SimpleMath""E.pairing(x,y)/> is a function which inputs  element < class"SimpleMath"x<span  < class=SimpleMath>G<span and  element <span class="SimpleMath">y</span> in <span class="SimpleMath">N</span> and java.lang.StringIndexOutOfBoundsException: Index 255 out of bounds for length 0

</li>
</ul>
<p>This<515.- NonabelianSymmetricKernel<h5

<pdivclass"">table=funcwidth"%"<>tdclass"tdleft><codeclass"func>#;NonabelianSymmetricKernel/>(<var =Arg>/>)<td< =tdright(nbsp;&;<td<tr<table></div>

<>Inputs   nilpotent group< "SimpleMath"><span  returns  abelian    Fourthhomotopygroup spanclass""><span   suspension">SSK,)/>theMacLane  spanclass"">K(,)/><p

<h5>15.1-7 NonabelianSymmetricKernel</h5>

<class""< =funcwidth%<>tdtdleft"< class=func>#;NonabelianSymmetricKernel/code>Arg><var </><tdclass"tdright"&;nbsp</>/>/>/>
<div class="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
<>Inputs    group< =S"<spanreturns   ofthe   group="SimpleMath">SG<span  doublesuspension < =S"SSKG1</>oftheEilenbergMacLanespace< "">(,)/./p

<p>For nonjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Alternatively, the

<<class"button"Examples/> spanclass=URL< "./www/SideLinks/About/.html"1/>/span />

<p><

<h5divclass"">tableclass"" =100"tr< "java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 258

<div class="func"><tablepInputs     infinitegroup< =SimpleMathG/>andreturns   spanclass""><span   followingcomponents./p
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NonabelianSymmetricSquare</code>( <var class="Arg">G</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs   or group< "SimpleMath">G/> andreturns   <class""><span   components<p


<ul>
<>p<class""Thomomorphism/>     < class"">  Gtilde )⟶<span nonabelian  of< =java.lang.StringIndexOutOfBoundsException: Range [196, 194) out of bounds for length 435

</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> 

</li>
</ul>pThe  secondvariablespan =SimpleMath><span     to< "SimpleMath"><span. this Todd-procedure beusedto  thesymmetric  when< "">/> issolvable<p
<>optional varible< "SimpleMath">m/>canbe toa    ofthesymmetric  spanclass"">(  G<span   help  spanclass=><span   but nilpotent especiallyif  estimated   spanclass""><span   accurate)as  boundisusedinthe solvable  algorithm<p>

<p>The

<p>This function should work for reasonably small solvable groups or extremely small non-solvable groups.</p>

<p><java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0

<p><a id="X829476FA82D5759B" name="X829476FA82D5759B"></a></p>

<h5

< =func>tableclassfwidth"%>><td "> =>#227;NonabelianTensorProduct/code(< ="">G/> <varArg><var><< >nbsp<<tr<table></>
<p>Inputs a finite group <java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0


<ul>
<li><p><span java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0

/>
<li><p>< class""> span =SimpleMathx/>in< =SimpleMath><span andanelement< "SimpleMath"><spaninspan=SimpleMath>/> andreturns<span ="SimpleMath">( ⊗)/>  thetensor <span class"SimpleMath">(G ⊗N)<span <p

</li>
</ul>
<p>This function p< "">Examples:/>< ="URL><a href".wwwSideLinksaboutNonabelian><a<> <p



p< id"" =X7C0DF7C97F78C666<a<p

<h5h5151-NonabelianTensorSquare<h5

<div<ivclass"">tableclass"unc ="100"<tr><td tdleft> class=func>#;NonabelianTensorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
< =func>tableclass=funcwidth"100%><tr><d class"tdleft>code""&8227NonabelianTensorSquare<code(varclass""><var,< =Arg></var>)/td>td class"tdright">(nbspfunction;<td>/tr></table>/div
pInputs  finite  nilpotent infinite group < class><span>andreturns a record<span="SimpleMath>T</span> with the  components<p


<ul>
<li><p><

<<l
<li>p><spanclass="SimpleMath">T.<span>agroup < ="SimpleMath"> :( ⊗ G ⟶G/>   tensorsquare  spanclass=SimpleMathG/> to< class"SimpleMath>G<span>. The kernelof <span class"SimpleMath"µ</span is  to the third homotopy group of the suspension <span class="SimpleMath">SK(G,1)</span> of an Eilenberg-Mac Lane space.</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="SimpleMath">(G ⊗ G)</span>. This might help when <span class="SimpleMath">G</span> is solvable but not nilpotent (especially if the <li><>< class"SimpleMath"Tpairing(,)/ is afunctionwhichinputstwoelements<class"SimpleMath>x,y/> in <span class="SimpleMath">G</>and returns tensor< class"SimpleMath/>  thetensorsquare span class=SimpleMath( ⊗G</span>./>

<p>The optional second variable <span class="SimpleMath">m</span> can also be set equal 

<p>This function should work for reasonably small solvable groups or

<p><strong class="java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0

<p><a

<h5

<
<p>Inputsafinitegroup<class"SimpleMath>G</span> and  subgroup <span class="impleMathN/>.Itreturns  group<class"SimpleMath">(GN,)<span>that theexactsequence/p

<>spanclass"">.⟶ H_3,Z)  H_3G/,)⟶ H_2(,,)⟶H_3(Z  H_3GN,Z  ..<span<p

<>Thisfunction  work for  small  groups spanclass""><span   smallnonnilpotentgroups./>

<>strongclass"button">Examples:/ < =URL>a href"./www/SideLinks//About/aboutSchurMultiplier.html">1</a></span> </p>

<p><

<>15.-2TensorCentre<h5

<div p>aid"" name=X854F2DB382723504>/><p
<h5>15.1-12 TensorCentre</h5

<p><< class="func"><table class=func"width"100%"<tr><tdclass"tdleft"<codeclass"func"&#;TensorCentre/code>(<varclass"Arg"G</var> )/td>td ="tdright"(  </td></tr><table<div>

<p

<h5>15

< ="">tableclass"" =100%><>td ="">codeclass"func">#8227; ThirdHomotopyGroupOfSuspensionB<code(< ="Arg">/> )/td><td="tdright"> function;<td><tr<table<div
divclass"">tableclass"" =100%>tr< ="tdleft">codeclass"">#8227;ThirdHomotopyGroupOfSuspensionB<code> varvar class=Argm<var <td< class""( function&;</>/tr<table<div>
<>  ornilpotentinfinite  spanclass"SimpleMath">G</span> and returns the abelian invariants of the third homotopy group <span class="SimpleMath">JG</span> of the< class"func">tableclass"" =100%>< =tdleft"<codeclass=func>#8227;</code(<var ="ArgG/>,<class"Arg"></>)/td><d ="tdright>( function )</td></tr></table></div>

<>Fornonnilpotentgroupsthe  of the < "">(G)/>isfarfrom  will  beimproved Asatemporary tothisproblem optional variable< class=SimpleMath"m</> canbe setequalto< class"SimpleMath"<span,andthenthefunctionefficiently 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 ⟶ B ⟶ JG ⟶ A ⟶ 0</span>.</p>

<p>Alternatively, the optional second varible <span class="SimpleMath">m</span> can<> nonnilpotent groups the of the spanclass"">G)/>  far from and soonbeimproved As  solutiontothisproblem,optionalsecondvariable< ="impleMath"><span  be  equal to< ="SimpleMath""></span,andthenthefunctionefficiently returns the abelian invariants of groups spanclass"SimpleMath"A/>and<spanclass"SimpleMath">B/span>suchthatthereisanexactsequence<span class"">  B ⟶A⟶0/./>

<

<p><a id="X7967760B7E0F1E5F" name="X7967760B7E0F1E5F"></a></

<h5

<
<p> anilpotentgroup<spanclass"SimpleMath"></span> aninteger< =SimpleMath></span>.It the< ="SimpleMath"><span>thtermoftheupperepicentral < =SimpleMath>1<span>&; spanclass"SimpleMath">Z_1^∗(G<span>&t span "SimpleMath">^∗(<span lt spanclass=SimpleMath>.<span./>

<p>The upper epicentral java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0

<p>

<


<


<div class="chlinkbot"><span

<hr />
<pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<
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