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 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
15
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorCosetAction(g,u);</span>
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15),\
  (1,4)(2,6)(3,5)(7,8)(10,12)(13,14) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(Range(last));</span>
"S5"
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorCosetAction(g,[u,p>computes the permutation that corresponds to the action of <var class="Arg">g</var> on the"rg>Omega</ (list java.lang.StringIndexOutOfBoundsException: Range [153, 152) out of bounds for length 693
<Imagelast)</>
120
</pre></div>

<p>The correspondence of points with cosets will, for performance reasons, depend on the method used. It is not guaranteed that it will be the same as used by <code class="code">RightTransversal</code> or <code class="code">RightCosets</code>.</p>

<p><a id="X8561DEBA79E01ABD" name="X8561DEBA79E01ABD"></a></p>

<h5>41.8-2 RegularActionHomomorphism<><aid"X81D4EA42810974A0" name="X81D4EA42810974A0"></a></p>

<div class
<p>returns <5>41.-PermutationCycle</>

<p><a id="X835317A7847477D4" name="X835317A7847477D4"></a></p>

<h5>41.8-3 AbelianSubfactorAction</h5>

<divclass="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AbelianSubfactorAction</code>( <var class="Arg">G</var>, <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr
<">\(\textit{} \geq \extit{N}\)</span> be subgroups of a common parent that are normal under <var class="Arg">G</var>, such that the 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(,76),,)

<p>Note: If only matrices for the action are needed, <code class="func">LinearActionLayerspan class="">ap><span <span class="GAPinput">PermutationCycle((1,2,3)(4,5)(6,7),[4..7],4);</span>


<div class="example"><pre>
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<span class="GAPprompt">></span> <span class="GAPinput">            (1,9,5,6,3,10)(2,11,12,8,4,7)
<"gap/> spanjava.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 101
[ 96, 48, 16, 4, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HasElementaryAbelianFactorGroup(c[3],c[4]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName(c[3],"my_group");;</span>
<"GAPprompt>gap&java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 108
[ [ (1,8,10,7,3,5)(2,4,12,9,11,6), (1,9,5,6,3,10)(2,11,12,8,4
    [ <an immutable 2x2 matrix over GF2>,
      <an immutable 2x2 matrix over GF2> ],
  MappingByFunction( my_group, ( GF(2)^
    2 ), function( e ) ... end, function( r ) ... end ),
  Pcgs([ (2,9,3,8)(4,11,5,10), (1,6,12,7)(4,10,5,11) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=Image(a[1],g);</span>
([ l;java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 49
  <an immutable 2x2 matrix overjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
=>&;/panspan"Size()span>
3
<span class="GAPprompt">gap></span> <span class="p>returns a list of the cycles (as lists of points) of action element  class=""<<java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111
(2,9,3,8)(4,11,5,10)
<span class="GAPprompt">gap></span> <span class="GAPinput">e in c[3];e java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 28
true
false
<

<B"=""></><p>

Induced  Elements andjava.lang.StringIndexOutOfBoundsException: Range [82, 74) out of bounds for length 86

<>Ifonlythepermutationimage of a single is  might worthtocreate , following yieldthe  and of  single element</>

pa ="X7807A33381DCAB26" name="X7807A33381DCAB26"></a></p>

<h5>41.9-1 <span class="Heading">java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 22

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Permutation</code>( <var class="Arg">g</var>, <var class="[2, java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
<div <p<a id="X87FDA6838065CDCB" name=X87FDA6838065CDCB"></a></p>
<p>computes the permutation that corresponds to the action of <var class="Arg">g</var> on the permutation domain <var class="Arg">Omega</var> (a list java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>If <var class="Arg">g</var> does notidth"%"<tr<class=tdleft">code "unc>#8227 /code(<arclass=varclass"Arg"Omega/>[,<varclass=Arg"act</>)</d> ">function;)/td/<table<>

<p><a id="X81D4EA42810974A0" name="X81D4EA42810974A0"></a></p>

<h5>41.9-2 PermutationCycle</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><<p =center"\[
<p>computes the permutation that represents the cycle of <var class="Arg">pnt</var> under the action of the element <var class="Arg">g</var>.


<div class="example"><pre>
<span class="GAPprompt">gap></span> <spanp> em index/em a  varclass""Gvar is as/>
(2,7,6)(3,4,8)
<span class="GAPprompt">gap></span> <span class="GAPinput">Permutation((1,2,3)(4,5)(6,7),[4..7]);</span>
(1,2)(3,4)
<span class="GAPprompt">gap></span> <span class="
(1,2)
</pre></div>

<p><a id="X80AF6E0683CA7F14" name="X80AF6E0683CA7F14"></a></p>

<h5>41.9-3 Cycle</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#S_4(an style='color: green'>6) /[()
<preturns alist thepointsin of<var "pnt/>underthe action    var ="">g</>./p

<p>124*x_1^+/*2*^+/4*java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60

<h5>41.9-4 CycleLength</h5>

<div class="func"><table class="func
<p>returns the <44110 <spanclass"Heading>  Actions</</>

<p><a id="<p>< id=XB15750851828" ""><a>/>

<h5>41.9-5 Cycles</h5>

<iv"unc"<table"unc"width"100%"<tr<class"java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 273
<p>returns a list of the cycles (as lists of points) of the action of the element <var class="Arg">g</var>.</p>

<p><a id="X83040A6080C2C6C6" name="X83040A6080C2C6C6"></a></p>

<h5>41.9-6 CycleLengths</h5>

<div class="func"><table class="func" width="div class">tableclass=" "%>trtd class=dleft"< "unc&8227;IsTransitive/code( <class="></>)</d< class="java.lang.StringIndexOutOfBoundsException: Range [175, 174) out of bounds for length 222
<p>returns the lengths of all the cycles under the action of the element < if actionimplied by  ,or<class="keyw">false</code> otherwise.</p>


<iv class=example">pre>
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 4, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CycleLength((1,2,3)(4,5)(6,7),[4..7],4);</span>
2
<span class="GAPprompt"divclass="xample"pre
[[ 4 5]  6 7] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CycleLengths((1,2,3)(4,5)(6,7),[4..7]);</span>
[ 2, 2 ]
</pre></div>

<p><a id="X87FDA6838065CDCB" name="X87FDA6838065CDCB"></a></p>

<5>.- span =Heading"CycleIndex</java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 55

<div class="funcf
<div class="func"><table class="func" width="100%"><tr><td<panclass=G>gt</ span ">(G );   Gis  [2. 4 </pan
<p>The <em>cycle index</em> of a permutation span =GAPprompt"gap&t<span> span ="APinput>(SL2java.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 123

<p class="center">\[
z(<var class="Arg">g</var>) = s_1^{{c_1}} s_2^{{c_2}} \cdots java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
\]</p>

<p>where <span class="SimpleMath">\(c_k\)</span> is the number of <span class="SimpleMath<div class=f"<class="func"width=100"<<class=">code"">#8227;<code(<class=Arg"G/>""Omegavar,<varclass"rg>ens/>,<class=Argacts<var>[,var =Arg><var> <td>td class=tdright>(nbsp;&)/<tr><table<div

<p>The <em>cycle index</em> of a group <var class="Arg">G</var> is <div class="func"><table class="func" width="100%"><tr><td class="tdleft">#;<code(< =>/var )/tdtd ">&attribute&bsp)/<tr>/<div>

<p class="center">\[
Z(<var class="Arg">G</var>) = \left( \sum_{{g \in <var class="Arg">G</var>}} z(g) \right) / java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\]</p>

<p>The indeterminates used by <p>For a permutation group <var class="Arg">G</var>, one may also invoke this as <code class="code">Transitivity(<var class="Arg">G</var>)</code>, which returns the degree 


<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 93
S_4(6c) = 1/2[2^3]S(3)
<span class="java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
1/24*x_1^6+1/8*x_1^2*x_2^2+1/4*x_1^2*x_4+1/4*x_2^3+1/3*x_3^2
</pre></div>

<p><a id="X850A84618421392A" name="X850A84618421392A"></a></p>

<>. <span class=HTests  /span>/java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

<p><a id="X79B15750851828CB" name="X79B15750851828CB"></a></p>

<h5>41.10-1 <span class="Heading<span class=GAPprompt&t;/> < class">java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 113

<div class="func"><table class="func" width
<class=func"<table class="width=100"<r<td "dleft">< class="func">‣ IsTransitive</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<<tablejava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 222
<p>returns <code class="keyw">true</code> if the action implied by the arguments is transitive, or <code class="keyw">false</code> otherwise.</p>

<p>We say that a group <var class="Arg">G</var> acts <em>transitively<div class=func">table class" "00%>tr<td "tdleft"<java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 341

<p>For a permutation group <var class="Arg">G</var>, one may also invoke this as <code class="code">IsTransitive(<var class="Arg">G</var>)</code>,<iv class"unc>< =" 100%<r>< class=tdleft"><ode class="func">#;/java.lang.StringIndexOutOfBoundsException: Range [121, 120) out of bounds for length 220


<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 98
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTransitive( G, [ 2 .. 4 ] );</span>
true
<span =GAPprompt"gt;/> <span class="GAPinput">IsTransitive( G  2,3  )  #  does act  2  ]<span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTransitive( G, [ 1 .. 4 ] );  # G has two orbits on [ 1 .. 4 ]</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">IsTransitive( G );  # G is transitive on [ 2 .. 4 ]</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTransitive( SL(2, 3), java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 55
false
<span divclass=func">table class="unc ""<r<class""> =func>8227; java.lang.StringIndexOutOfBoundsException: Range [112, 110) out of bounds for length 216
java.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 4
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 12

<p><a id="X8295D733796B7A37" name

<h5>41.10-2 <span class="Heading">Transitivity</span></h5>

<divclass=Arg><var> <class=Arg/>],<var class=A"></ar>] </d ">&operation&bsp;)</td>/</<div
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Transitivity</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
divclass="<table class=func "100"<tr< class"tdleft><code""8227;java.lang.StringIndexOutOfBoundsException: Range [114, 113) out of bounds for length 223
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>An action is <em><span class="SimpleMath">\(k\)</span>-transitive</em> if every <span class="SimpleMath">\(k\)</span>-tuple of points can be mapped simultaneously to every other <span class="SimpleMath">\(k\)</span>-tuple.</p>

<>For apermutationgroup varclass="rg"><var,one     class"code>java.lang.StringIndexOutOfBoundsException: Range [113, 112) out of bounds for length 437


<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">IsTransitive(g,[1..5]);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Transitivity(g,[1..4]);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Transitivity(g);</span>
2
</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

<"></a></>

<4110 "java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 56

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RankAction</code>( <div "func><java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 221
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color:red'>p>java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 144
<p>returns the rank of the transitive (see<>An  is em></em>ifit is transitive (ee< class=f></>(<a href"html>class="41.1<span</>)actionadmitsjava.lang.StringIndexOutOfBoundsException: Range [211, 210) out of bounds for length 363


<div class="example"><pre>
<class=GAPprompt"gap&t</> <span class="GAPinput">RankAction(g,Combinations([1..4],2),OnSets);</span>
4
</pre></div>

<p><a id="X7B77040F8543CD6E" name="X7B77040F8543CD6E"></a></p>

<>.104 span=Heading<>java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSemiRegular</code>( <var class="Arg"><ivclass=">>
<div class="func"><table class="func" width="100%"><trjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSemiRegular</code>( <java.lang.StringIndexOutOfBoundsException: Range [0, 127) out of bounds for length 0
<p>returns <code class="keyw">true</code> if the action implied by the arguments is semiregular, or <code class="keyw">false</code> otherwise.</p>

<> java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 89

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

<p><a id class="func"="unc" width="%"><r<tdjava.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 365

<h5>41.10-5 <span class="Heading">IsRegular</span></h5>

< =func"table class=fjava.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for length 337
< ">table class=func ="%>;/(< =Arg></>)/>< ">&pnbsp;)<td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsRegular</code>( <var class="Arg">xset</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the action implied by the arguments is regular, or <code class="keyw">false</code> otherwise.</p>

<p>An action is <em>regular</em> if it is both semiregular (see <code class="func">IsSemiRegular</code> (<a href="chap41_mj.html#X7B77040F8543CD6E"><span class="RefLink">41.10-4</span></a>)) and transitive (see <code class="func">IsTransitive</code> (<a href="chap41_mj.html#X79B15750851828CB"><span class="RefLink">41.10-1</span></a>)). In this case every point <var class="Arg">pnt</var> of <var class="Arg">Omega</var> defines a one-to-one correspondence between <var class="Arg">G</var> and <var class="Arg">Omega</var>.</p>

<p>For a permutation group <var class="Arg">G</var>, one may also invoke this as <code class="code">IsRegular(<var class="Arg">G</var>)</code>, which tests whether the group is regular with respect to its natural action on the set of points moved by it. For example the group [[1  ,  ,3]  ,5],  ,7  


<div class="example"><pre>
<<pre></div>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRegular(g,Arrangements([1..4],3),OnTuples);</span>
false
</pre></div>

<p><a id="X7CB1D74280F92AFC" name="X7CB1D74280F92AFC"></a></p>

<>.- <span=Heading></java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 51

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Earns</code>( <var class="Arg">G</var>, <var div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalBlocks</code>( <var class="Arg">xset</var<class"dright"(nbspattributenbsp;</><tr>/table<div
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Earns</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the elementary abelian regular (when acting on <var class="Arg">Omega</var>) normal subgroups of <var class="Arg">G</var>.</p>

<p>At the moment only java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0

<p>

<41.- spanclass"</pan>/>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPrimitive</code>( <var class="java.lang.StringIndexOutOfBoundsException: Range [0, 136) 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; IsPrimitive</code>( <var class
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPrimitive</code>( <var class="Arg">xset</var> )</td><td class="div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesMinimalBlocks</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>] )</td><td class="tdright">( operation )</td></tr><div=f"<able=func"=100"<>td=tdleft"<codeclass>#; </(varclass=Arg">/ /> "dright"(&anbsp;)/>/r><table/>
<p>returns <code class="keyw">true</code> if the action implied by the arguments is primitive, or <code class="keyw">false</code> otherwise.</p>

<p>An action is <em>primitive</em> if it is transitive (see <code class="func">IsTransitive</code> (<a href="chap41_mj.html#X79B15750851828CB"><span class="RefLink">41.10-1</span></a>)) and the action

<p>For a permutation group <var class="Arg">G</var>, one may also invoke thisdiv class=example"><pre>

<p>For an explanation of the meaning of all the inputs, please refer to  <a href="chap41_mj.html#X83661AFD7B7BD1D9"><span class="RefLink">41.1</span>[ [1,2 ] [1 3  ,[1, ],[  ] [, 6 , [ ,7 ,

<p><em>Note:</em> This operation does not tell whether a matrix group is primitive in the sense of preserving a direct sum of vector spaces. To do this use <code class="code">IsPrimitiveMatrixGroup</code> or <code class="code">IsPrimitive</code> from the package <strong class="pkg">IRREDSOL<<pre><div


<div class="example"><pre>
<span class="GAPprompt">gap></span
true
</pre></div>

<p><a id="X7E9D3D0B7A9A8572" name="X7E9D3D0B7A9A8572"></a></p>

<h4>41.11 spanclass=="> Systems/>/4>

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

>java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 62

<[ 1,8 ] [ ,2,3  ,[ ,4,5 8 ,[1, 6,7,8 ,  ,3]java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Blocks</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">seed</var>][, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var[1, 2,  [,4 ]  1 7 ,  ,6]]
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Blocks</code>( <var class="Arg">xset</var>[, <var class="Arg">seed</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a block system for the transitive (see <code class="func">IsTransitive</code> (<a href="chap41_mj.html#X79B15750851828CB"><span class="RefLink">41.10-1</span></a>)) action of <var class="Arg">G</var> on <var class="Arg">Omega</var>. If <var class="Arg"Group( (,8(,)4)(7 )

<p>The result is undefined if the action is not transitive.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=TransitiveGroup(8,3);</span>
E()=[x][x]2
<span class="GAPprompt">gap></span> <span class="GAPinput">Blocks(g,[1..8]);</span>
[5 ], [ 6,, 7 ]
<span class="GAPprompt">gapGroup([ (,7, 5,6, 5,) (23)(,4(5,) (
[ [ 1, 4 ], [ 2, 7 ], [ 3, 6 ], [ 5, 8  (15,,)2,,,) ]java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
</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><a id="X79936EB97AAD1144" name="X79936EB97AAD1144"></a></p>

<h5>41.p considering   sometimes of SimpleMath>(G\<span-set/> used. Thisis span ="">\\Omega\)endowed with anaction  span class="\(G\)/>.The  of  <pan class=SimpleMath"\G\</pan>-set the  thoseof span "">\<span>  concepts like equality   of <span"impleMath"\G\<>do theunderlying span=>()/span>but the actionas <p

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MaximalBlocks</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">seed</var>][, <var class="Arg">gens</var>, <var class="Arg">acts</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
div "unc"< class=func"width=100">tr< class="tdleft"< class=func>&#>8227; MaximalBlocks<code> var =""xset/var>[, <var class="Arg">seed</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a block system that is maximal (i.e., blocks are maximal with respect to inclusion) for the transitive (see <code class="func">IsTransitive</code> (<a href="chap41_mj.html#X79B15750851828CB"><span class="RefLink">41.10-1</span></a>)) action of <var class="Arg">G</var> on <var class="Arg">Omega</var>. If <var class="Arg">seed</var> is given, a block system is computed in which <var class="Arg">seed</var> is a subset of one block.</p>

< java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 63


<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
[ [ 1, 2, 3, 8 ], [ 4 .. 7 ] ]
</pre></div>

<p><a id="X7941DB6380B74510" name="X7941DB6380B74510"></a></p>

<h5>41.11-3 <span class="Heading">RepresentativesMinimalBlocks</span></h5>

<div class="func"><tableh5>.121/h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RepresentativesMinimalBlocks</code>( <var class="Arg">xset</vardivclass=func<class=""100%"<>java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 222
<p>computes a list of block representatives for all minimal (i.e blocks are minimal with respect to inclusion) nontrivial block systems for the transitive (see <code class="func">IsTransitive</code> (<a href="chap41_mj.html#X79B15750851828CB"><span class="RefLink">41.10-1</span></a>)) action of <var class="Arg">G</var> on <var class="Arg">Omega</var>.</p>

<p>The result is undefined if the action is not transitive.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativesMinimalBlocks(g,[1..8]);</span>
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ],
  [ 1, 8 ] ]
</pre></div>

<p><a id="X835658B07B28EF3B" name="X835658B07B28EF3B"></a></p>

<h5>41.11-4 AllBlocks</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllBlocks</code>( <var class="Arg">G</var> )</td><td ass=Arg"Omega<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>
<p>computes a list of representatives of all block systems for a permutation group <var class="Arg">G</var> acting transitively on the points moved by the group.</p>

<p>Each representative in the returned list is sorted and contains the smallest point moved by <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllBlocks(g);</span>
[ [ 1, 8 ], [ 1, 2, 3, 8 ], [ 1, 4, 5, 8 ], [ 1, 6, 7, 8 ], [ 1, 3 ],
  [ 1, 3, 5, 7 ], [ 1, 3, 4, 6 ], [ 1, 5 ], [ 1, 2, 5, 6 ], [ 1, 2 ],
  [ 1, 2, 4, 7 ], [ 1, 4 ], [ 1, 7 ], [ 1, 6 ] ]
</pre></div>

<lt;:[ 1, 2,3,4 ]&java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27


<div class="example"><pre>
<span  1,,,)2,) 142)(43, (,4)2,3) ]&tjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
Group([ (1,8)(2,3)(4,5)(6,7) ])
</pre></div>

<p>If <code class="code">bs</code> is a partition of the action domain, given as a set of sets, the stabilizer under the action <code class="func">OnSetsDisjointSets</code> (<a href="chap41_mj.html#X7E23686E7A9D3A20"><span class="RefLink">41.2-8</span></a>) returns the largest subgroup which preserves <code class="code">bs</code> as a block system.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2<ivclass="func"<table class=func"width="%><r<class"">< classfunc>8227 ActingDomaincode <ar=Arg</ar></d< classt>&;attributenbsp></tr<>/div
<span class="GAPprompt">gap></span> <span class="GAPinput">bs:=[[1,2,3,4],[5,6,7,8]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(g,bs,OnSetsDisjointSets);</span>
Group([ (6,7), (5,6), (5,8), (2,3), (3,4)(5,7), (1,4),
  (1,5,4,8)(2,6,3,7) ])
</pre></div>



<h4>41.12 <span class="Heading">External Sets</span></h4>

<p>When considering group actions, sometimes the concept of a <em><span class="SimpleMath">\(G\)</span>-set</em> is used. This is a java.lang.StringIndexOutOfBoundsException: Range [0, 135) out of bounds for length 0

<p>This concept is implemented in <strong class="pkg">GAP</strong> via <em>external sets</em>.</p>

<p>The constituents of an external set are stored in the attributes <code class="func">ActingDomain</code> (<a href="chap41_mj.html#X7B9DB15D80CE28B4"><span class="RefLink">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Most operations for actions are applicable as an attribute for an external set.</p>

<</code> (<a href="chap41_mjhtml#X7FB656AE7A066C35"><spanclass"RefLink"41.2-9</<a>./>

<p>Many, as classes or cosetss ">/java.lang.StringIndexOutOfBoundsException: Range [109, 108) out of bounds for length 362

<p>External sets also are implicitly underlying action homomorphisms, see <code class="func">UnderlyingExternalSet</code> (<a href="chap41_mj.html#X8190A8247F29A5C7"><span class="RefLink">41.12-16</span></a>) and <code class="func">SurjectiveActionHomomorphismAttr</code> (<a href="chap41_mj.html#X7A3D87DE809FBFD4"><span class="RefLink">41.12-17</span></a>).</p>

<p><a id="X8264C3C479FF0A8B" name="X8264C3C479FF0A8B"></a></p>

<h5>41.12-1 IsExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsExternalSet</code>( <var class="Arg">obj</var> )</td><td class="java.lang.StringIndexOutOfBoundsException: Index 169 out of bounds for length 62
<p>An <em>external set<eclass=="unc width="%><><tdclass""><code class""#;/java.lang.StringIndexOutOfBoundsException: Range [124, 123) out of bounds for length 231

<p><a id="X7C90F648793E47DD" name="X7C90F648793E47DD"></a></p>

<h5>41.12-2 ExternalSet</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
> beeitheraproper,  domainwhichis represented asdescribedin< href=".#X7BAF69417BB925F6">span class="efLink">124<span></a>and<a =chap30_mj.html#X8050A8037984E5B6"><span class="RefLink">30</span></a>, or (to use less memory but with a slower performance) an enumerator (see <code class="func">Enumerator</code> (<a href="chap30_mj.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) ) of this domain.</p>

<p>The result is undefined if <var class="Arg">G</java.lang.StringIndexOutOfBoundsException: Range [0, 53) out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g<h5>4112-8 IsExternalOrbit</h5>
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=ExternalSet(g,[1..4]);</span>
<xset:[ 1, 2, 3, 4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=ExternalSet(g,g,OnRight);</span>
<xset:[ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2),
  (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbits(e);</span>
[ [ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,4,2),
      (1,2,3, (1,,3,4) (2,3,),(13) (14,3, (12, ] ]
</pre></div>

<p><a id="X7B9DB15D80CE28B4" name="X7B9DB15D80CE28B4"></a></p>

<5>4112-3 ActingDomainh5>

<div class="func"><table class="<p>The result is undefined if <var=Arg">G</var> does not act on <var class="Arg">Omega</var>.</p>
<p>This attribute returns the group with which the external set <var class="Arg">xset</var> was defined.</p>

<p><a id="X86153CB087394DC1" name="X86153CB087394DC1"></a></p>

<h5>41.12-4 FunctionAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FunctionAction</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr<p>< id"X7BAFF02B7D6DF9F2"name=X7BAFF02B7D6DF9F2"></a></p>
<p>is the acting function with which the external set <var class="Arg">xset</var> was defined.</p>

<p><a ="86" name="X86A0CC1479A5932A"></a></p>

<h5>41.12-5 HomeEnumerator</h5>

<iv class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#n style='color: green'>8227; HomeEnumerator</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an enumerator of the action domain with which the external set <var class="Arg">xset</var> was defined. For external subsets, this is in general different from the <code class="func">Enumerator</code> (<a href="chap30_mj.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2<<iv ">>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ActingDomain(e);</span>
Group([ (1,2,3), (2,3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">FunctionAction(e)=<p> id="867262A82FDD592" name"X867262FA82FDD592"></a>/>
true
<">HomeEnumerator(e);/span>
[ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2),
  (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]
</pre<div

<p><a id="X879DE63C7858453C" name="X879DE63C7858453C"></a></p>

<h5>41.12-6 IsExternalSubset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsExternalSubset</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>An external subset is the restriction of an external set to a subset of the domain (which must be invariant under the action). It is again an external set.</p>

<p><a id="X87D1EA1486D86233" name="X87D1EA1486D86233"></a></p>

<h5>41.12-7 ExternalSubset</h5>

<div class="func"><table class="func"<//>
<p>

<p>The result is <

<p><a id="X7E081F568407317F" name="X7E081F568407317F"></a></p>

<>1.- /h5

<div class="func"><tableclass"func"width">tr<td class"tdleft"<ode class="unc>&8227;IsExternalOrbit</>(<var class=Arg></ar )/td>td ="dright"(nbsprepresentationnbsp;)<td/tr>table/>
<p>An external orbit is an external subset consisting of one orbit.</p>

<>aid="7FB656AE7A066C35FB656AE7A066C35" name="X7FB656AE7A066C35"></a></p>

<h5>41.12-9 ExternalOrbit</h5>

<div class="func"<iv class=">pre>
<pthe subseton the orbit of < class"Arg"><. class="func">Representative/code(<ahref=chap30_mj.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>) value of this external set is <var class="Arg">pnt</var>.</p>

<p>The result is undefined if <var class="Arg">G</var> does not act on <var class="Arg">Omega</var>.</p>


<div class="example"><pre>
< class=GAPprompt">></span><span class="GAPinput">e:=ExternalOrbit="APinput">e:=ExternalOrbit(g,g,(1,2,3));</span>
(1,2,3)^G
</pre></div>

<p><a id="X7BAFF02B7D6DF9F2" name="X7BAFF02B7D6DF9F2"></a></p>

<h5>41.12-10 StabilizerOfExternalSet</h5>

< ""<class""width=100%"<< class=""codeclass">&#227;StabilizerOfExternalSetcode var =Arg><var </d>td=tdright(nbsp;attributenbsp;)</<tr</>/java.lang.StringIndexOutOfBoundsException: Index 234 out of bounds for length 234
<p>computes the stabilizer of the <code class="func">Representative</code> (<a href="chap30_mj.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>) value of the external set <var class="Arg">xset</var>. The stabilizer will have the acting group of <var class="Arg">xset</var> as its parent.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Representative(e);</span>
(1,2,3)
<span class="GAPprompt">gap><h5>.1214 </h5>
Group([ (1,2,3) ])
<pre>/div>

<p><a id="X867262FA82FDD592" name="X867262FA82FDD592"></a></p>

<h5>41.12-11 <span class="Heading">ExternalOrbits</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227;ExternalOrbits<code(<var class=A"</var> <var class=Arg"Omega<var>,class=Arg">ens<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="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ExternalOrbits</code>( <var class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<divclass="func">tableclass=func =100%<><td =""< =java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 229


<div class="example"><pre>
<span(1,,
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
</pre></div>

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

<h5>41.12-12 <span class="Heading">ExternalOrbitsStabilizers</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ExternalOrbitsStabilizers</code>( <var class="Arg">G</var>, <var class="Arg<p>he underlying set of anaction homomorphism <var class="Arg">acthom</var> is the external set on which it was defined.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ExternalOrbitsStabilizers</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>In addition to <code class="func">ExternalOrbits</code> (<a href="chap41_mj.html#X867262FA82FDD592"><span class="RefLink">41.12-11</span></a>), this operation also  Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 94


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=ExternalOrbitsStabilizers(g,AsList(g));</span>
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HasStabilizerOfExternalSet(e[3]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">StabilizerOfExternalSet(e[3]);</span>
Group([ (2,4,3) ])
</pre></div>

<p><a id="X8048AE727A7F1A2F" name="X8048AE727A7F1A2F"></a></p>

<h5>41.12-13 CanonicalRepresentativeOfExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CanonicalRepresentativeOfExternalSet</code>[2, 4,1] , 4, ][3 1, 2 ],2],3,4]3 2, 1 ] ..&t;
<p>The  an external < "">java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 714

<p><a id="X8071A8D784DC8325" name="X8071A8D784DC8325"></a></p>

h5.2-4CanonicalRepresentativeDeterminatorOfExternalSet</5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227;  [2,4, 1],,4,3 ] [3, ,2 ,[,1 4],  3,2,  ,
<p>returns a function that takes as its arguments the acting group and a point. This function returns a list of length 1 or 3, the first entry being the canonical representative and the other entries (if bound) being the stabilizer of the canonical representative and a conjugating element, respectively. An external set is only guaranteed to be able to compute a canonical representative if it has a <code class="func">CanonicalRepresentativeDeterminatorOfExternalSet</code>.</p>

<p><a id="X85E9A6A77B8D00B8" name="X85E9A6A77B8D00B8"></a></p>

<h5>41.12-15 ActorOfExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ActorOfExternalSet</code>( <var class="Arg">xset</var<
<p>returns an element mapping <code class="code">Representative(<var class="Arg">xset</var>)</code> to <code class="code">CanonicalRepresentativeOfExternalSet(<var class="Arg">xset</var>)</code> under the given action.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=Subgroup(g,[(1,2,3)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=RightCoset(u,(1,2)(3,4));;</span>
<=GAPprompt"gt;/> < class="GAPinput">CanonicalRepresentativeOfExternalSet(e);</span>
(2,4,3)
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 6
(1,3,2)
</>
(2,4,3)
</pre></div>

<p><a id="X8190A8247F29A5C7" name="X8190A8247F29A5C7"></a></p>

<h5>41.12-16 UnderlyingExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; UnderlyingExternalSet</code>( <var class="Arg">acthom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The underlying set of an action homomorphism <var class="Arg">acthom</var> is the external set on which it was defined.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=UnderlyingExternalSet(hom);</span>
<xset:[[ 1, 2, 3 ],[ 1, 2, 4 ],[ 1, 3, 2 ],[ 1, 3, 4 ],[ 1, 4, 2 ],
[ 1, 4, 3 ],[ 2, 1, 3 ],[ 2, 1, 4 ],[ 2, 3, 1 ],[ 2, 3, 4 ],
[ 2, 4, 1 ],[ 2, 4, 3 ],[ 3, 1, 2 ],[ 3, 1, 4 ],[ 3, 2, 1 ], ...]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print(s,"\n");</span>
[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 2 ], [ 1, 3, 4 ], [ 1, 4, 2 ],
  [ 1, 4, 3 ], [ 2, 1, 3 ], [ 2, 1, 4 ], [ 2, 3, 1 ], [ 2, 3, 4 ],
  [ 2, 4, 1 ], [ 2, 4, 3 ], [ 3, 1, 2 ], [ 3, 1, 4 ], [ 3, 2, 1 ],
  [ 3, 2, 4 ], [ 3, 4, 1 ], [ 3, 4, 2 ], [ 4, 1, 2 ], [ 4, 1, 3 ],
  [ 4, 2, 1 ], [ 4, 2, 3 ], [ 4, 3, 1 ], [ 4, 3, 2 ] ]
</pre></div>

<p><a id="X7A3D87DE809FBFD4" name="X7A3D87DE809FBFD4"></a></p>

<h5>41.12-17 SurjectiveActionHomomorphismAttr</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SurjectiveActionHomomorphismAttr</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an action homomorphism for the external set <var class="Arg">xset</var> which is surjective. (As the <code class="func">Image</code> (<a href="chap32_mj.html#X87F4D35A826599C6"><span class="RefLink">32.4-6</span></a>) value of this homomorphism has to be computed to obtain the range, this may take substantially longer than <code class="func">ActionHomomorphism</code> (<a href="chap41_mj.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>).)</p>


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Messung V0.5 in Prozent
C=100 H=100 G=100

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