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<?xml version=script type=text/java.lang.StringIndexOutOfBoundsException: Range [31 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (WPE) - Chapter 3: Tutorial</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheetdiv cjava.lang.StringIndexOutOfBoundsException: Range [31, 29) <=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for len <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap3" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter <div class="chlinkprevnexttop"> <a <>java.lang.StringIndexOutOfBoundsException: Range [2 <=java.lang.StringIndexOutOfBoundsException: Range [26, 21) out of bounds for len <p><a id="X81932F777898AD72" name="X81932F777898AD72 class="ContSect"><span class="toclin <div class="ChapSects"><a href="chap3.html#X81932F777898AD72">3 <span class="Heading">Tuto <class=">java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds f </span> </div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7 </span>=""<class=">span class="ocss>java.lang.StringIndexOutOfBoundsException: Range [ </div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a <span> </span> </div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span </spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length </divclass="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7 </span> </div> <div class="ContSect"><span class="tocline"><span class="java.lang.StringIndexOutOfBound </span> </div> <div java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 <span> </div> </div> <java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 <pter of thatshow toworkwith in ="kg>GAP</trong>in with package>WPE<strong><p> <p><a id="X820D3CEF8368DB04" name="X820D3CEF8368DB04"></a></p> <h4>3.1 <span class="Heading">Creating Wreath Product Elements</span>< =GAPprompt>&t/ spancla <p>In this section we present an example session which demonstrates how we can create wreath prod <p>In the following we will work with the wreath product <span class="SimpleMath">G = Alt(5) ≀ Sym(4 <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("WPE");;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">K := AlternatingGroup(5);;</span> < class="java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds f <span class="GAPprompt">gap></span> <span class="GAPinput">G := WreathProduct(K, H);</span> &t; group ofsize 311040000 generators> </pre></div> < <p class"pcenter">| | =|lt5)^4 ⋅|Sym() =604⋅24 =311040000 ./pjava.lang.StringIndexOutOfBoun <p>Suppose we would like to input the wreath product element</p> <p class="pcenter">g = ( (1,5,2,4,3), (1,3,5,2,4), (1,5,3,4,2), (1,4,5); (1,3)(2,4) )<java.l <p>as an element of <span class="SimpleMath">G</span>. The method <code class="code">WreathProduc <div class="example"><pre> <span class="GAPprompt">gap></span> "java.lang.StringIndexOutOfBoundsException: Range [ java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for leng (1,15,3,11,5,12)(2,14)(4,13)(6,18,span classGAPprompt&t/> =GAPinput>last ;/java.lang. classGAPprompt>g;</span> spanjava.lang.StringIndexOutOfBoundsException: Range [51, 5 true </pre></div> <p>On the other hand, the method <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ListWreathProductElement(G, g [ (1,5,2,4,3), (1,3,5,span class"APprompt"gap;/><spanclass"GAPinput">Territory(List)<sp < =G>><span><span =GAPinput> =gList<span> true </pre></div < hasjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for l <p>Moreover, all functions that work for <code class="code">IsWreathProductElement</code> can a <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Territory(gList);</span> [ 1, 2, 3, 4 ] </pre></div> <p>If the wreath product element is "sparse", i.e. has only a few (, (23,(,(52,,124 java.lang.St <div class= <<432<=H> Product</<h4 <span class="GAPprompt">></span> <span class="GAPinput"> * (1,5,2,4,3) ^ Embedding(G,4)</span> <=">java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for le (1,6,17,4,9,19,3,8,16,5,10,20,2,7,18) <"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for len [ (), (1,2,3), (), (1,5,2,4,3), (1,2,4) ] <span class="GAPprompt">gap></span>"java.lang.StringIndexOutOfBoundsException: Range [ true </><> <p><a id="X7C85A3718343D842" name="X7C85A3718343D842"></a></p> <h4>3.2 <span class="Heading">Displaying Wreath Product span class="GAPprompt">gap></span> <span <p>In this section we present an example session which demonstrates how we can display wreath pro <p>Suppose we are given some element < <div class="example"><pre> < =>><java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for <"gt/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for <span > <span class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for <</span> <span class="GAPprompt">gap></span> <span class="GAPinput">W := Image(java.lang.StringIndexO <span class=" (1,5,2,4,3), (1524),(53,,(,,) 1324 <span class="GAPprompt">gap></span> <span =G"L: (W, w);</span>> true </pre></div> <p>First we translate span class="GAPprompt">gap> =GAPinput> =( ;/> <div class="example"><pre> <span class=GAPprompt">gap></span> <span class="GAPinput">w := g ^ iso;</span> < </div class"xample"><pre> <p>If we span ="">gap&;/> <span class""Displayw<span> <div class="( (1,5,2,4,3), (1,3,(,,,,) (,5; (1,3,24) ) <span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 36 out [,4,5), (1,3,4 java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds f <span class="GAPprompt">gap& ="gapg;/ span ="(,(:= false);/> [ (1,5,2, 1: (1,,,4,java.lang.StringIndexOutOfBoundsException: Index 16 out of boun <span class="GAPprompt">gap><: 1324java.lang.StringIndexOutOfBoundsException: Index 14 true </pre></div> <p>Usually, we want to display this element in a nice format instead 152,3, 1,,,413,,(45;1324) <spa class"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds <java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for lengt 1 2 3 4 top 1,,,4,3, (3,5,2,4 (,5,3,,),(1,,5) 1,,2,) ) </pre></div> <p>There are many display options available for adjusting the display behaviour for wreath prod <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(w, rec(horizontal := false 1: (1,5,2,4,3) 2: (1,3,5,2,4) 3: (1,5,3,4,2) 4: (1,4,5) top: (1,3,2,4) <span class="GAPprompt">gap></span> <span class="GAPinput">Display(w);</span> 1 2 3 4 top ( (1,5,2,4,3), (1,3,5,2,4), (1,5,3,4,2), (1,4,5); (1,3,2,4<pre><> </pre></div> < also changethe global display options viathe command. <div class="example"><pre> <class=G">&;/>< ="GAPinput>(rec(horizontal= ))/> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(w);</span> 1: (1,5,2true 2: (1,3,5,2,4) 3: (span class"GAPprompt">gap>t;/span> <span class="APinput">b := TopComponentOfWreathProduct 4: (1,4,5) top: (1,3,2,4) </pre></div> <tr <div class="example"><pre> <span classpa id"8345CC25855C9406"=""></a> <span class="GAPprompt">gap&<h4>3.3 <span class="Heading"> in Wreath Products</span></h4> 1 2 3 4 top ( (1,5,2,4,3), (1,3,5,2,4), (1,5,3,4,2), (1,4,5); (1,3,2,4) ) </pre></div> <p>But sometimes, it is sufficient to just look at some components of a java.lang.StringIndexOu <div class="example"><pre> <span class=(w, 3);</span> (1,5,3,4,2) <span class="span class="GAPprompt">gap></span> <span class="GAPinput">H := Group([ (1,2,3,4,5 true <span class="GAPprompt">gap></span> <span class="GAPinput">b := TopComponentOfWreathProductE (1,3,2,4) <> <span class=java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bound true </<span class="GAPprompt>gap&t;/span span class=GAPinput"iso: IsomorphismWreathProduct(P;/ <p><a id="X8345CC25855C9406" <h4>3.3 <span class="Heading">Computing in Wreath Products</span></h4> <p>As div class"example><pre> <p>We include in the following sections examples for each computational problem listed in <a href <java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length <span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("WPE");;</span> <pan class=GAPprompt">gap&;/> <span class=GAPinput"K:= Group[ (,,,4,) (,,,;#F5<span <span class="GAPprompt">gap></span> <span class="GAPinput<span class=GAPprompt>&<span>< class= <G = WreathProduct(K, H);</span> <permutation group of size 768000000 with 4 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">P := WreathProduct(K, SymmetricGro <permutation group of size 46080000000 with 4 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(P, G);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">iso := IsomorphismWreathProduct(P </></div> <p>Moreover, we fix the <pre></ivjava.lang.StringIndexOutOfBoundsException: Index 12 out <div class="example"><pre> <span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException <span class="GAPprompt">gap></span> <span class="GAPinput">y := (1,12,26,8,3,14,28,7,2,13,27 <span class="GAPprompt">gap></span> <span class="GAPinput">Display(x ^ iso);</span> 1 2 3 5 top ( (,,2,5) (,4,5,) ,,42) (1,,,) (,,,,2(1,,4,;(1,5,3,2(46 ) <span class="GAPprompt">gap></span> <span class="GAPinput">Display(y ^ iso);</span> 1 2 3 4 5 6 top ( (1,2,,5) (,(,5,,,2) 1,,,4,(,2)(,5) (,,,5) (,2(4,5) ) <span <div class"xample><pre> true <span class="GAPprompt">gap></span> <span class="GAPinput">not x in G and not y in Gfail true </pre></div> <p <h4>3.4 <span class=<pan class="APprompt>gap&&;</span> <span class="APinput"c:=RepresentativeAc <p>We now demonstrate how to solve the conjugacy problem for <span class="SimpleMath<pan class=" <p>We continue the <strong class="pkg">GAP</strong> session from Section <a href="chap3.html#X83 <p>To check in <strong class="pkg">GAP</strong> whether two elements java.lang.StringIndexOutO <div class="example"><pre <classGAPprompt>;spanspan "GAPinput">RepresentativeActionG ,y;/> fail <pre>/iv <p>The output <div class="example"><pre> <span class="GAPprompt">gap></span> <span class= (2,5)(,4(6,8,,)11,2925)1226,,28,22,15,27,2414,3023 <java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 1 2 3 4 5 6 top ( (2,5)(3,4), (1,3,4<iv class="xample"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x<pan ="GAPprompt">gap&t</span>< clas true </pre></div> <p>We java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for len <p><a id="X7D474F8F87E4E5D9" name="X7D474F8F87E4E5D9"></a></p> <h4>3.5 <span class="<div class="example"><pre <p>Enumerate representatives of all conjugacy453(,26(,,9,,,,827)112122(,)1424(1525) <p>We continue the <strong class="pkg">GAP</strong> session from Section <a href="chap3.java.lan <p>To java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length <div class="example"><pre> <span class="GAPprompt">gap <span class="GAPprompt">div ""<> 1960 </pre></div> <p>We see that there are <span class="SimpleMath,,,301028827)(11,)1222)(13,23)(),) <div class="example"><pre> <spanclass=GAPprompt";/span><span class=GAPinput"> =CC1617;</pan (2,4,5,3)(6,26)(7,29 2 4 5 top </pre></div> <p <div span=GAPprompt"gap><span> < =java.lang.StringIndexOutOfBoundsException: Range [61, 60) 60000 </pre></div> <ative ofthis class,we the.</p <div class="<> the strong =p></strong> session <a href=cjava.lang.StringIndexOutOfBoundsExcept <span class="GAPprompt">gap></span> <span class="GAPinput">a := Representative(A);</span> (2,4,5,3)(6,26)(7,29,9,30,10,28,8,27<p>To computein<strong ="pkg">GAP<strong>the of elemen <span class="GAPprompt">gap></span> <span class="GAPinput">Display(a ^ iso);</span> 1 2 3 4 5 6 top ( (2,4,5,3), (2,4,5,3), (), (), (), </re></> </></> <p>java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0 <p><a id="spanclass="APprompt>apgt;span><span =GAPinput>izeC;/> <h43.6 span class"">Centralizer<span/> <izer of span ="SimpleMath></> in< =SimpleMath"G/pan,i.compute setofspanclassS">_x)><p> <p>We continue the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds f <p>Tospanclass""&<span><class""> Displayg^iso;/> java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for leng <span class="GAPprompt">gap></span> <span class="GAPinput">C Group([ (16,20)(17,19)(26,27)(28,30), (16,19,20,17)(26,28,27,30), (1,4,5,2)(6,9,10,7)(12,13,15,14)(21,25,23,24) ]) < <p>We can compute additional information about the centralizer on the go. For example, we can ask < <div java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for le <span java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for l 16 </pre></div> <p>The generators of a centralizer are always given in a sparse format, e.g. all cycles in the wrea <div java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for l <span class="GAPprompt">gap></span> <span class="GAPinput">for g in GeneratorsOfGroup(C) do</sp <pan ="">gt<span span "GAPinput" g;/ <span class="GAPprompt< ">&;><class": java.lang.StringIndexOutOfBoundsException: Range [86 1 2 3 4 5 6 top ( (), (), ()/></div 1 2 3 4 5 6 top (),(,,,) ) 1 2 3 4 5 6 top ( ((1,4,52), (14,,2), (,3,,4),,,4,(; () </pre></div> <p" name=""></></p> <>37< class="">Cycle Polynomial</span></h4> <p>Compute the cycle index polynomial of <span java.lang.StringIndexOutOfBoundsException: <div class="example"><pre> <span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <span class="GAPprompt">gap></<span class="GAPprompt">gap></span=GAPinput>(ExtRepPolynomial < =">g;<>< =" =[(,)]; #2< <span class="GAPprompt">gap></span> <span class="GAPinput">G_impr := WreathProduct(K, H);;</sp <span class=" [ 1, 1, 2, 1, 3, 1 ,16 ,2 ,2 ,18 ,3 3 ]118 6 <span class="GAPprompt">gap></span> <span class="GAPinput"> [ 1, 4 2, 1 ] 1/,[,172 ] 72 </pre></div> <>To<strongclass=pkg>GAP/strong>thecycleindex a a group we use <code class="func">CycleIndex</c <div class="example"><pre> span class"APinput"c_prod:CycleIndex)<java.lang.StringIndexOutOfBoundsException: R 1/72*x_1^6+1/12*x_1^4*x_2+1/18*x_1^3*x_3+1/8*x_1^2*x_2^2+1/6*x_1*x_2*x_3 +1/12*x_2^3+1/4*x_2*x_4+1/18*x_3^2+1/6*x_6 </pre></div> <java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(span class="GAPprompt">ga [ [ 6, 1 ], 1/6, [ 3, 2 ], 1/18, [ 2, 1, 4, 1 ], 1/4, [ 2< "java.lang.StringIndexOutOfBoundsException [ 1, 1, 2, 1, 3, 1 ], 1/6, [ 1, 2, 2, 2 ], 1/8, [ 1, 3, 3, 1 ], 1/18, , ,1] 1/12,[ 1,6 , 1/ ] </pre></div> <p>The way how to read this representation is roughly the following. The list consists of alterna <p>If we want to consider the wreath product given in product action, we need to use the command <cod <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">G_prod := WreathProductProductAct <span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(G_prod);</span> 9 <span class="GAPprompt">gap></span> <span class="GAPinput">c_prod := CycleIndex(G_prod);</spa 1/72*x_1^9+1/6*x_1^3*x_2^3+1/8*x_1*x_2^4+1/4*x_1*x_4^2+1/9*x_3^3+1/3*x_3*x_6 </pre></div> <p>However, we do not need to create the wreath product in order to compute the cycle index of the gr <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">c1 := CycleIndexWreathProductImpr <span class="GAPprompt">gap></span> <span class="GAPinput">c_impr = c1;</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">c2 := CycleIndexWreathProductProd <span class="GAPprompt">gap></span> <span class="GAPinput">c_prod = c2;</span> true </pre></div> <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <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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