xversion10"encoding="UTF8"?

<script
         "http://www.3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<script type="text/javascript"
  src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (Images) - Chapter 2: Minimal and Canonical Images</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" />
<="anualj ="ext/avascript>/cript
<script type="text/javascript">overwriteStyle();</scriptrel""type"ext/css"href=". java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
</head
<bodydiv class"hlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   


<div class="chlinktop"><span class="chlink1">Goto <p><a id="X7B4A093B7FBA4C23" name="X7B4A093B7FBA4C23

<div class="<div class="ContSect"><span"ocline"><span class="nocss"> </span><a href="chap2_mj.html#X7DA24AB4803F68DA<span>

<p id="mathjaxlink" class="pcenter"><a href="chap2.html">[MathJax off]</a></<span class="ContSS"><br /><span class="nocss">  </span><a href="chX848A9B117C3EEF48">2.1-2 IsMinimalImageLessThan</a></span>
<>a ="7B4A093B7FBA4C23" name="X7B4A093B7FBA4C23"></a></p>
<div class="ChapSects"><a href="chap2_mj.html#X7B4A093B7FBA4C23">2 <span class=<pan class="ContSS"><br /><span class="nocss">  </span></div>/div>
<div class=
</p>Givena group G and action ,theminimalimage of an object O is the smallest image of O under any element of G, under the action A.</p>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.java.lang.StringIndexOutOfBoundsException: Range [0, 88) out of bounds for length 0
<span class="ContSS"><br /><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<spanclass=="APprompt">gap></span> <span class="GAPinput">G := Group((1,2,3)(4,5<pan class=GAPprompt">gap></span> <span class="GAPinput">List(G, g -> java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 65
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X803A8BB57A63C577">2.1-4 ImagesAdvancedConfig</a></span>
</div></div>
</div>

<> span class"Heading"and <>/

<p<pan class=GAPpromptgap&t</span><panclass="GAPinput">Minimum(List(G, g -> OnSets([2,3,5,7], g) ) );</span>

<p>As a more concrete example, let us consider the

<


<div class="example"><pre>
<<pan  class="APprompt"gap>span <panclass="GAPinput">LoadPackage("images", false);</span>
<span class=GAPprompt">gap></span> <span class="GAPinput">List(G, g -> OnSets([2,3,5,7], g) );</[1,2, 4 9 ]
[ [ 2, 3, <pre></iv>
   7, 8],[3, 5,7, 9 ] [1,5, 6,8 ] [ 3 ,5,7 ]
  [ 2, 4, 6, 9 ] ]
<span class="GAPprompt">span class="GAPprompt">gap></span> <span class="GAPinput">MinimalImage(G, [2,3,5,7], [1, 2,4 9]
 1,2 4,9 ]
</pre></div>

<p>This is very inefficient,[1 2,4 9 ]


<div class="example"><pre>
<span class="java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 14
ue
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 0
[ 1, 2, 4, 9 ]
</pre></div>

<p>p><strong class="button">WARNING:</strong> The value of MinimalImage java.lang.StringIndexOutOfBoundsException: Range [0, 74) out of bounds for length 0


<div class="example"><pre>
<span 
<4>2. <pan class="eading">Function documentation</span></h4>
<span 
[ 1,2, 4, 9 ]
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 104

</pre></div>

<p>In this situation,

<p by ,thispackageprovides CanonicalImage (,,)returns someimage of O byan element Gunderthejava.lang.StringIndexOutOfBoundsException: Range [142, 141) out of bounds for length 454

<p><strong class="button">WARNING:</strong> The value of MinimalImage will remain identical between versions of GAP and the Images package, unless bugs are discovered. This is NOT true for CanonicalImage.</p>

<p><a id="X7DA24AB4803F68DA" name="X7DA24AB4803F68DA"></a></p>

<h4>2.1 <span class="Heading">Function documentation</span></h4>

<p><a id="X7E9EFA0E7A997586" name="X7E9EFA0E7A997586"></a></p>

<h5>2.p><code class=>MinimalImage</ode>returns the minimal image of <var class="Arg">pnt</var> under the group <var class="Argjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 0
div="> class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMinimalImage</code>( <var class="Arg">G</var>, <var class="Arg">pnt</var>[, <var class="Arg">act</var>][, <var class="Arg">Config</var>] )</td><td class="tdright">( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 class="Arg">G/ar,<class=Arg"pnt</ar>[, var class="Arg">act</var>][, <var class="Arg">Config</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">MinimalImage</code> returns the minimal image of <var class="Arg">pnt</var> under the group <var class="Arg">G</var>. <code class="p><code class="func">IsMinimalImageLessThan</code> checks if the minimal image of <var class="Arg">A</var> under the group <var class="Arg">G</var> is smaller>B</ar>.</p>

<p><code class="func">MinimalImagePerm</code> returns the permutation which maps <var class="Arg">pnt</var> to its minimal imagejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The option <var class="Arg">Config</var> defines a number of advanced configuration   options, which are described in 'ImagesAdvancedConfig'.</p>

<p

<h5>2.1-2 IsMinimalImageLessThan</h5>

<div class="func"><table class="func" width="100%d class"unc"><table class=func"width=100"<<tdclass="tdleft"><code class="func">‣ IsCanonicalImage</code>( <var class="Arg">G</var>[class="func>tableclass=func"width="00%>tr>td class="tdleft><ode class="func>#8227 CanonicalImagePermcode( <varclass=ArgG/ar[<varclass""pnt<var],<var class=Arg>act/ar][, <arclass=Arg"Config</ar>)td< class=tdright">&bspfunction )/d>/>/>/>
<>codeclass"unc"IsMinimalImageLessThan/ode    minimalimage <arclass=Arg"></ar> under the group <var class="Arg">G</var> is smaller than <var class="Arg">B</var>.</p>

<p>It returns MinImage.Smaller, MinImage.Equal or MinImage.Larger, if the minimal image of <var class="Arg">A</var> is smaller, equal or larger than <var class="Arg">B</var>.</p>

<p>The option <var class="Arg">Config</var> defines a number of advanced configuration   options, which are described in 'ImagesAdvancedConfig'.</p>

<p><a id="X780189E3873B7A42" name="X780189E3873B7A42"></a></p>

<h5>2.1-3 CanonicalImage</h5>

<div class="func
<div class=func">table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCanonicalImage</code>( <var class="Arg">G</var>[, <var class="Arg
<java.lang.StringIndexOutOfBoundsException: Range [91, 4) out of bounds for length 318
<p>code class"func"CanonicalImage</code> returns a canonical image of <var class="Arg">pnt</var> under the group <var class="Arg">G</var>. <code class="func">IsCanonicalImage</code> returns a boolean which is <code class="keyw">true</code> if <code class="func">CanonicalImage</code> would return <var class="Arg">pnt</var> (so the value is it's own minimal image).</p>

<p><code class="func">CanonicalImagePerm</code> returns the permutation which maps <var class="Arg">pnt</var> to its minimal image.</p>

<p>By default, these functions use the fastest algorithm for calculating canonical images, which is often changed in new versions of the package. The option <var class="Arg">Config</var> defines a number of advanced configuration   options, which are described in 'ImagesAdvancedConfig'. These include the ability to choose the canonicalising algorithm used.</p>

<p><a id="dd><p>The search used while building the image. There are many configuration options available. We shall list here just the three most useful ones. A full list is java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>2.1-4 ImagesAdvancedConfig</h5>

<div class="func<trong class="<="code"CanonicalConfig_FixedMinOrbit</t>
<d>  set the of integersgivenby the MinOrbitPermfunction./>


<dl>
<dt>java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
< while building the image.There aremany configuration options available. We shall list here just the three most useful ones. A full list is in the paper "Minimal and Canonical Images" by the authors of this package.</p>


<dl>
<dt><strong class="Mark"><code class="code">CanonicalConfig_Minimum<
<d<p>exicographically smallest set -- same as using MinimalImage.</p>

</dd<>By default,during search we perform cheap checks to try to find extra elements of the stabilizer. Pass java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<</code></strong></dt>
<dd><p>Lexicographically smallest set under the ordering of the integers <dd<p>Return the calculated value of <code class="code">Stabilizer(<var class="Arg">G</var>,<var class="Arg">O

</dd>
<dt><strong class="Mark"><code class="code">CanonicalConfig_RareRatioOrbitFixPlusMin</code></strong></dt>
<>p>   algorithm()<p

</dd>
</dl>
</dd>
<dt>< class="footbyahrefh:/wwmath.-.de~rankL/"GAPDoc2HTML<a>/java.lang.StringIndexOutOfBoundsException: Range [112, 111) out of bounds for length 112
<dd><p>The group <code class="code">Stabilizer(<var class="Arg">G</var>,<var class="Arg">O</var>,<var class="Arg">A</var>)</code>, or a subgroup of this group; see <code class="func">Stabilizer</code> (<a href="../../../doc/ref/chap41_mj.html#X86FB962786397E02"><span class="RefLink">Reference: Stabilizer</span></a>). If this group is large, it is more efficient to pre-calculate it. Default behaviour is to calculate the group, pass <code class="code">Group(())</code> to disable this behaviour. This is not checked, and passing an incorrect group will produce incorrect answers.</p>

</dd>
<dt><strong class="Mark"><code class="code">disableStabilizerCheck</code> (default <code class="keyw">false</code>)</strong></dt>
<dd><p>By default, during search we perform cheap checks to try to find extra elements of the stabilizer. Pass true to disable this check, this will make the algorithm MUCH slower if the stabilizer argument is a subgroup.</p>

</dd>
<dt><strong class="Mark"><code class="code">getStab</code> (default <code class="keyw">false</code>)</strong></dt>
<dd><p>Return the calculated value of <code class="code">Stabilizer(<var class="Arg">G</var>,<var class="Arg">O</var>,<var class="Arg">A</var>)</code>. This may return a subgroup rather than the whole stabilizer.</p>

</dd>
</dl>

<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap1_mj.html">[Previous Chapter]</a>    <a href="chap3_mj.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><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="chapInd_mj.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>