|
|
|
|
SSL chap41_mj.html
Sprache: HTML
|
|
| products/Sources/formale Sprachen/GAP/doc/ref/chap41_mj.html |
 |
version.encoding= "-8"?
<! DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
httpwwwwo/java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bound s for length 61
<html xmlns=http://www.w3.org/xhtml" xml:="en">
<head>
<script type="text/javascript"
src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (ref) - Chapter 41: Group Actions</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap41" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chap10_mj.html">10</a> <a href="chap11_mj.html">11</a> <a href="chap12_mj.html">12</a> <a href="chap13_mj.html">13</a> <a href="chap14_mj.html">14</a> <a href="chap15_mj.html">15</a> <a href="chap16_mj.html">16</a> <a href="chap17_mj.html">17</a> <a href="chap18_mj.html">18</a> <a href="chap19_mj.html">19</a> <a href="chap20_mj.html">20</a> <a href="chap21_mj.html">21</a> <a href="chap22_mj.html">22</a> <a href="chap23_mj.html">23</a> <a href="chap24_mj.html">24</a> <a href="chap25_mj.html">25</a> <a href="chap26_mj.html">26</a> <a href="chap27_mj.html">27</a> <a href="chap28_mj.html">28</a> <a href="chap29_mj.html">29</a> <a href="chap30_mj.html">30</a> <a href="chap31_mj.html">31</a> <a href="chap32_mj.html">32</a> <a href="chap33_mj.html">33</a> <a href="chap34_mj.html">34</a> <a href="chap35_mj.html">35</a> <a href="chap36_mj.html">36</a> <a href="chap37_mj.html">37</a> <a href="chap38_mj.html">38</a> <a href="chap39_mj.html">39</a> <a href="chap40_mj.html">40</a> <a href="chap41_mj.html">41</a> <a href="chap42_mj.html">42</a> <a href="chap43_mj.html">43</a> <a href="chap44_mj.html">44</a> <a href="chap45_mj.html">45</a> <a href="chap46_mj.html">46</a> <a href="chap47_mj.html">47</a> <a href="chap48_mj.html">48</a> <a href="chap49_mj.html">49</a> <a href="chap50_mj.html">50</a> <a href="chap51_mj.html">51</a> <a href="chap52_mj.html">52</a> <a href="chap53_mj.html">53</a> <a href="chap54_mj.html">54</a> <a href="chap55_mj.html">55</a> <a href="chap56_mj.html">56</a> <a href="chap57_mj.html">57</a> <a href="chap58_mj.html">58</a> <a href="chap59_mj.html">59</a> <a href="chap60_mj.html">60</a> <a href="chap61_mj.html">61</a> <a href="chap62_mj.html">62</a> <a href="chap63_mj.html">63</a> <a href="chap64_mj.html">64</a> <a href="chap65_mj.html">65</a> <a href="chap66_mj.html">66</a> <a href="chap67_mj.html">67</a> <a href="chap68_mj.html">68</a> <a href="chap69_mj.html">69</a> <a href="chap70_mj.html">70</a> <a href="chap71_mj.html">71</a> <a href="chap72_mj.html">72</a> <a href="chap73_mj.html">73</a> <a href="chap74_mj.html">74</a> <a href="chap75_mj.html">75</a> <a href="chap76_mj.html">76</a> <a href="chap77_mj.html">77</a> <a href="chap78_mj.html">78</a> <a href="chap79_mj.html">79</a> <a href="chap80_mj.html">80</a> <a href="chap81_mj.html">81</a> <a href="chap82_mj.html">82</a> <a href="chap83_mj.html">83</a> <a href="chap84_mj.html">84</a> <a href="chap85_mj.html">85</a> <a href="chap86_mj.html">86</a> <a href="chap87_mj.html">87</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
classjava.lang.StringIndexOutOfBoundsException: Range [31, 29) out of bounds for length 265
<p id="mathjaxlink" class="pcenter"><a href="chap41.html">[MathJax off]</a></p>
<p><a id="X87115591851FB7F4" name="X87115591851FB7F4"></a></p>
<divspanclass="ContSS"<br /><class="nocss">&;& /span><a href="chap41_mj.html#X7A3D87DE809FBFD4">41.12-17 SurjectiveActionHomomorphismAttr</a></span>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X83661AFD7B7BD1D9">41.1 <span class="Heading">About Group Actions</span></a>
</pan>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss
</pan>
<div class="ContSSBlock">
<span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7960924D84B5B18F">41.2-2 OnRight</a></span>
<span class="java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 0
<</span><a href="chap41_mj.html#85AA04347CD117F9>.24OnSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X832CC5F87EEA4A7E">41.2-5 OnTuples</a></span>
<span class=ContSS>br/<class"ocss" nbsp;/>a="hap41_mj.html#X80DAA1D2855B1456">41.2-6 OnPairs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7C10492081D72376">41.2-7 OnSetsSets</a></span>
< ="ontSS"<br/<span class="nocss"> </span><a href="chap41_mj.html#X7E23686E7A9D3A20">41.2-8 OnSetsDisjointSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7ADE244E819035FF">41.2-9 OnSetsTuples</a></span>
<span class="ContSS"><br /><span class="nocss">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span ">41.2-11 OnTuplesTuples</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X86DC2DD5829CAD9A">41.2-12 OnLines</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7FA394D27E721E2B">41.2-13 OnIndeterminates</a></span>
<span class="ContSS"><br /><span =""&; /span><href=chap41_mj.tml#7A8D76586F1F06E>.-14Permuted<a</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X85124D197F0F9C4D">41.2-
</div<>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X82181CA07A5B2056">41.3 <span class="Heading">Action on canonical representatives</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X81E0FF0587C54543">41.4 <span java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 0
</>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X80E0234E7BD79409">41.4-1 Orbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X86BCAE17869BBEAA">41.4-2 Orbits</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X86BC8B958123F953">41.4-3 <span class="Heading">OrbitsDomain</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X799910CF832EDC45">41.4-4 OrbitLength</a></span>
<span class="ContSS"><br /><span class="nocss">&<h4>41.1 <span class="Heading">About Group Actions</spanh4>
</span>
<span<>Thewhich used by the operations for group actions is quite flexible. For example we can call the operation <code class="func">OrbitsDomain</code> (<a href="chap41_mj.html#X86BC8B958123F953"><span class="RefLink">41.4-3</span></a>) for the orbits of the group <var class="Arg">G</var> on the domain <var class="Arg">Omega</var> in the following ways:</p>
</span>
</div></div>
< class="Heading">Stabilizers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7C34EC437EF598BF">41.5-1 OrbitStabilizer</a></span>
<span class/dd>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X78C3A8568414BC44">41.5-3 OrbitStabilizerAlgorithm</a></span>
<div<div
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X7A9389097BAF670D">41.6 <span class="Heading">Elements with Prescribed Images</span></a>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X857DC7B085EB0539">41.6-1 RepresentativeAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X87F73CCA7921DE65">41.7 <span class="Heading">The Permutation Image of an Action</span></a>
</pan>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X78E6A002835288A4">41.7-1 <span class="Heading">ActionHomomorphism</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X85A8E93D786C3C9C">41.7-2 Action</a></span>
<span class"ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X86FF54A383B73967">41.7-3 SparseActionHomomorphism</a></span>
</div></div>
<div =ContSect"<span class"tocline>span class="nocss"> </span><a href="chap41_mj.html#X7FED50ED7ACA5FB2">41.8 <span class="Heading">Action of a group on itself</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X784D417D87F4E58D">41.8-1 FactorCosetAction</a></span>
<span class="ContSS"><br /><span class="nocss"> <ddjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<pan class=ContSS">br/><pan class="nocss"> </span><a href="chap41_mj.html#X835317A7847477D4">41.8-3 AbelianSubfactorAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"<><a id=="X81B8F9CD868CD953" name="X81B8F9CD868CD953"></a></p>
<span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</panjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X81D4EA42810974A0">41.9-2 PermutationCycle</a></span>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X80AF6E0683CA7F14">41.9-3 Cycle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7F559E897B333758">41.9-4 CycleLength</a></span>
<="><br ><span class=nocss">nbsp&;/span>ahref"chap41_mj.html#73387A7FD8AE5E">41.- Cycles/>/>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X83040A6080C2C6C6">41.9-6 CycleLengths</a></span>
<span class="ContSS"><br /><
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X850A84618421392A">41.10 <span class="Heading">Tests for Actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X79B15750851828CB">41.10-1 <span class="Heading">IsTransitive</span></a>
</span>
<span class="ContSS"><br /><span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mjjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 47
</span>
<span class="ContSS"><br /><span OnIndeterminates( polypair[ ];
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7CF02C4785F0EAB5">41.10-5 <span class="Heading">IsRegular<end;
/>
<span class="ContSS"><br /><span class="nocss">  
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X84C19AD68247B760">41.10-7 <span class="Heading">IsPrimitive</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocssss="tdleft"><code class="func">‣ OnPoints</code>( <var class="Arg">pnt</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 0
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X79936EB97AAD1144">41.11-2 <span class="Heading">MaximalBlocks</span></a>
</spanspanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 100
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7941DB6380B74510">41.11-3<="GAPprompt"gap>/span spanclass=APinput>(g 1,OnPoints);/java.lang.StringIndexOutOfBoundsException: Range [93, 92) out of bounds for length 93
</span>
<span class="java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 0
</div></div>
<div class="ContSect"><
</spandiv=func>tableclass=func =100%<><td class class="">class=func"&8227; </java.lang.StringIndexOutOfBoundsException: Range [115, 114) out of bounds for length 242
java.lang.StringIndexOutOfBoundsException: Index 242 out of bounds for length 25
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X8264C3C479FF0A8B">41.12-1 IsExternalSet</a></span>
<span class=""<r />spanclass">&; </span< href"html#">41122 ExternalSet</a>/>
<span class="ContSS"><br /><span class="nocss"> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<spanclass"ContSS><r /span class="nocss"> </span><a href="chap41_mj.html#X86153CB087394DC1">41.12-4 FunctionAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X86A0CC1479A5932A">41.12-5 HomeEnumerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X879DE63C7858453C">41.12-6 IsExternalSubset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X87D1EA1486D86233">41.12-7 ExternalSubset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7E081F568407317F">41.12-8 IsExternalOrbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7FB656AE7A066C35
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7BAFF02B7D6DF9F2">41.12-10 StabilizerOfExternalSet</a></span><pan class=GAPprompt"&;/span> <span class="GAPinput">Orbit( g, (), OnRight );</span>
<span class="ContSS"><br /><span class="nocss3]
</span>
<span class="ContSS
</span>
<span class="ContSS"><br /><span classh54123h5
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X8071A8D784DC8325">41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X85E9A6A77B8D00B8">41.12-15 ActorOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X8190A8247F29A5C7">41.12-16 UnderlyingExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7A3D87DE809FBFD4">41.12-17 SurjectiveActionHomomorphismAttr</a></span>
</div></div>
</div>
<h3>41 <span class=
<p>A <em>group action</em> is a triple <span class="span class="GAPprompt">gap></span> <span class="GAPinput [ );</span>
<p>In <strong class="pkg">GAP</strong>, <span class="SimpleMath">\(\Omega\)</span> can be a duplicate-free collection (an object that permits access
<p>The acting function <span class="SimpleMath">\(\mu\)</span> is a binary <strong class="pkg">GAP</strong> function that returns the image <span class="SimpleMath">\(\mu( x, g )\)</span> for a point <span class="SimpleMath">\(x \in \Omega\)</span> and a group element <span class=124java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<p>In <strong class="pkg">GAP</strong>, groups always act from the right, that is <span class="SimpleMath">\(\mu( \mu( x, g ), h ) = \mu( x, gh )\)</span>.</p>
<p><strong class="pkg">GAP</strong> does not test whether the acting function <span class="SimpleMath">\(\mu\)</span> satisfies the conditions for a group operation but silently assumes that is does. (If java.lang.StringIndexOutOfBoundsException: Index 205 out of bounds for length 50
<p>The first section of this chapter, <a href="chap41_mj.html#X83661AFD7B7BD1D9"><span class="RefLink">41.1</span></a>, describes the various ways how operations java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>Functions for several commonly used action are already built into <strong class="pkg">GAP</strong>. These are listed in section <a href="chap41_mj.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span
<p>The sections <a href="chap41_mj.html#X87F73CCA7921DE65"><span class="RefLink">41.7</span></a> and <a href="chap41_mj.html#X7FED50ED7ACA5FB2"><span class="RefLink">41.8</span></a> describe homomorphisms and mappings associated to group actions as well as the permutation group image of an action.</p>
<p>The <p>et <var class="Arg">set</var> be a proper set (see <a href="chap21_mj.html#X80ABC25582343910"><span class="RefLink">21.19</span></a>). <code class="func">OnSets</code> returns the proper set formed by
<p>Finally section <a href="chap41_mj.html#X7FD3D2D2788709B7"
<p><a <div class"xample"><pre>
<h4>41.1 <span class="Heading">About Group Actions</span></h4>
<p>The syntax which is used by the operations for 1,2 ]
<dl>
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">\(( G, \Omega[, \mu] [(,) 13 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
<dd><p>The acting function <span class="SimpleMath">\(\mu\)</span> is optional. If it is not given, the built-in action <code class="func">OnPoints</code> (<a href="chap41_mj.html#X7FE417DD837987B4"><span class="RefLink">41span classgapgt;/pan>< class="APinput"Orbit( g,[1,2] OnSets )<>
/java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<dt>41.2-5 OnTuples/>
<dd><p>This second version of <code class="func">OrbitsDomain</code> (<a href="chap41_mj.html#X86BC8B958123F953"><span<
<p>Here <span class="SimpleMath">\(gens\)</span> must be a set of generators of <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(actspLet var class"></> be a . < class="func>OnTuples/> the by of points<java.lang.StringIndexOutOfBoundsException: Range [138, 137) out of bounds for length 427
<p>The advantage of this notation is that <strong class="pkg">GAP</strong> does not need to construct this homomorphism <span class="SimpleMath">\(\varphi\)</span> and the range group <var class="Arg">H</var> as <strong class="pkg">GAP</strong> objects. (If a small group <span class="SimpleMath">\(G\)</span> acts via complicated objects <span class="SimpleMath">\(acts\)</span> this otherwise could lead to performance problems.)</p>
<><strong="">APstrong> not whetherthemappingspanclass"SimpleMath">\(gens\mapsto acts\</pan actually inducesa homomorphism and resultsare unpredictable if this is not the case.</p>
</dd>
<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
<dd><p>A third variant is to call the operation with span class="GAPprompt">gap></span> <span class="GAPinput2)<
</dd>
</dl>
<p>For operations like <code class="func">Stabilizer</code> (<a href="chap41_mj.html#X86FB962786397E02"><span class="RefLink">41.5-2</span></a>)[13,2)]
<p><a id="X81B8F9CD868CD953" name="X81B8F9CD868CD953"></a></p>
<h4>41.2 <span class="Heading">Basic Actions</span></h4>
<p><strong class="pkg">GAP</strong> already provides acting functions for the more common actions of a group. For built-in operations such as <code class="func">Stabilizer</code> (<a href="chap41_mj.html#X86FB962786397E02"><span class="RefLink">41.5-2</java.lang.StringIndexOutOfBoundsException: Range [0, 257) out of bounds for length 61
<p>If one needs an action for which no acting function is provided by the library it can be implemented via a <strong class="pkg">GAP</strong> function that conforms to the syntax</p>
<p><code class="code">actfun( omega, g )</code></p>
<p>where <code class="code">omega</code> is an element of the action domain, <code class="code">g</code> is an element of the acting group, and the return value is the image of <code class="code">omega</code> under <code class="code">g</code>.</p>
<p>For example one could define the following function that acts on pairs of polynomials via <code class="func">OnIndeterminates</code> (<a href="chap41_mj.html#X7FA394D27E721E2B"><span class="RefLink">41.2-13</span></a>):</p>
<div </div>
OnIndeterminatesPairs:= function( polypair, g )
[OnIndeterminates polypair[1,g ),
OnIndeterminates( polypair[2], g ) ];
end;
</pre></div>
<p>Note that this function <em>must</em> implement a
<p><a id="X7FE417DD837987B4java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 197
<h5>41.2-1 OnPoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OnPoints</code>( <var class="Arg">pnt</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="code"><var class="Arg">pnt<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnPoints( 1, (1,2,3) );</span>
2
< class="GAPprompt>&;span>< =Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 96
(2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );; ofsets.Forthe thatthesets ,itis possibleto code ""OnSetsDisjointSets<code> (a href=.html#X7E23686E7A9D3A20"<spanclass="RefLink"41.8/span></a>). <var class="Arg">set</var> must be a sorted list whose entries are again sorted lists, otherwise an error is triggered (see <a href="chap41_mj.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
=">g;/span> span class="GAPinput">Orbit( g, 1, OnPoints );</span>
[ 1, 2, 3, 4 ]
</pre></div>span=GAPpromptgap&t;/span span ">( g 1 2][ 3 4 ,);/span>
<p><a id[ [ 1 , 3, ]],[[1, ] [ 2 ]]java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
<h5>41.2-2 OnRight</h5>
<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns<class=code"<var ="rgpnt/>* class="Arg></var></ode>. This is for example the action of a group on its elements via right multiplication, or the action of a group on the cosets of a subgroup. The action of a matrix group on vectors from the right is described by both <code class="func">OnPoints</code> (<a href="chap41_mj.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) and <code class="func">OnRight</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnRight( [ 1, 2 ], [ [ 1, 2 ], [ 3, 4 ] ] );</span>
java.lang.StringIndexOutOfBoundsException: Index 210 out of bounds for length 9
<span class="GAPprompt">gap></span> <span class="GAPinput">OnRight( (1,2,3), (2,3,4) );</span>
(1,3)(2,4)
<panclass=GAPpromptgap&;< <""g=Group(1,3(,,));/java.lang.StringIndexOutOfBoundsException: Range [100, 99) out of bounds for length 100
< "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 93
[ (), (1,2,3), (2,3,4), (1,3,2), (1,3)(2,4), (1,2)(3,4), (2,4,3),
(1,4,2), (1,4,3), (1,3,4), (1,2,4), (1,4)(2,3) ]
</pre></div>
<p><a id="X832DF5327ECA0E44" name="X832DF5327ECA0E44"></a></p>
<h5>41.2-3 OnLeftInverse</h5>
<div=func"<table func"width="100%"><tr><td class="tdleft"><code class="func">8227; OnLeftInverse</code>( <var class="Arg">pnt</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns <span class="SimpleMath">\(\textit{g}^{{-1}}\)</span> <code class="code">* <var 1,4] [ , ]]
<p><code class="func">OnLeftInverse</code> is used for example in the representation of a right coset as an external set (see <a href="chap41_mj.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a>), that is, a right coset <span class=" class">gt;</><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 118
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnLeftInverse( [ 1, 2 ], [ [ 1, 2 ], [ 3, 4 ] ] );</span>
[ 0, 1/2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OnLeftInverse( (1,2,3), (2,3,4) );</span>
(1,2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 99
[ (), (1,3,2), (2,4,3), (1,2,3), (1,3)(2,4), (1,2)(3,4), (2,3,4),
12,,4) 13,) (,,3),(1,,),14)2,3)]
</pre></div>
<p><a id="X85AA04347CD117F9" name="X85AA04347CD117F9"></a></p>
<h5>41.2-4 OnSets</h5>
<div class="func"><table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 26
<> <var ">java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 525
<p><code class="func">OnSets</code> is for example used to compute the action of a permutation group on blocks.</p>
<p>(<code class="func">OnTuples</code> (<a href="chap41_mj.html#X832CC5F87EEA4A7E"><span class="RefLink">41.2-5</span></a>) is an action on lists that preserves the ordering of entries.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnSets( [ 1, 3 ], (1,2,3) );</span>
[ 1,
<span class="GAPprompt">gap></span> <span class="span class="GAPprompt">gap></span> <span class="GAPinput(g,[[ 1, 2 ], [ 1, 2 ] [3 4 ]], OnTuplesSets );/>
[ (2,3), (1,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit( g, [ 1, 2 ], OnSets );</span>
[ [1 2 ] [[ 2 3, [ 1, , [3 4 ] [ 1 4 ,[ , 4 ]
</pre></div>
<p><a id="X832CC5F87EEA4A7E" name="X832CC5F87EEA4A7E"></a></p>
<h5>41.2-5 OnTuples</h5>
<d java.lang.StringIndexOutOfBoundsException: Range [100, 64) out of bounds for length 243
<pLet var class=A"tup</var> be a list. <code class="func">OnTuples</code> returns the list formed by the images of all points <span class="SimpleMath">\(x\)</span> of <var class="Arg">tup</var> via the action function <code class="func">OnPoints</code> (<a href="chap41_mj.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>), applied to <span class="SimpleMath">\(x\)</span> and <var class="Arg">g</var>.</p>
<p>(<code class="func">OnSets</code> (<a href="chap41_mj.html#X85AA04347CD117F9"><span class="RefLink">41.2-4</span></a>) is an action on lists that additionally sorts the entries of thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnTuples( [ 1, 3 ], (1,2,3) );</span>
[ 2, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OnTuples( [ (2,3), (1,2) ], (1,2,3) );</spandivclass="example"><pre>
[ (1,3), (2,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
<panclass="GAPprompt">gap></span> <span class="GAPinput">Orbit( g, [ 1, 2 ], OnTuples );</span>
[3, 4 ], [ 2, 1 ]
[ 1, 4 ], [ 4, 1 ], [ 4, 2 ], [ 3, 2 ], [ 2, 4 ], [ 4, 3 ] ]
</pre></div>
<p><a id="X80DAA1D2855B1456" name="X80DAA1D2855B1456"></a></p>
<5>41.-6 OnPairs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="[ [1 4] [2,3 ]], [ [ 4, 1], 3 2]],
<p>is a special case of <code class="func">OnTuples</code> (<a href="chap41_mj.html#X832CC5F87EEA4A7E"><span class="RefLink">41.2-5</span></a>) for lists <var class="Arg">tup</var> of length 2.</p>
<p><a id="X7C10492081D72376" name="X7C10492081D72376"></a></p>
<h5>41.2-7 OnSetsSets</h5>
<div class="func"><table class="func" width="100%"><tr><td class
<pimplements the action on sets ofsets the special case that the sets are pairwise disjoint, it is possible to use <code class="func">OnSetsDisjointSets</code> (<a href="chap41_mj.html#X7E23686E7A9D3A20"><span class="RefLink">41.2-8</span></a>). <var class="Arg">set</var> must be a sorted list whose entries are again sorted lists, otherwise an error is triggered (see<>Let varclass="Arg"vec<var>bea <normed/> vector, that is,its nonzero the identity of the relevant field, see <code class="func">NormedRowVector</code> (<a href="chap23_mj.html#X785DC60D8482695D"><span class="RefLink">23.2-1</span></a>). The function <code class="func">OnLinesjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<ivclass="xample"<re
<span class="GAPprompt">gap></span> <span spanG"&;/span>span "h=Actiongl,(gl,,,nLines;<span
[[1,4] [ ,3]]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
< class>&t<span><span class="GAPinput">Orbit( g, [ [ 1, 2 ], [ 3, 4 ] ], OnSetsSets );</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ]<5>412-13OnIndeterminates/h5>
[ [ 1, 3 ], [ 2, 4 ] ] ]
</re></>
<p><a id="X7E23686E7A9D3A20" name="X7E23686E7A9D3A20"></a></p>
<h5>41.2-8 OnSetsDisjointSets</h5>
<div class="func"><table <p>A permutation <var class=Apermutation < classclass="Arg">perm</var> acts on the multivariate polynomial <var class="Arg">poly</var> by permuting the indeterminates as it permutes points.</p>
mplements on setsof pairwise disjoint sets (see also< class="func">OnSetsSets</code> (<a href="chap41_mj.html#X7C10492081D72376"><span class="RefLink">41.2-7</span></a>)). <var class="Arg">set</var> must be a sorted list whose entries are again sorted lists, otherwise an error is triggered (see <a href=<pan class="APprompt>>span> <span class="GAPinput">x:=Indeterminate(Rationals,1);; y:=Indeterminate(Rationals,2);;</span>
<> idX7ADE244E819035FFname="X7ADE244E819035FF"></a></p>
<h5>41.2-9 OnSetsTuples</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OnSetsTuples</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements "func"< " =100"<rjava.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 245
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnSetsTuples( [ [ <p>The followingexampledemonstratescodeclass="unc>/> ( href="htmlX7B5A19098406347Ajava.lang.StringIndexOutOfBoundsException: Range [123, 122) out of bounds for length 221
[ [ 1, 4
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit( g, [ [ 1, 2 ], [ 3, 4 ] ], OnSetsTuples );</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ],
[ [ 1, 3 ], [ 4, 2 ] ], [ [ 2, 4 ], [ 3, 1 ] ],
[ [ 2, 1 ], [ 4, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ] ]
</pre></div>
<p><a id="X7FF556CD7E6739A9" name="X7FF556CD7E6739A9"></a></p>
<41.- OnTuplesSets/5java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
<java.lang.StringIndexOutOfBoundsException: Range [16, 4) out of bounds for length 247
<p>implementstheaction on tuplesof sets. <var class="Arg">set</var> must be a list whose entries are again sorted lists, otherwise an error is triggered (see <a href="chap41_mj.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnTuplesSets( [ [ 2, 3 ], [ 3, 4 ] ], (1,2,3) );</span>
[ [ 1, 3 ], [ 1, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit( g, [ [ 1, 2 ], [ 3, 4 ] ], OnTuplesSets );</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ],
[ ,4 ] [ 3, ,[1,2 ]java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
[ [ 1, 4 ], [ 2, 3 ] ], [ [ 2, 4 ], [ 1, 3 ] ] ]
/</
<p><a id="X844E902382EB4151" name="X844E902382EB4151"></a></pp> "82181"name=""/>>
<h5>41.2-11 OnTuplesTuples</h5>
< class=">table "uncwidth=100%"<tr< =tdleft"<="unc">8227 OnTuplesTuples/ode var=Arg>/, var classArg><var </<tdclass="(nbspfunction&;)/td><>/table></div>
<p>implements the action on tuples of tuples.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnTuplesTuples( [ [ 2, 3 ], [ 3, 4 ] ], (1,2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 3, 1 ], [ 1, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(2,3,4));;</span>
<span class="GAPprompt`Action on canonical representatives''.
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ],
[ [ 1, 3 ], [ 4, 2 ] ], [ [ 3, 1 ], [ 2, 4 ] ],
[ [3 4 ] [1 2 ]], [ [2, 1],[ 4, 3 ] ],
[ [ 1, 4 ], [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 2 ] ],
[ [ 4, 2 ], [ 1, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ],
[ [ 2, 4 ], [ 3, 1 ] ], [ [ 4, 3 ], [ 2, 1 ] ] ]
</re>/iv>
<p><a id="X86DC2DD5829CAD9A" name="X86DC2DD5829CAD9A"></a></p>
<h5>41.2-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OnLines</code>( <var class="Arg">vec</var>, <var class="Arg">g</var> )</td><
<p>Let << class="">ec/> be <em</em>row vector, is, its first nonzero entry is normedtothe of relevant field see<code ="func">NormedRowVector/code (<="hap23_mj.#X785DC60D8482695D"<span class=""23.-1<span><a>) The <code class="func>OnLines/code the rowvectorobtained firstmultiplying <var class="Arg">vec</var> from the right with <var class="Arg">g</var> (via <code class="func">OnRight</code> (<a href="chap41_mj.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>)) and then normalizing the resulting row vector by scalar multiplication from the left.</p>
<p>This action corresponds to the projective action of a matrix group on one-dimensional subspaces.</p>
<p>If <var class="Arg">vec</var> is a zero vector or is not normed then an error is triggered (see <a href="chap41_mj.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnLines( [ 1, 2 ], [ [ 1, 2 ], [ 3, 4 ] ] );</span>
[ 1, 10/7 ]
<span ="">gap>/span> span class="GAPinput">gl:=GL(2,5);;v:=[1,0]*Z(5)^0;</span>
[ Z(5)^0, 0*Z(5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Actionh5414- Orbit<h5>
Group([ (2,3,5,6), (1,2,4)(3,6,5) ])
</pre>/iv
<p><a id="X7FA394D27E721E2B" name="X7FA394D27E721E2B"></a></p>
<h5>41.2-13 OnIndeterminates</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OnIndeterminates</code>( <var class="Arg">poly</var>, <var class="Arg">perm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>A permutation <var class="Arg">perm</var> acts on the multivariate polynomial <var class="Arg">poly</var> by permuting the indeterminates as it permutes points.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Indeterminate(Rationals,1);; y:=Indeterminate(Rationals,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnIndeterminates(x^7*y+x*y^4,(1,17)(2,28));</span>
x_17^7*x_28+x_17*x_28^4
<span class="GAPprompt">gap></span> <span class="[1 3, 2 4]
Group([ (1,2), (3,4) ])
</pre></div>
<p><a id="X7BA8D76586F1F06E/pre></div
<h5.-/h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">p> id=X86BCAE17869BBEAAname="></a></java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
< exampledemonstrates"func"Permutedcode < =chap21_mjhtmlX7B5A19098406347A>spanclass=RefLink2120-7/></>)beingused toimplement adomain:/p
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
< class=GAPprompt>&;/span> <spanclass=GAPinput">dom:[ a" "b" "" ;<span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(g,dom,Permuted);</span>
[ [ "a", "b", "c" ], [ "c", "a", "b" ], [ "b", "a", "c" ],
[ "b", "c", "a" ], [ "a", "c", "b" ], [ "c", "b", "a" ] ]
</pre></div>
<
<h5
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 0
<div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0
<p>implements the operation of a matrix group on subspaces of a vector space. <var class="Arg">bas</var> must be a list of (linearly java.lang.StringIndexOutOfBoundsException: Range [0, 144) out of bounds for length 0
<p><var class="Arg">bas</var> must be given in Hermite normal form, otherwise an error is triggered (see <a href="chap41_mj.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<p><a id="X82181CA07A5B2056" name="X82181CA07A5B2056"></a></p>
>3 ""java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 78
<p>A variety of action functions assumes that the objects on java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0
<p>If orbit seeds or domain elements are not given in the required form <strong class="pkg">GAP</strong> will issue an error message:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(SymmetricGrouppN that java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 107
Error, Action not well
``Action <div class=example"<>
</pre></div>
<p>In this case the affectedspan class="APprompt">ap></span> <span class="GAPinput">Orbits(g,[1..5]);</span>
<p>(A similar
<>a= "java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
<h4>41.4 <span class="Heading">Orbits</span></h4>
th>\G\</ java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 435
<p><a id="X80E0234E7BD79409" name="X80E0234E7BD79409"></a></p>
<h5>41.4-1 Orbit</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Orbit</code>( <var class="Arg">G</var>[, <var class="Arg">Omega</var>], <var class="Arg">pnt</var>[, <var class="Arg">[ 2 4 3 ][ , 2 [43, ,
< is given.</p>
<p>(Note that the arrangement of points in this list is not defined by the operation.)</p>
<p>The orbit of <var class="Arg">pnt</var> will always contain one element that is <em>equal</em> to <var class="Arg">pnt</var>, however for performance reasons this element is not necessarily <em>identical</em> to <var class="Arg">pnt</var>, in particular if <var class="Arg">pnt</var> is mutable.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ 1, 3, 2, 4 ]
<pan=gapg;/ span">(,1,]java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 91
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2,java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 340
<pre>/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<p>(See Section <a href="chap41_mj.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>
<p><a id="X86BCAE17869BBEAA" name="X86BCAE17869BBEAA"></a></p>
<h5>41.4-2 Orbits</h5>
<div class="func"><table class="funcd =f< "java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 223
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Orbits</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Orbits</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></java.lang.StringIndexOutOfBoundsException: Index 209 out of bounds for length 0
<p>returns a duplicate-free list of the orbits of the elements in <var class="Arg">seeds</var> under the action <var class="Arg">act</var> of <var class="Arg">G</var> or under <code class="func">OnPoints</code> (<a href="chap41_mj.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) if no action function is given.</p>
<>For permutation group <varclass="rgG</var>, one may also invoke as< class=code"Orbits(var class""><var></> all the orbits of its natural action on the set of points moved by it. For example the group <span class="SimpleMath">\(\langle (1,2,3), (4,5) \rangle\)</span> has the orbits <span class="SimpleMath">\(\{1,2,3\}\)</span> and <span class="SimpleMath">\(\{4,5\}\)</span>.</p>
<p>(Note that the arrangement of orbits or of points within one orbit is not defined by the operation.)</p><div =""<table=f "%<tr><td class="tdleft"><code class="func">‣ OrbitLengthsDomain</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X86BC8B958123F953" name="X86BC8B958123F953"></<Thisjava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 298
<h5>41.4-3 <span class="Heading">OrbitsDomain</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitsDomain</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitsDomain</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitsDomain</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table><<span class="APprompt>gap&</span>span class=GAPinput">OrbitLength(g,[1,2,3,4],OnTuples);</span>
< list theorbits of < java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 328
<p>This operation is often faster than <code class="func">Orbits</code> (<a href="chap41_mj.html#X86BCAE17869BBEAA"><span class="RefLink">41.4-2</span></a>). The domain <var class="Arg">Omega</var> must be closed under the action of <var class="Arg">G</var>, otherwise an error can occur.</p>
<p>For a permutation group <var class="Arg">G</var>, one may also invoke this as <code class="code">OrbitsDomain(<var class="Arg">G</var>)</code>, which returns all the orbits of its natural action on the java.lang.StringIndexOutOfBoundsException: Index 208 out of bounds for length 5
<p>(Note that the arrangement of orbits or of points within one orbit is not defined by the operation.)</p>
<div class="example"><pre>
<=gapg;<span <span "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 96
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbits(g,[1..5]);</span>
[ [ 1, 3, 2, 4 ], [ 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitsDomain(g,Arrangements([1
[ [ [ 1, 2, 3 ], [ 3, 1, 2 ], [ 1, 4, 2 ], [ 2, 3, 1 ], [ 2, 1, 4 ],
[ 3, 4, 1 ], [ 1, 3, 4 ], [ 4, 2, 1 ], [ 4, 1, 3 ],
[ 2, 4, 3 ], [ 3, 2, 4 ], [ 4, 3, 2 ] ],
[ [ 1, 2, 4 ], [ 3, 1, 4 ], [ 1, 4, 3 ], [ 2, 3, 4 ], [ 2, 1, 3 ],
[ 3, 4, 2 ], [ 1, 3, 2 ], [ 4, 2, 3 ], [ 4, 1, 2 ],
[ 2, 4, 1 ], [ 3, 2, 1 ], [ 4, 3, 1 ] ] ]
="gap&;/span>< =""(,2)^2[123),(,(,),/>
<span class="GAPprompt">></span> <span class="GAPinput">[[[Z(2)^0,Z(2)^0],[Z(2)^0,0*Z(2)]],[[Z(2)^0,0*Z(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ < GF2 length2&;],
[ <an immutable GF2 vector of length 2>,
<an immutable GF2 vector of length 2>,
<an immutable GF2 vector of length 2> ] ]
</pre></div>
<p>(See Section <a href="chap41_mj.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about
<p><a id="X799910CF832EDC45" name="X799910CF832EDC45"></a></p>
41.-4</
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitLength</code>( <var class="Arg">G</var>[, <java.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 27
<p>computes the length of the orbit of <var class="Arg">pnt</var> under the action function <var class="Arg">act</var> or <code class="func">OnPoints</code> (<a href="chap41_mj.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X8032F73078DF2DDB" name="X8032F73078DF2DDB"></a></p>
<h5>41.4-5 <span class="Heading">OrbitLengths</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitLengths</code>( <var class="Arg[ 2, 4 : ( (,)3,)))
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 27
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitLengths</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the lengths of all the orbits of the elements in <var class="Arg">seeds</var> underp> standardjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 348
<p>For a permutation group <var class="Arg">G</var>, one may also invoke this as <code class="code">OrbitLengths(<var class="Arg">G</var>)</code>, which returns the lengths of all the orbits of its natural action on the set of points moved by it. For example the java.lang.StringIndexOutOfBoundsException: Range [381, 268) out of bounds for length 381
/a>
<h5>41.4-6 <span class="Heading">OrbitLengthsDomain</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitLengthsDomain</
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitLengthsDomain</code>( <var class="Arg">G</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< class"unc>table =func width="00%><tr><dclass="dleft">code class="unc">&8227;OrbitLengthsDomain</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the lengths of all the orbits of <var class="Arg">G</var> on <var class="Arg">Omega</var>.</p>
<p>This operation is often faster than <code class="<p><a ="X7A9389097BAF670D name""</a<p
<p>For a permutation group <var class="Arg">G</var>, one may also<h4>41.6 span class="Heading">Elements with Prescribed Images</span></h4>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitLength(g,[1,2,3,4],OnTuples);</span>
12
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitLengths(g,<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func8227;RepresentativeAction</code>(<var class="Arg"><var>[<var class="">Omega</var>], <var class="Arg">d</var>, <var class="Arg">e</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class"example">pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(5,6,7));;</span>
< class=GAPprompt>ap;</pan> <pan class="APinput"OrbitLengthsDomaing,1,,3);/span
[ 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitLengthsDomain(g);</span>
< class="APprompt>&;/>< class="GAPinput">epresentativeAction(,,,OnPoints)/span
</pre></div>
<p><a id="X797BD60E7ACEF1B1<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(g,(1,2,3),(2,4,3));</span>
<h4>41.5 <span class="Heading">Stabilizers</span></h4>
<p>The <em>stabilizer</em> of a point <span class="SimpleMath">\(x\)</span> under the action of a group <span class="SimpleMath">\(G\)</span> is the set of all those elements in <span class="SimpleMath">\(G\)</span> which java.lang.StringIndexOutOfBoundsException: Index 224 out of bounds for length 4
<p><a id="X7C34EC437EF598BF" name="X7C34EC437EF598BF"></a></p>
<5>41.-1 OrbitStabilizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitStabilizer</code>( <var class="Arg">G</var>[, <(,,)
<p>computes the orbit and the stabilizer of <var class="Arg">pnt</var> simultaneously in a single orbit-stabilizer algorithm.</p>
<p>The stabilizer will have <var class="Arg">G</var> as its parent.</p>
<p><a id="X86FB962786397E02" name="X86FB962786397E02"></a></p>
<h5>41.5-2 Stabilizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Stabilizer</code>( <var class="Arg">G</var>[, <var class="Arg">Omega</var>], <var class="Arg">pnt</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( function )</td></tr></table></div>
< in<class=Arg"G/var>> of the point <varclass">pnt</var> that is thesubgroup ofthose elements of <var class="Arg">G</var> that fix <var class="Arg">pnt</var>. The stabilizer will have <var class="Arg">G</var> as its parent.</p>
<div class="example"><pre>
<<pWhen group< class="SimpleMath">\(G\)</span>acts adomain <span class="SimpleMath">\(\Omega\)</span>, an enumeration of <span class="SimpleMath">\(Omega\)</span> yields a homomorphism from <span class="SimpleMath">\(G\)</span intothe symmetric group on <span class="SimpleMath">\(\{ 1, \ldots, |\Omega| \}\)</span>. In <strong class="pkg">GAP</strong>, the enumeration of <span class="SimpleMath">\(\Omega\)</span> is provided by the <code class="func">Enumerator</code> (<a href="chap30_mj.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) value of <span class="SimpleMath">\(\Omega\)</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<span class="GAPprompt">gap></spanh5>1.71<span class"Heading">ActionHomomorphism></h5>
Group([ (1,3,2) ])
<span class="GAPprompt">gap></div=""><table class="func" width="100%"><tr><td class="tdleft"><code class="func">8227; ActionHomomorphism</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>][, <var class="Arg">"surjective"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
Group([ (1,3,2), (2,4,3) ])
</pre></div>
<p>The stabilizer of a set or tuple of points can be computed<div ="func"><table =func width=100%>tr<td classtcode class="unc"&8227 </code var class=Arg"action</><td<td "dright(n;function )</td></tr></table></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(g,[1,2],OnSets);</span>
23,]
<span
pBydefault returnedby< =""ActionHomomorphism notnecessarily surjective (java.lang.StringIndexOutOfBoundsException: Range [123, 122) out of bounds for length 444
<span class="GAPprompt">gap></span> <span class="GAPinput">orbstab:=OrbitStabilizer(g,[1,2],OnSets);</span>
rec(
orbit := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ],
[ 2, 4 ]
< classGAPprompt">gap&;/span> <span class="GAPinput">Parent(orbstab.stabilizer);</span>
Group([ (1,3,2), (2,4,3) ])
</pre></div>
<p>(See Section <a href="chap41_mj.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>
<p>The standard methods for all these actions are an orbit-stabilizer algorithm. For permutation groups backtrack algorithms are used. For solvable groups an orbit-stabilizer algorithm java.lang.StringIndexOutOfBoundsException: Index 187 out of bounds for length 94
<p><a78C3A8568414BC44>/>/>
<h5>41.5-3 OrbitStabilizerAlgorithm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitStabilizerAlgorithm</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>, <var class="Arg">blist</var>, <var class="Arg">gens</var>, <var class="Arg">acts</var>, <var class="Arg">pntact</varGroup(
<p>This operation should not be called by a user. It is documented however for purposes to extend or maintain the group actions 24, 17(2,)(3,(410)5,1)6,)13,5),)(,8(,
<code =func">OrbitStabilizerAlgorithm</ode> > performs an orbit stabilizer algorithm for the group <var class="Arg">G</var> acting with the generators <var class="Arg">gens</var> via the generator images <var class="Arg">gens</var> and the group action <var class="Arg">act</var> on the element <var class="Arg">pnt</var>. (For technical reasons <var class="Arg">pnt</var> and <var class="Arg">act</var> are put in one record with components <code class="code">pnt</code> and <code class="code">act</code> respectively.)</p>
<p>The <var class="Arg">pntact</var> record may carry a component <var class="Arg">stabsub</var>. If java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<p>The <var class="Arg">pntact</var> component also may contain the boolean entry <code class="code">onlystab</code> set to <code class="keyw">true</code>. In this case the <code class="code">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<The argument <class="">Omega</var> (which may be replaced by <code class="keyw">false</code> to be ignored) is the set within which the orbit is computed (once the orbit is the full domain, the orbit calculation may stop). If <var class="Arg">blist</var> is given it must be a bit list corresponding to <var class="Arg">Omega</var> in which elements which have java.lang.StringIndexOutOfBoundsException: Index 372 out of bounds for length 216
<p><a id="X7A9389097BAF670D" name="X7A9389097BAF670D"></a></p>
<h4>41.6 <span class="Heading">Elements with Prescribed Images</span></h4>
<>a =XDC7B085EB0539 X857DC7B085EB0539>a>/>
<h5>41.6-1 RepresentativeAction</h5>
<div class="func"><table class="func" java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0
<p>computes an element of <var class="Arg">G</var> that maps <var class="Arg">d</var> to <var class="Arg">e</var> under the given action and returns <code class="keyw">fail</code> if no such element exists.</p>
<divdiv =>>
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 96
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(gspanclassGAPprompt>&;/> <span =">(g,sList(g),OnRight)</span
(1,3)(2,4)
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 0
(1,3)(2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction
(24)
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(g,(1,2,3),(2,3,4));</span>
ail
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(g,Group((1,2,3)),Group((2,3,4)));</span>
(24java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<span class="GAPprompt">gap></span> <span class="GAPinput"> RepresentativeAction(g,[1,2,3],[1,2,4],OnSets);</span>
(2,4,3)
<</ < "APinput">(,123[,,]OnTuples;span>
fail
</pre></div>
<p>(See Section <a href="chap41_mj.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>
<p>Again the standard method for < class=func"RepresentativeAction/> is an orbit- ,for java.lang.StringIndexOutOfBoundsException: Range [137, 136) out of bounds for length 192
<p><a id="X87F73CCA7921DE65" name="X87F73CCA7921DE65"></a></p>
<h4>41.7 <span class="Heading">The Permutation Image of an Action</span></h4>
<p>When a group <span class="SimpleMath">\(G\)</span> acts on a domain <spanspan class="GAPprompt">gap></span> <span class="GAPinput">h:=Group(Z(3)*[[[1,1],[0,1]]]);</span>
<For an action homomorphism, the operation <code class="func">UnderlyingExternalSet</code> (<a href="chap41_mj.html#X8190A8247F29A5C7"><span class="RefLink">41.12-16</span></a>) will return the external set on <span class="SimpleMath">\(\Omega\)</span> which affords the action.</p>
=X78E6A002835288A4>a/p>
<h5>41.7-1 <span class="Heading">ActionHomomorphism</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<divjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 230
<>computes ahomomorphism from <var class="Arg">G</var> into the symmetric group on <span class="SimpleMath">\(|\textit{Omega}|\)</span> points that gives the permutation action of <var class="Arg">G</var> on <var class="Arg">Omega</var>. (In particular, this homomorphism is a permutation equivalence, that is the permutation image of a group element is given by the positions of points in <var class="Arg">Omega</var>.)</p>
<p>The result is undefined if <var class="Arg">G</var<>a =" "784D417D87F4E58D<>/
<p>By default the homomorphism returned by <code class
<p>The =>java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 251
<p>(See Section <a href="chap41_mj.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
span">&;<><spanclass=G">:ActionHomomorphism,[.43,nTuples/pan
<action homomorphism>
<span class="GAPprompt">gap></span> <span class="GAPinput"<div ">pre
Group(
[ (1,9,13)(2,10,14)(3,7,15)(4,8,16)(5,12,17)(6,11,18)(19,22,23)(20,21,
24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,15)(14,16)(17,18)(19,
21)(20,22)(23,24) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Range(hom));Size(Image(hom));</span>
620448401733239439360000
6
<span class="GAPprompt">gap&(1,4)(2,6)(3,5)(7,8)(10,12)(1314) ]
<span class="GAPprompt">></span> <span class="GAPinput">"surjective");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Range(hom));</span>
6
</pre></div>
<>WhenWhenacting on a domain, the operation <code class="func">PositionCanonical</code> (<a href="chap21_mj.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>) is used to determine the position of elements in the domain. This can be used to act on a domain given java.lang.StringIndexOutOfBoundsException: Range [0, 277) out of bounds for length 3
<p><a id="X85A8E93D786C3C9C" name="X85A8E93D786C3C9C"></a></p>
<h5>p>"X8561DEBA79E01ABD"name="X8561DEBA79E01ABD"></a></p>
<div class="func"><table class="func" width="100%"><tr>
< class="unc"><table class=func" width=100%"tr<td class=tdleft"<code class=func">='color: green'>8227; Action</code>( <var class="Arg">xset</var> )</td><td class="tdright">( function )</td></tr></table></div>
java.lang.StringIndexOutOfBoundsException: Range [14, 10) out of bounds for length 196
<p>Note that (for compatibility reasons to be able to get the action homomorphism) this image group internally stores the action homomorphism. If <var class="Arg">G</var> or <var class="Arg">Omega</var> are extremely big, this can cause memory problems. In this case compute only generator images and form the image group yourself.</p>
<p>(See Section <a href="chap41_mj.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>
<p>The following code shows for example how to create the regular action of a group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span classh5418-3AbelianSubfactorAction
< =""gapgt;/<spanclass"GAPinputA(,(,java.lang.StringIndexOutOfBoundsException: Range [89, 88) out of bounds for length 97
Group([ (1,5,3)p> < class"rg">G</var> be a group and <span class="SimpleMath">\(\textit{M} \ \textit{})<span> be subgroups common parentthat are under var ="G</> such thatthe subfactor <span class="SimpleMath">\(\textit{M}/\textit{N}\)</span> is elementary abelian. The operation <code class="func">AbelianSubfactorAction</code> returns a list <code class="code">[ <var class="Arg">phi</var>, <var class="Arg">alpha</var>, <var class="Arg">bas</var> ]</code> where <var class="Arg">bas</var> is a list of elements of <var class="Arg">M</var> which are representatives for a basis of <span class="SimpleMath">\(\textit{M}/\textit{N}\)</span>, <var class="Arg">alpha</var> is a map from <var class="Arg">M</var> into a <span class="SimpleMath">\(n\)</span>-dimensional
>
<p><span class="GAPprompt">></span> <span class="GAPinput"> (1,9,5,6,3,10)(2,11,12,8,4,7));;</span>
<5>41.7-3 /java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; SparseActionHomomorphism</code>( <var class="Arg">G</var>, <var class="Arg">start</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width my_group,(GF2^
<p><code class="func">2)functione) . r .end )java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
e domain notsurjectivelythis yield ofjava.lang.StringIndexOutOfBoundsException: Range [128, 127) out of bounds for length 182
<p>The operation <code class="func">SparseActionHomomorphism</code> will only useGroup([ <an immutable 2x2 matrix over GF2>,
<p>The operation <code class="func">SortedSparseActionHomomorphism</code> in contrast will sort the orbit and thus produce an image group which does<span class"GAPprompt">">ap><span><span class="APinput">Size(mat);</span>
<div class="example"><pre>
< "GAPprompt">apgt;/span><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 100
Group([ [ [ Z(3), Z(3) ], [ 0*Z(3), Z(3) ] ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=ActionHomomorphism(h,GF(3)^2,OnRight);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(hom);</span>
Group([ (2,3)(4,9,6,7,5,8) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=SparseActionHomomorphism(h,[Z(3)*[1,0]],OnRight);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(hom);</span>
Group([ (1,2,3,4,5,6) ])
</pre></div>
<p><a id="X7FED50ED7ACA5FB2" name="X7FED50ED7ACA5FB2"></a></p>
<h4>41.8 <span class="Heading">java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 20
<p>Of particular importance is the java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<p><a id="X784D417D87F4E58D" name="X784D417D87F4E58D"></a></p>
<h5>41.8-1 FactorCosetAction</h5>
<="tdleft"><code=func" FactorCosetAction</code>( <var class="Arg">G</var>, <var class="Arg">U</var>[, <var class="Arg">N</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" p>If only the permutation image of a singleneeded, it might not be worth to create the action homomorphism, the following operations yield the permutation image and cycles of a single element.</p>
<p>This command computes the action of the group <<p><a id="X7807A33381DCAB26" name="X7807A33381DCAB26>/>
<div <h541.9-1 <span class="Heading">Permutation</span></h5>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4,5),(1,2));;u:=SylowSubgroup(java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 | | |