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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap40_mj.html">[Previous Chapter]</a> <a href="chap42_mj.html">[Next Chapter]</a> </div>
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<div class="ChapSects"><a href="chap41_mj.html#X87115591851FB7F4">41 <span class="Heading">Group Actions</span></a>
<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>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X81B8F9CD868CD953">41.2 <span class="Heading">Basic Actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7FE417DD837987B4">41.2-1 OnPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7960924D84B5B18F">41.2-2 OnRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X832DF5327ECA0E44">41.2-3 OnLeftInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X85AA04347CD117F9">41.2-4 OnSets</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 /><span class="nocss"> </span><a href="chap41_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>
<span class="ContSS"><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"> </span><a href="chap41_mj.html#X7FF556CD7E6739A9">41.2-10 OnTuplesSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X844E902382EB4151">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 class="nocss"> </span><a href="chap41_mj.html#X7BA8D76586F1F06E">41.2-14 Permuted</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X85124D197F0F9C4D">41.2-15 OnSubspacesByCanonicalBasis</a></span>
</div></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 class="Heading">Orbits</span></a>
</span>
<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"> </span><a href="chap41_mj.html#X8032F73078DF2DDB">41.4-5 <span class="Heading">OrbitLengths</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X8520E2487F7E98AF">41.4-6 <span class="Heading">OrbitLengthsDomain</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X797BD60E7ACEF1B1">41.5 <span 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="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X86FB962786397E02">41.5-2 Stabilizer</a></span>
<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>
</span>
<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>
</span>
<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 class="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"> </span><a href="chap41_mj.html#X8561DEBA79E01ABD">41.8-2 RegularActionHomomorphism</a></span>
<span class="ContSS"><br /><span 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"> </span><a href="chap41_mj.html#X807AA91E841D132B">41.9 <span class="Heading">Permutations Induced by Elements and Cycles</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7807A33381DCAB26">41.9-1 <span class="Heading">Permutation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X81D4EA42810974A0">41.9-2 PermutationCycle</a></span>
<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>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7F3B387A7FD8AE5E">41.9-5 Cycles</a></span>
<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 class="nocss"> </span><a href="chap41_mj.html#X87FDA6838065CDCB">41.9-7 <span class="Heading">CycleIndex</span></a>
</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="nocss"> </span><a href="chap41_mj.html#X8295D733796B7A37">41.10-2 <span class="Heading">Transitivity</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X8166A6A17C8D6E73">41.10-3 <span class="Heading">RankAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7B77040F8543CD6E">41.10-4 <span class="Heading">IsSemiRegular</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7CF02C4785F0EAB5">41.10-5 <span class="Heading">IsRegular</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7CB1D74280F92AFC">41.10-6 <span class="Heading">Earns</span></a>
</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="nocss"> </span><a href="chap41_mj.html#X7E9D3D0B7A9A8572">41.11 <span class="Heading">Block Systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X84FE699F85371643">41.11-1 <span class="Heading">Blocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X79936EB97AAD1144">41.11-2 <span class="Heading">MaximalBlocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7941DB6380B74510">41.11-3 <span class="Heading">RepresentativesMinimalBlocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X835658B07B28EF3B">41.11-4 AllBlocks</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X7FD3D2D2788709B7">41.12 <span class="Heading">External Sets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X8264C3C479FF0A8B">41.12-1 IsExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7C90F648793E47DD">41.12-2 ExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7B9DB15D80CE28B4">41.12-3 ActingDomain</a></span>
<span class="ContSS"><br /><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">41.12-9 ExternalOrbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7BAFF02B7D6DF9F2">41.12-10 StabilizerOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X867262FA82FDD592">41.12-11 <span class="Heading">ExternalOrbits</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X7A64EF807CE8893E">41.12-12 <span class="Heading">ExternalOrbitsStabilizers</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X8048AE727A7F1A2F">41.12-13 CanonicalRepresentativeOfExternalSet</a></span>
<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 /><div="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap40_mj.html">[Previous Chapter]</a> <a href="chap42_mj.html">[Next Chapter]</a> </div>
< class>brspan " nbsp</java.lang.StringIndexOutOfBoundsException: Range [69, 65) out of bounds for length 161
</div></div>
</div>
<h3>41 <span class=/pan
<p>A<>
<p>In <strong class="pkg">GAP</strong>, <span class="SimpleMath">\(\Omega\)</span> can be a duplicate-free collection (an object that permits access to its elements via the <span class="SimpleMath">\(\Omega[n]\)</span> operation, for example a list), it does not need to be sorted (see <code class="func">IsSet</code> (<a href="chap21_mj.html#X80CDAF45782E8DCB"><span class="RefLink">21.17-4</span></a>)).</p>
<p>The acting function <span class="SimpleMath">\(\mu\)</span class="ContSS"><br /><span class="nocss"> #"412- java.lang.StringIndexOutOfBoundsException: Range [124, 122) out of bounds for length 133
<p>In <strong class="pkg">GAP</strong>, groups span"ontSS"< >span =n>nbsp;<span>a href"java.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bounds for length 134
<p><strong class="pkg">GAP</strong> does not test whether the acting function <span class="SimpleMath">\(\mu\)</span> satisfies the conditions for aspanclass"ontSS" >java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 145
<p>The first section of this chapter, <a href="chap41_mj.html#X83661AFD7B7BD1D9"><span class="class="ContSS"><br /><span class="nocss"> </span><a href="chap41_mj.html#X844E902382EB4151span
spanbr/spanclass"ocss>nbsp;</<a "htmlXB"41.14 <>
<p>The sections <a href="chap41_mj.html#X87F73CCA7921DE65"><span class="RefLink">41.7</span></a> and <a href="chap41_mj.html#X7FED50ED7ACA5FB2"><><div
<p>The other sections then describe operations to compute orbits, stabilizers, as well as properties of actions.</p>
<p>Finally section <a href="chap41_mj.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a> describes the concept of <q>external sets</q> which represent the concept of a <em><span class="SimpleMath">\(G\)</span>-set</em> and span
<p><a id="X83661AFD7B7BD1D9" name="X83661AFD7B7BD1D9"></a></p>
<></h4
< syntax isjava.lang.StringIndexOutOfBoundsException: Range [49, 45) out of bounds for length 367
<dl>
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">\(( G, \Omega[, \mu] )\)</span></strong></dt>
<dd><p>The acting function <span class="SimpleMath">\(\mu\)</span> is optional. If it is not given, the built-in action <code div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41_mj.html#X797BD60E7ACEF1B1">41.5 <span
<ddjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">\(( G, \Omega, gens, acts[, \mu] )\)</span><<>/>
<dd><p>This second version of <code class="func">OrbitsDomain</code> (<a href="chap41_mj.html#X86BC8B958123F953"><span class="RefLink">41.4-3</span></a>) permits one to implement an action induced by a homomorphism: If the</span>
<p>Here <span class="SimpleMath">\(gens\)</span> must be a set of generators of <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">/
<p>The advantage of this notation is that <strong java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 7
<p><strong class="pkg">GAP</strong> does not test whether the mapping=java.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 151
</dd>
<dt><strong classclass=>="<java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 185
<dd><p>A third variant is to call the operation with an external set, which then provides <span class="SimpleMath">\(G\)</span>, <span class="SimpleMath">\(\Omega\)</span> and <span java.lang.StringIndexOutOfBoundsException: Index 186 out of bounds for length 7
<dd>
</dl>
<p>For operations like <code class="func">< "> ><java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 149
<aidjava.lang.StringIndexOutOfBoundsException: Range [33, 27) out of bounds for length 62
<h4>41.2 <span class="Heading"/
<p><strong class="pkg">GAP</strong> already provides />
<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 span class"ontSS"/=nocss>nbsp;nbsp<> =.#FB387"41.-</<span>
<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 class="example"><pre>
OnIndeterminatesPairs:= function( polypair, g )
return [ OnIndeterminates( polypair[1], g ),
2], g)]java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
;
</pre></div>
<p>Note that this function <em>must</em> implement a group action </pan>
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 7
<h5>41.2-1 OnPoints</h5>
tdleftjava.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 243
<p>returns <code class="code"><var class="Arg">pnt</var> ^ <var class="Arg"</span>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnPoints( 1, (1,2,3) );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">OnPoints( (1,2), (1,2,3) );</span>
(2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
<span class">&;/span><span class=""Orbit g,1 OnPoints )<span>
[ 1, 2, 3, 4 ]
</pre></div>
<p><a id="X7960924D84B5B18F" name="X7960924D84B5B18F"></a></p>
<h5>41.2-2 OnRight</h5>
< class""><table ""width"%">trtdclass"dleft">code ">#8227 OnRightcode>( <var class="Arg">pnt</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<pclass="ContSS><r />span class="nocssnbsp&;/>a=chap41_mj.#X7C90F648793E47DD41.- />/span>
<div class="example"><pre class=""> />spanjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 142
<span class="GAPprompt">gap></span> <span class="GAPinput">OnRight( [ 1, 2 ], [ [ 1, 2 ], [ 3, 4 ] ] );</span>
[ 7, 10 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OnRight( (1,2,3), (2,3,4) );</span>
(1,3)(2,4)
java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 100
< class=">gapgt<span java.lang.StringIndexOutOfBoundsException: Range [45, 44) 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),
,) java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
</pre></div>
<p><a id="X832DF5327ECA0E44" name="java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 7
<h5>41.-3 OnLeftInverse</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OnLeftInverse</code>( <var class="Arg">pnt</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <span class="SimpleMath">\(\textit{g}^{{-1}}\)</span> <code class="code">* <var class="Arg">pnt</var></code>. Forming the inverse is necessary to make this a java.lang.StringIndexOutOfBoundsException: Index 175 out of bounds for length 161
<
<div class="example"><pre>
<">OnLeftInverse( [ 1, 2 ], [ [ 1, 2 ], 3,4]]java.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119
<span class="GAPprompt">gap></span> <span class="GAPinput">OnLeftInverse( (1,2,3), (2,3,4) );</span>
(,,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 0
[ (), (1,3,2), (2,4,3), (1,2,3), (1,3)(2,4), (1,2)(3,4), (2,3,4),
(1,2,4), (1,3,4), (1,4,3), (1,4,2), (1,4)(2,3) ]
</pre></div>
<p><a id="X85AA04347CD117F9" name="X85AA04347CD117F9"></a></p>
<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; OnSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
><java.lang.StringIndexOutOfBoundsException: Range [18, 17) 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
<=">java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
<span class
[1 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 4
[ 23)(1,)]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
<="GAPprompt">></pan span"" g [, ,;/span
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ]
</pre></div>
<p</dd>
<h55 OnTuples<h5
<div class="func"><table class="func" width="100%"><tr><td class="tdleft">
<>Let<="rg"tup</ar alist code =""OnTuples<codereturnsthelistformed the imagesofall 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 classpkg></does testwhether < =SimpleMath mapsto\/>actually ahomomorphismthe java.lang.StringIndexOutOfBoundsException: Range [176, 175) out of bounds for length 218
<div class="example"><pre>
<span">OnTuples( [ 1, 3 ], (1,,3) );/span>
[ 2, 1 ]
<span class="java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 5
[(,), (,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="java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 0
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 1 ], [ 3, 4 ], [ 2, 1 ],
[ 1, 4 ], [ 4, 1 ], [ 4, 2 ], [ 3, 2 ], [ 2, 4 ], [ 4, 3 ] ]
>
<p>return (polypair] )java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
<h5>41.2-6 OnPairs</h5>
<div class="func"><table class=
<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="java.lang.StringIndexOutOfBoundsException: Range [0, 169) out of bounds for length 0
<p><a id="X7C10492081D72376" name="X7C10492081D72376"></a></p>
<h5>41.2-7 OnSetsSets<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table span"GAPprompt"gap&t;/>span class"APinput">OnPoints( (1,2), (1,2,3) );</span>
<p>implements the action on sets of special case arepairwise disjoint it is possible use < class=func>OnSetsDisjointSets</code(<"chap41_mjhtml>span class">.-<java.lang.StringIndexOutOfBoundsException: Range [246, 245) out of bounds for length 463
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnSetsSets( [ [ 1, 2 ], [ 3, 4 ] ], (1,2,3) );</span>
[ [ 1, 4 ], [ 2, 3 ] ]
<span class="GAPprompt">gap></span> <span<span class=GAPpromptgap&t<span<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 93
< class"GAPprompt">&t</span><spanclass="GAPinput"Orbit(g,[[,2 ] [ , 4]] OnSetsSets <span
[[,2],[ ,4 , , 4 , [,3] ,
[ [ 1, 3 ], [ 2, 4 ] ] ]
</pre></div>
<p><a id="X7E23686E7A9D3A20" name="X7E23686E7A9D3A20 code class="code>class""><var>*var "g/</java.lang.StringIndexOutOfBoundsException: Range [91, 89) out of bounds for length 468
<h5>41.2-8 OnSetsDisjointSets</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 0
<p>implements the action on sets of pairwise disjoint sets (see also <code class="func">OnSetsSets</code> (<a href="chap41_mj.html#X7C10492081D72376"><span class="RefLink">41.2-7</span></a>)). <var class="Arg">
<<s "APprompt">><span> <pan class="APinput>: Group( (12,3),), (2,34 );</span>
<h5>41.2-9 <panclass=GAPprompt">gap></span> <span class="GAPinput">Orbit( g, (), OnRight );</span>
<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 the action class">tableclass=""java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 248
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnSetsTuples( [ [ 1, 2 ], [ 3, 4 ] ], (1,2,3) );</span>
[[1 ], [23 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3), (2,3,4) );;</span>
span=GAPprompt"gap&;<//pan 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
</pre></div>
<p><a id="X7FF556CD7E6739A9" name="X7FF556CD7E6739A9"></a></java.lang.StringIndexOutOfBoundsException: Range [0, 61) out of bounds for length 7
<h5>41.2-10 OnTuplesSets</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OnTuplesSets<pan class="GAPprompt">gap></span> <span class="GAPinput">Orbit( g, (), OnLeftInverse );</span>
<p>implements the action on tuples of sets. <var class="Arg">set</var> must be a list whose entries are again sorted lists, otherwise an error is triggered (see <a(1,2)(,,4, (43 (1,,2, (,),)java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OnTuplesSets( [ [ 2, 3 ], [ 3<Letclass="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 the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 1, 3 ], [ 1, 4 ] ]
<span,2]
<">Orbit [ 1 2,[3, 4] , ;</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ],
[ [ 1, 3 ], [ 2, 4 ] ], [ [ 3, 4 ], [ 1, 2 ] ],
[ [ 1, 4 ], [ 2, 3 ] ], [ [ 2, 4 ], [ 1, 3 ] ,2, [, ] 1,3][, 4,[,4] 2,]]
</pre></div>
<p><a id="X844E902382EB4151" name="X844E902382EB4151"></<div class="func"><table class="func" width="100%"><tr><tclass="tdleft"><code class="func">‣ OnTuples</code>( <var class="Arg">tup</var>, <var class="Arg">g</var> )</td><td class="tdright">( function&>< "rg>java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 427
<h5>41.2-11 OnTuplesTuples</h5>
<div class="func"><table class="func" width="100%"><tr><java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 0
<p>implements the action on tuples of tuples.</p>
< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">OnTuplesTuples( [ [ 2, 3 ], [ 3, 4 ] ], (1,2,3) );</span>
[ [ 3, 1 ], [ 1, 4 ] ]
<span class="GAPprompt">gap><< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 100
<span class="GAPprompt">gap></span> <[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 1 ], [ ,
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ],
[ [ 1, 3 ], [ 4, 2 ] ], [ [ 3, 1 ], [ 2, 4
<4126java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 23
[ , , ,3 ]] [ 4, , ,2 ,
[ [ 4, 2 ], [ 1, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ],
[ [ 2, 4 ], [ 3, 1 ] ], [ [ 4, 3 ], [ 2, 1 ] ] ]
</pre></div>
<p><a id="X86DC2DD5829CAD9A" name="java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0
<h5>41.2-12 OnLines</h5>
<div class="func"><table class=p>implements onof . Forjava.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 463
p< =Arg></ aem><emrowvector that is firstentry isnormedto java.lang.StringIndexOutOfBoundsException: Range [128, 125) out of bounds for length 669
<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 class="GAPprompt">gap></span> <span class
[ Z(5)^0, 0<iv "><>
< class="APprompt"gapgt;><spanclass="GAPinput">:=Action(,OrbitglvOnLines)O)</span>
Group([ (2,3,5,6), (1,2,4) 1 4 ,[2 java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
</pre></div>
<p><a id="X7FA394D27E721E2B" name="X7FA394D27E721E2B"></spanclass="GAPprompt"gap&;/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 116
<41.13 <java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
<div class="func"><table class="func" width="100%"><tr></<div
<p> permutationvar java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 167
<<p>i theaction setssee codejava.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 403
<classG"gap&</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 132
<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></<>a="" java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 62
Group([ (1,2), (3,4) ])
</pre></div>
<p><a id="X7BA8D76586F1F06E" name="X7BA8D76586F1F06E"></a></p>
<h5>41.2-14 Permuted</h5>
<divclass=func>table class="unc"width"100%><r><td class="tdleft"><code class="func">‣ Permuted</code>( <var class="Arg">list</var>, <var class="Arg">perm</var> )</td><td class="tdright">( method )</td></tr></table></div>
pThe < class""Permuted<code<ahrefchap21_mj.tml#X7B5A19098406347A"><span class="RefLink">21.20-17</span></a>) being used to implement a permutation action on a domain:</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">dom:=[ "a", "b", "c" ];;</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>
<p><a idh5>41210</>
<h5>41.2-15 div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class=" ofjava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 247
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OnSubspacesByCanonicalBasis<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; OnSubspacesByCanonicalBasisConcatenations</code>( <var class="Arg">basvec</var>, <var class="Arg">mat</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the operation of a matrix group on subspaces of a vector space. <var class="Arg">bas [ 1, 3],[2,4 ], [3,4] [1,2],
<p><var class="Arg">bas</var> must be given in Hermite normal form, otherwise</pre>/iv>
<p<a id=X82181CA07A5B2056 ="82181CA07A5B2056"<a>/p
<h4>41.3 <span class="Heading">Action on canonical representatives</span></h4>
<p>A varietydiv=func">table class=f" =100">td class=>code class=f>#;</>(<var class""set<var>< class=""g/>)</d> class=tdright"&;nbsp</<tr<java.lang.StringIndexOutOfBoundsException: Range [243, 242) out of bounds for length 249
<p>If orbit seeds or domain elements are not given in the required form <strong class="pkg">GAP</strong> will
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(SymmetricGroup(5),[[2,4],[1,3]],OnSetsSets);</span>
Error, Action not well-defined. See the manual section
`java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 40
</pre></div>
<p>In this case the affected domain elements have to be [, , , ,3 java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
<p>(A similar error could arise if a user-defined action />/iv>
<p><a id="X81E0FF0587C54543" name="X81E0FF0587C54543"></a></p>
<h4>41.4 <span class="Heading">Orbits</java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
<p>If a group <span class="SimpleMath">\(G\)</span> acts on a set <span class="SimpleMath">\(\Omega\)</span>, the set of all images of <span class="SimpleMath">\(x \in \Omega\)</span> under elements of <span class="SimpleMath">\(G\)</span> is called the <em>orbit</em> of <span class="SimpleMath">\(x\)</span>. The set of orbits of <span class="SimpleMath">\(G\)</span> is a partition of <span class="SimpleMath">\(\Let var =Arg><var bea<>normed> that entryis theidentity the , codeclassfunc></>a hrefchtml"< =RefLink>2/>/>)Thefunctioncode ="<>returns from java.lang.StringIndexOutOfBoundsException: Range [383, 382) out of bounds for length 669
<p>classGAPprompt>;><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 98
<>.-/>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Orbit</code>( <var/>/>
<p>The orbit of the point <var class="Arg">pnt</var> is the list of all java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<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 class="GAPinput">Orbit(g,1);</span>
[1, 3,,4 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<span class="GAPprompt">gap
[
<pre>
<p>(See>41214 Permuted<h5>
<>a=" "X86BCAE17869BBEAA//>
<h5>41.4-2 Orbits</h5>
<div class="p>The following <code class=>Permuted</> (ahref=chap21_mj.#"< class"">.20-7<span/ used apermutationactionon :>
<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" width="100%"><tr><td class="tdleft"><span""gapgt<span < "">dom=","b,c];/java.lang.StringIndexOutOfBoundsException: Range [93, 92) out of bounds for length 93
<p>returnsjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<p>For a permutation
<p>(Note that the arrangement java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p><a id="X86BC8B958123F953" name="X86BC8B958123F953"></a></p>
<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 class="Arg">acts</var>][, <var class="Arg">act</var>] )</td>
<div
<div class="func"><<h441.3<span class=Heading"Action on canonical representatives</span></h4>
<p>returns a list of thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This operation is often faster than <code class="func">Orbitsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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
<>(ote that the arrangement oforbits or of points within one orbit is not defined by the operation.)</p>
">prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span
<class">
[ [ 1, 3, 2, 4 ], [ 5 ] ]
<span class="GAPprompt">gap></span> p> id"X81E0FF0587C54543" name=X81E0FF0587C54543"></a></p>
[ [ [ 1, 2, 3 ], [ 3, 1, 2 ], [ 1, 4, 2 ], [ 2, 3, 1 ], [ 2, 1, 4 ],
[ 3, 4, 1 ], [ 1, 3, 4 ], [ 4">()/pan> acts on a set <span class="SimpleMath">\(\Omega\)</span>, the set of all images of <span class="SimpleMath">\(x \in \Omega\)</span> under elements of <span class="SimpleMath">\(G\)</span> is called the <em>orbit</em> of <span class="java.lang.StringIndexOutOfBoundsException: Index 293 out of bounds for length 0
,,3 ] 3, 2,4 ], [, 3 2]]java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
[ [ 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 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitsDomain(g,GF(2)^2,[(1,2,3),(1,4)(2,3)],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[[Z(2)^0,Z(2)^0],[Z(2)^0,0*Z(2)]],[[Z(2)^0,0*Z(2)],[0*Z(2),Z(2)^0]]]);</span>
[ [ <an immutable GF2 vector of length 2> ],
[ <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 specific actions.)</p>
<p><a id="X799910CF832EDC45" name="X799910CF832EDC45"></a></p>
<h5>41.4-4 OrbitLength</h5>
<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>[, <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">( operation )</td></tr></p>The orbit of the point <var class="Arg">pnt</var> is the list of all images of <var class="Arg">pnt</var> under the action of the group <var class="Arg">G</var> w.r.t. 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</span></a>) if no action function
<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</span></a>) if no action function is given.</p>
<p><a id="X8032F73078DF2DDB" name="X8032F73078DF2DDB"></a></p>
<h5>41cpan class"GAPprompt">gap&t<span><span class=GAPinput"Orbitg,12]OnSets);</span>
<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">G</var>, <var class="Arg">seeds</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation/>div>
<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">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<ivclass"unc">table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#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> under the action <var class="Arg">act</var> of <var class="Arg">G</var>.</p>
<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 group <span class="SimpleMath">\(\langle (1,2,3), (5,6) \rangle\)</span> has the orbit lengths 2 and 3.</p>
<p><a id="X8520E2487F7E98AF" name="X8520E2487F7E98AF"></a></p>
<h5>41.4-6 <span class="Heading">OrbitLengthsDomain</span></h5>
<div classp>apermutation class="">,may this code=><=ArgG<></code>whichreturnsjava.lang.StringIndexOutOfBoundsException: Range [172, 169) out of bounds for length 428
<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> )</td><td class="tdright">( attribute )</td></tr></table></div>
classfunc> class="unc"width=100"><tr>java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 229
<p>computes the lengths of all the orbits of <var class="Arg">G</var> on <var class="Arg">Omega</var>.</p>
<> operation is often faster than <code class="func">OrbitLengths</code> (<a href="chap41_mj.html#X8032F73078DF2DDB"><span class="RefLink">41.4-5</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">OrbitLengthsDomain(<var class="Arg">G</var>)</code>, which returns the length of all the orbits of its natural action on the set of points moved by it.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
classG"gapgt;><spanclass"java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 103
12
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitLengths(g,Arrangements([1..4],4),OnTuples);</span>
[ 12, 12 ]
<span class=<>returns aof <arclass="Arg">G</var> on the domain <var class="Arg">Omega</var> (given as lists) under the action <var class="Arg">act</var> or under <code class="func">OnPoints</code> (<a href="chap41_mj.html#X7FE417DD837987B4"><span class="RefLink">41.java.lang.StringIndexOutOfBoundsException: Index 277 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitLengthsDomain(g,[1,2,3]);</span>
[ 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitLengthsDomain(g);</span>
[ 3, 3 ]
</pre></div>
<p><a id="X797BD60E7ACEF1B1" name="X797BD60E7ACEF1B1"></a></p>
<h4>41.5 <span
<p>The <em>stabilizer</em> of a point <span class="SimpleMath">\(xspan class"GAPprompt">&t</> class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<p><a id="X7C34EC437EF598BF" name="X7C34EC437EF598BF"></a></p>
<h5>41.5-1 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>[, <var class="Arg">Omega</var>], <var class="Arg">pnt</var>[, <var class="Arg">gens</var>java.lang.StringIndexOutOfBoundsException: Index 239 out of bounds for length 57
<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><span class=GAPprompt>gt; span class=GAPinput>OrbitsDomaingGF(^,(,,3,(,)(3]<span
<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">[[;animmutable vectorof 2gt;,
<p
<div class="example"><pre>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">stab:=Stabilizer(g,4);</span>
Group([ (1,3,2) ]<h5>41.-4 OrbitLengthh5>
<span class="GAPprompt">gap></span> <span class="GAPinput">Parent(stab);</span>
Group([ (1,3,2), (2,4,3) ])
</pre></div>
<p>The stabilizer of a set or tuple of points can be computed by specifying an action of sets or tuples of points.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(g,[1,2],OnSets);</span>
Group([ (1,2)(3,4) ])
<span
Group(())
<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 ],
4 ]],stabilizer : Group[ 1,2(3,)]) java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
<span class="GAPprompt">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>
<Thestandard methods for all these actions are an orbit-stabilizer algorithm. For permutation groups backtrack algorithms are used. For solvable groups an orbit-stabilizer algorithm for solvable groups, which uses the fact that the orbits of a normal subgroup form a block system (see <a href="chapBib_mj.html#biBSOGOS">[LNS84]</a>) is used.</p>
<p><a id="X78C3A8568414BC44" name="X78C3A8568414BC44"></a></p>
<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</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This operation should not be called by a user. It <p><a id="X8520E2487F7E98AF" name="X8520E2487F7E98AF"><></p>
<p><code java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 63
<p>The <var class="Arg">pntact</var> record may carry a component <var class="Arg">stabsub</var>. If given, this must be a subgroup stabilizing <em>all</em> points in the domain and can be used to abbreviate stabilizer
<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">orbit</code> component may be omitted from the result.</p>
<p>The argument <var class="Arg">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> div =f"<class="""< "< f"&;java.lang.StringIndexOutOfBoundsException: Range [120, 119) out of bounds for length 229
pid"name=X7A9389097BAF670D"<><>
41<java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 74
<p><a id="X857DC7B085EB0539" name="X857DC7B085EB0539"></a></p>
<h5>41.6-1 RepresentativeAction</h5>
<">&#; /code( G/[ class="rgjava.lang.StringIndexOutOfBoundsException: Range [180, 179) out of bounds for length 401
<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>
=>java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<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">RepresentativeAction(g,1,3);</spanspan "">></pan<pan class">(,123];<>
(1,3)(2,4)
<span class=""gapgt<span span ">epresentativeActiong13OnPoints)<span>
(1,3)(2,4)
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 109
(1,2,4)
<span class="GAPprompt">gap></span
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(g,Group((1,2,3)),Group((2,3,4)));</span>
(1,2,4)
<span class="GAPprompt">gap></<54151 java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
(43)
<span class="GAPprompt">gap></span> <span class="GAPinput"> RepresentativeAction(g,[1,2,3],[1,2,4],OnTuples);</span>
fail
</pre></div>
<p>(See Section <a href="chap41_mj.html#X81B8F9CD868CD953">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Again the standard method for <code class="func">RepresentativeAction</code> is an orbit-stabilizer algorithm, for permutation groups and standard actions a backtrack algorithm is used.</p>
<p><a id="p>computes the stabilizer var "Arg"</of =Arg"pntvar,thatis java.lang.StringIndexOutOfBoundsException: Range [126, 125) out of bounds for length 268
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
>a span)span> on domain> java.lang.StringIndexOutOfBoundsException: Range [267, 248) out of bounds for length 712
<p>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>
<p><a id="X78E6A002835288A4" name="X78E6A002835288A4"></a></p>
<>17- span=Heading</spanh5
< classfunc"tablejava.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 384
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ActionHomomorphism</code>( <var class="Arg">xset</var>[, <var class="Arg">"surjective"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
classclass""="<>< ="dleft"><">#;ActionHomomorphism>(<var=Arg"action<var>)/>td class=t">&bsp;java.lang.StringIndexOutOfBoundsException: Range [200, 199) out of bounds for length 230
<p>computes a homomorphism 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
<poup([ 1,2)(3,) )
<>By the homomorphism <ode class"unc></code> is not surjective (ts <code class="func">Range</code> (<a href="chap32_mj.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>) value is the full symmetric group) to avoid unnecessary computation of the image. If the optional string argument <code class="code">"surjective"</code> is given, a surjective homomorphism is created.</p>
<p>The third version (which<panclass="gt< spanjava.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 96
<p>(See Section <a href="chap41_mj.html#X81B8F9CD868CD953"><span class="RefLink">41.2</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
<span id="X" name="X78C3A8568414BC44"</</>
<action homomorphism>
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(hom);</spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ (1,9,13)(2,10,14)(3,7,15)(4,8,16)(5,12,17)(6,11,18)(19,22,23)(20,21,
),(,)28()(,10),1)(612(13,)141617,)19,
21)(20,22)(23,24) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Range(hom<p>< class=func/java.lang.StringIndexOutOfBoundsException: Range [66, 62) out of bounds for length 532
620448401733239439360000
6
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples,</span>
<span class="GAPprompt">></span> <span class="GAPinput">"surjective");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Range(hom)java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
6
</pre></div>
<p>When acting on a domain, the operation <code class="func">PositionCanonical</code> (<a href="chap21_mj.html#X7B4B10AE81602D4E"><span class="RefLinkjava.lang.StringIndexOutOfBoundsException: Index 150 out of bounds for length 0
<p><a id="X85A8E93D786C3C9C" name="X85A8E93D786C3C9C"></a></p>
<h5>41.7-2 Action</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><p>The <var Argjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 1213
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Action</code>( <var class="Arg">xset</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the image group of <code class="func">ActionHomomorphism</code> (<a href="chap41_mj.html#X78E6A002835288A4"><span class="java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 0
<p>Notep><id"857DC7B085EB0539"name="X857DC7B085EB0539">/a<p
<p
<p>The following code shows for example how to create the regular action of a group.</p>
<divclass"example"<pre>
<span class="GAPprompt">gap></<spanclass="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
< =""gapgt;/pan <spanclass="APinput"ActiongA(g)>
Group([ (1,5,3)(2,6,4), (1,6)(2,5)(3,4) ])
</pre></div>
<p><a id="X86FF54A383B73967" name="X86FF54A383B73967"></a></p>
<h5>41.7-3 SparseActionHomomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="(,,4)
<div class="func"><table class="func" width="100%"><tr><td class="tdleftfjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<p><code class="func">SparseActionHomomorphism</code> computes the action homomorphism (see <code class="func">ActionHomomorphism</code> (<a href="chap41_mj.html#X78E6A002835288A4"><span class="RefLink">41.7-1</(11,,)
<p>If <var class="Arg">G</var> acts on a very large domain not surjectively this may yield a permutation image of java.lang.StringIndexOutOfBoundsException: Range [0, 127) out of bounds for length 117
<p>The operation <code class="span> spanclass="APinput RepresentativeActiong,1,,],12,,OnTuples)</ code""></ode is -stabilizeralgorithm, permutationgroups and standard actions a backtrack algorithm is used.</p>
<p>The operation <code class="func">SortedSparseActionHomomorphism</code> in
<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 100
Group([ [ [ Z(3), Z(3) ], [ 0*Z(3), Z(3) ] ] ])
<spanp>java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 284
<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></""</a></java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 62
Group([ (1,2,3,4,5,6) ])
</pre></div>
<p><a id="X7FED50ED7ACA5FB2" name="X7FED50ED7ACA5FB2"></a></p>
<h4>41.8 <span class="Heading">Action of a group on itselfdiv class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActionHomomorphism</code>( < java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 426
<p>Of particular importance is
<<id="X784D417D87F4E58D name="784">/a</>
<h5>41.8-1 FactorCosetAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="
<div class="unc"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#n style='color: green'>8227; FactorCosetAction</code>( <var class="Arg">G</var>, <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This command computes the action of the group << class="APprompt"gapgt;<span class"APinputhom:ActionHomomorphism(gArrangements([.4],),nTuples);</pan>
<divclass=example"<>
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 6
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),\
,14 java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(Range(last));</span>
"S5"
<span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 1
<span class="GAPprompt">gap></span> <span class="p>When java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 624
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
<><a id=" java.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for length 62
<h5""table=""100"<>td class=">code="java.lang.StringIndexOutOfBoundsException: Range [93, 91) out of bounds for length 216
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RegularActionHomomorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</<p>returns the image group of <code class="func">ActionHomomorphism</code> (<a href="chap41_mj.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>) called with the same parameters.</p>
<p>returns an isomorphism from <var class="Arg">G</var> onto thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X835317A7847477D4" name="X835317A7847477D4"></a></p>
<>.83 </h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleftspan class="GAPprompt"gap>span> < =GAPinput">ctiong,sListg)OnRight);</span>
<Letvar class="Arg">G</var> be a group and <span class="SimpleMath">\(\textit{M=AgeqtextitN\/be ofa normalunder< classArg">G/var> such java.lang.StringIndexOutOfBoundsException: Range [204, 203) out of bounds for length 1329
<p>Note: If only matrices for the action are needed, <code class="func">LinearActionLayer</code> (<a href="chap45_mj.html#X7C2135B98732BBC3"><span class="RefLink">45.14-3</span></a>) might be faster.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,8,10,7,3,5)(2,4,12,9,11,6),</pre></div>
<java.lang.StringIndexOutOfBoundsException: Range [101, 5) out of bounds for length 110
<span class="GAPprompt">gap></span> <span class="GAPinput">c:=ChiefSeries(g);;List(c,Size);</span>
[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap<541.3SparseActionHomomorphism<h5>
true
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=AbelianSubfactorAction(g,c[3],c[4]);</span>
[ [ (1,8,10,7,3,5)(2,4,12,9,11,6), (1,9,5,6,3,10)(2,11,12,8,4,7) ] ->
[ <an immutable 2x2 matrix over GF2>,
<an immutable 2x2 matrix over GF2> ],
MappingByFunction( ()java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
),( ) .. 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 <span class="GAPprompt">gap></span> <span classnot surjectively this may yield a permutation imageof substantially smaller degree than by action on the whole domain.</p>
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 49
&java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
=>/>spanGjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 79
3
<span class="GAPpromptspanclass=G&<span span class="GAPinput">h:=Group(Z(3)*[[[1,1],[0,1]]]);</span>
(2,9,3,8)(4,11,5,10)
<span class="GAPprompt">gap></span> <span class=
true
false
</pre><
<p><a id="X807AA91E841D132B" name="X807AA91E841D132B"></a></p>
<h4>41.9 <span class="Heading">Permutations Induced by div class="func"><table class="func" width="100%"><tr><td class class">8227java.lang.StringIndexOutOfBoundsException: Range [119, 118) out of bounds for length 279
< element is java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 209
<></a>p>
<91 java.lang.StringIndexOutOfBoundsException: Range [17, 16) 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; Permutation</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>java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<div class="func"><table class="func" width""
<p permutation domain <var class=A"Omega</ar> ( listof objects that are permuted). If an external set <var class="Arg">xset</var> is given, the permutation domain is the <code class="func">HomeEnumerator</code> (<a href="chap41_mj.html#X86A0CC1479A5932A"><span class="RefLink">41.12-5</span></a>) value of this external set (see Section <a href="chap41_mj.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a>).<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Image());/span
<p>If <var class="Arg">g</var> does not leave the domain invariant, or does not map the domain injectively then <code class="keyw">fail</code> is returned.</p>
< =java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62 | | |
