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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "?xml version="1." ="UTF"? /java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for len <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (LAGUNA) - Chapter 2: A sample calculation with LAGUNA</title> <meta http-equiv="content-type"java.lang.StringIndexOutOfBoundsException: Index 0 <" /> <link rel"tylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript">< </javascript">overwriteStyle();</script> </head> <body class="chap2" onload="jscontent()"> <div class="div class="ChapSects"><a href="chap2.html#X7AAB1F18838B1DAB">2 </iv> <div class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for <p id="mathjaxlink" class="pcenter"><a href="chap2_mj.html">[MathJax on]</ajava.lang.Stri < id =X7"<a></p> <div class="ChapSects"><a href="chap2.html#X7AAB1F18838B1DAB">2 <span class="Heading">A sam </div> <h32< "> sample <strong class=pkg"LAGUNAstrong/>> <p the behindthe < ="<strong> wepresent sample calculation to to show what< "kg>java.lang.StringIn <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">K := GF( 2 );</span> GF <<The <code=""<code> attributes areconsequences its. can <p> <pc <<="example"<pre> &algebra-one over GF(2) with4generators&t </pre></divtrue <p>The group algebra <code class="code">KG</code> has some properties and attributes that are dire <div class="example"><pre> <span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 35 out java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 4 <span class="GAPprompt"><pc group of size 16 witgenerators> true <span class="GAPprompt">gap></span> <span class="GAPinput"> class="GAPprompt">gap></span> <span cl true <span class="GAPprompt">gap></span> <span class &; size 4&java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45 <;</span =G>(KG)/ GF(2) </pre></div> <Since="ode><code> is naturally a group algebra, the information provided by <code class="code">L <div class="example"><pre> <java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0 GF(2) <=G">gap&t;span>span ""(KG ;/> GF(2) </pre></div> <p>Let [, ] <div class="example"><pre> <span Z2))f1 ((2^) java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds [ f1, f2 ] <span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0 [ (Z(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length </pre></div> <p>Now we construct an element <code java.lang.StringIndexOutOfBoundsException: Index 40 <div class="example"><pre> <span <class""g;</> span=> +a ;< (Z((2))&ig;.+Z20*+((2)0*+(2^**f2 (Z(2)^0)*f2 <span class="GAPprompt">gap& 0*lt;g;of.. <pan"">gt<><=GAPinput> : ( e+a)* e +b) <span> (Z(2)^0)*<identity </pre></div> <p>We mayspan class"GAPprompt">apgt;</span> <span class="GAPinput">CoefficientsBySupport( x ) <iv ="<re> <span class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for [&;&t ..,f2, f1*f2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">CoefficientsBySupport( x );</span> 2^0, Z)0 Z()0 (20 ] <span p>We can also the < =c"<java.lang.StringIndexOutOfBoundsException: Range [72, 71) ou 4 <span Z( =e<java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length </pre></div> <p>We can also java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for <div java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for le <class=""gapg;<>< ">( ;/> ) < </pre></div> <ce =c"x</code> is,<code class="code">x</code> is not invertible, but <code class=code"1<code> . is to <div class="example"><pre> <java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0 (Z =""gt;/ ""> , ;/java.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bou true </pre></div> <p><strong class="pkg">LAGUNA</strong> can calculate the inverse(2^0)f1*2*f3*f4 <div class="example"><pre(2^0)& <span java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length (Z()^)f1+Z2^0)2+(Z(2)^0)*f3+((2)0)f4+Z)^)*1*2(()java.lang.StringIndexOutOfBounds 0)*f1*f3+(Z(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for Z2^)*2f3*f4 <span class="GAPprompt">gap></span> <span class="GAPinput">y * y^-1;</span> Z2^)<identity&t;of .. </pre></div> <p>We mayspan =GAPprompt>g;/> span ="y= (y )<span> <span="java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds fo <span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException (Z(2)^0)*f1+(Z(2)^0)*f1*f2+(Z(2)^((2^)&tigt;..(()0*((2^*f3f4 < java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for leng false <span class="GAPprompt">gap></span> <span class="GAPinput">IsSymmetric( y );</span> false <span class="GAPprompt">gap></span> <spanjava.lang.StringIndexOutOfBoundsException: Rang ((20*lt;&t ..(()0*+(Z)0*2** <span falsediv class=example>pre> </pre></div> <p>Now we calculate some important ideals of <code class="code">java.lang.StringIndexOutOfBo <div class="<two-sided ideal in <algebra-with-one over GF(2), with 4 generators>, <span class="GAPprompt">gap (dimension 15>java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20 <two-sided ideal in <algebra-< class=GAPprompt"gap&</> <pan ="GAPinput"RadicalOfAlgebra( KG (dimension 15 <pre>/div> <pIt well that the of< =code><code is a a . 'theoryon can index without .is <java.lang.StringIndexOu < <p>It is well-known that the augmentation java.lang.StringIndexOutOfBoundsException: Ind <div class="example">java.lang.StringIndexOutOfBoundsException: Index 0 out of boun <java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0 9 </pre></div> <p>On the other handclassGjava.lang.StringIndexOutOfBoundsException: Range [23, 22) o <div class="example"><pre> <span class="GAPprompt">gap>< ""gapgt<java.lang.StringIndexOutOfBoundsException: Ran <pan""gap;/ =">[2]<span> <algebra of dimensionjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of <span class="GAPprompt">gap></span> <span class="GAPinput">List(s [ <p seethat this is nilpotency ofthe =""KG.<java.lang.StringIndexOutOfBoundsException: Ra 9 </pre></div> <p>div ""<> <> letswork with the unit group of <code class="code">KG</code>. First we calculate the normalized u <div class="p>In order to makeour computation in the normalised unit group efficient, we calcul <span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0 &; 32768 15generatorsg; </pre></div> <p>In order to make our computation in <divp< "kg">GAP/> given a commutator orderuse carry , need isomorphisms < ="NormalizedUnitGroup/ <span class="GAPprompt">gap></span> <span class="GAPinput">W := PcNormalizedUnitGroup( KG );</s <pc group of size 32768 with 15p>first maps class=""NormalizedUnitGroup()/ the presented< ">P)<c </pre></div> p< ""GAP/> efficient and and given -.In usealgorithmsto computation the need < =c>/> code=c"java.l <p>The first isomorphism maps <code class="code">java.lang.StringIndexOutOfBoundsExceptio <div class="example"><pre> <class""gap&;/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of boun MappingByFunction( <group of size 32768 with 15 generators>, <pc group<span class="GAPprompt">g 32768 with 15 enerators&t; function ) ... end ) <span class="GAPprompt">gap></span>&;dentityg;of.. , ,, *f2f4 * f4,f8, true <span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0 &ti;.., f1 ,f4 ,f2, *,f2*4 * f4 </pre></div> <p>The second <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">f := java.lang.StringIndexOutOfBo <"APprompt"gap;<>< ="">Image(f)=V;/span> true </pre></div <p><div class"example">pre <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">cc := ConjugacyClasses( <pan class=" <span class="APprompt"gap&t<span> <span ="APinput"Length )/> 848 <span class="GAPprompt">gap></span> <span class="GAPinput">Representative( cc[ Length f1*f2**f6*f10f13 <span class="GAPprompt">0)f2*+(()^)f1*f3+((2)^0)*1*f3*java.lang.StringIndexOutOfBou ((2^)&t;&; .+Z)0)*f1*f2(Z(2)0*1*+Z()^0)f1f4+Z2^ 0)*f2*f3+(Z(2)^0)*f1*f2*f3+(Z(2)^0)*f1*f3*f4 <>/iv <p>Having span ">&;<> span =">LowerCentralSeries ;/ <div class="example"><pre> <span class="GAPprompt">gap></ f10*f11, f12*f13, f13*f15, f14*f13 f15 ) [>, Group([ f4*f8, f5*f7*f11*Group([ f12*f15, f13*f15, f14*f15 ]), Group([ <identity> of f10*f11java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for leng Group([ f8, f9*f15, f10*f11, f12*f15, f13*f15, f14*f15 ]), Group <// <p> G>>/ <="">:= ( ;<span <[ *13f14f15, f13**f15 *f12f15 ** <">gapgt;span span java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of <pan class="APprompt"gapgt;</span> <span class="GAPinput">m := MinimalGeneratingSet( C );</sp [ f8*f13 0)*1*f2*f3+(Z)^0)f1*f2f4+(Z(2)^0)*f11*f2*f3*f4, <> <span class="List( m, &;^ )<span> [ (Z(2)^0)*<identity> of ...+(Z(2 (2^0**f2+Z(^)f2f4 Z2)0*ltig; .+(^ 0**f2*+Z2^)f1f2*+Z2^)f2**, (Z(2)^0)*f3+(Z(2)^0)*f4+(Z( )*((2))f1f2*(()^0*f2((2^)*f4Z2^ 0)***f4Z2^0**f2*f4 Z)^)ltigt .+Z2^ )+(20*(2^)f1f2+Z2))f1*+2^**+(2^ 0) (20*f4(2^)*1f2*(()^)f2*((20*1**+((^ )f2+Z2^)java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds 0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f2*f3+(Z(2)^0)*f2(20*+Z2^)f3(()) Z(2)^0)*f3*0)*f1f3*((2^)f2f3* java.lang.StringIndexOutOfBoundsException: Index 35 0)* Z(pWe calculating algebra < "ode"KG<code example no ./> 0)*java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 </pre <p> LAGUNA algebra of dimension 16 over GF(2) 16 (); <div class="example"><pre> <span class="GAPprompt">gap></span> <<Lie algebra of dimension 9 over GF(2)> <Lie algebra of dimension 16 over GF(2l;Liejava.lang.StringIndexOutOfBoundsException: R <java.lang.StringIndexOutOfBoundsException: Range [30, 5) out of bounds for lengt # LAGUNA package: Computing theLie derivedsubalgebra.. <Lie algebra of dimension 9 over GF(2)> < =">&;/><class=G> =( )<span> <Lie algebra of dimension 7 5 <""gapgt<span span">( KG )/java.lang.StringIndexOutOfBoundsException: Range [99, 98) o 5 <span class="GAPprompt">gap></span> <span class="GAPinput">LieUpperNilpotencyIndex( KG );</sp 5 <spanjava.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for le false <span class="GAPprompt">gap></span> <span class="GAPinput">IsLieSolvable( L );</span> #java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for len true <span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 26 out of false <span class="GAPprompt">gap></span> <span class="GAPinput">IsLieCentreByMetabelian( L );</spa true </pre></div> <div class="chlinkprevnextbot">nbsp< hrefhtml>java.lang.StringIndexOutOfBoundsExceptio <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html>
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