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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap40.html">[Previous Chapter]</a> <a href="chap42.html">[Next Chapter]</a> </div>
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<div class="ChapSects"><a href="chap41.html#X87115591851FB7F4">41 <span class="Heading">Group Actions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X83661AFD7B7BD1D9">41.1 <span class="Heading">About Group Actions</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X81B8F9CD868CD953">41.2 <span class="Heading">Basic Actions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7FE417DD837987B4">41.2-1 OnPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7960924D84B5B18F">41.2-2 OnRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X832DF5327ECA0E44">41.2-3 OnLeftInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X85AA04347CD117F9">41.2-4 OnSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X832CC5F87EEA4A7E">41.2-5 OnTuples</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X80DAA1D2855B1456">41.2-6 OnPairs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7C10492081D72376">41.2-7 OnSetsSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7E23686E7A9D3A20">41.2-8 OnSetsDisjointSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7ADE244E819035FF">41.2-9 OnSetsTuples</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7FF556CD7E6739A9">41.2-10 OnTuplesSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X844E902382EB4151">41.2-11 OnTuplesTuples</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X86DC2DD5829CAD9A">41.2-12 OnLines</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7FA394D27E721E2B">41.2-13 OnIndeterminates</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7BA8D76586F1F06E">41.2-14 Permuted</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X85124D197F0F9C4D">41.2-15 OnSubspacesByCanonicalBasis</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X82181CA07A5B2056">41.3 <span class="Heading">Action on canonical representatives</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X81E0FF0587C54543">41.4 <span class="Heading">Orbits</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X80E0234E7BD79409">41.4-1 Orbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X86BCAE17869BBEAA">41.4-2 Orbits</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X86BC8B958123F953">41.4-3 <span class="Heading">OrbitsDomain</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X799910CF832EDC45">41.4-4 OrbitLength</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X8032F73078DF2DDB">41.4-5 <span class="Heading">OrbitLengths</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X8520E2487F7E98AF">41.4-6 <span class="Heading">OrbitLengthsDomain</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X797BD60E7ACEF1B1">41.5 <span class="Heading">Stabilizers</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7C34EC437EF598BF">41.5-1 OrbitStabilizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X86FB962786397E02">41.5-2 Stabilizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X78C3A8568414BC44">41.5-3 OrbitStabilizerAlgorithm</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X7A9389097BAF670D">41.6 <span class="Heading">Elements with Prescribed Images</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X857DC7B085EB0539">41.6-1 RepresentativeAction</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X87F73CCA7921DE65">41.7 <span class="Heading">The Permutation Image of an Action</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X78E6A002835288A4">41.7-1 <span class="Heading">ActionHomomorphism</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X85A8E93D786C3C9C">41.7-2 Action</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X86FF54A383B73967">41.7-3 SparseActionHomomorphism</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X7FED50ED7ACA5FB2">41.8 <span class="Heading">Action of a group on itself</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X78C37C4C7B2BDC44">41.8-1 FactorCosetAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X8561DEBA79E01ABD">41.8-2 RegularActionHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X835317A7847477D4">41.8-3 AbelianSubfactorAction</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X807AA91E841D132B">41.9 <span class="Heading">Permutations Induced by Elements and Cycles</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7807A33381DCAB26">41.9-1 <span class="Heading">Permutation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X81D4EA42810974A0">41.9-2 PermutationCycle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X80AF6E0683CA7F14">41.9-3 Cycle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7F559E897B333758">41.9-4 CycleLength</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7F3B387A7FD8AE5E">41.9-5 Cycles</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X83040A6080C2C6C6">41.9-6 CycleLengths</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X87FDA6838065CDCB">41.9-7 <span class="Heading">CycleIndex</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X850A84618421392A">41.10 <span class="Heading">Tests for Actions</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X79B15750851828CB">41.10-1 <span class="Heading">IsTransitive</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X8295D733796B7A37">41.10-2 <span class="Heading">Transitivity</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X8166A6A17C8D6E73">41.10-3 <span class="Heading">RankAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7B77040F8543CD6E">41.10-4 <span class="Heading">IsSemiRegular</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7CF02C4785F0EAB5">41.10-5 <span class="Heading">IsRegular</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7CB1D74280F92AFC">41.10-6 <span class="Heading">Earns</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X84C19AD68247B760">41.10-7 <span class="Heading">IsPrimitive</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.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.html#X84FE699F85371643">41.11-1 <span class="Heading">Blocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X79936EB97AAD1144">41.11-2 <span class="Heading">MaximalBlocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7941DB6380B74510">41.11-3 <span class="Heading">RepresentativesMinimalBlocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X835658B07B28EF3B">41.11-4 AllBlocks</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.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.html#X8264C3C479FF0A8B">41.12-1 IsExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7C90F648793E47DD">41.12-2 ExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7B9DB15D80CE28B4">41.12-3 ActingDomain</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X86153CB087394DC1">41.12-4 FunctionAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X86A0CC1479A5932A">41.12-5 HomeEnumerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X879DE63C7858453C">41.12-6 IsExternalSubset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X87D1EA1486D86233">41.12-7 ExternalSubset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7E081F568407317F">41.12-8 IsExternalOrbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7FB656AE7A066C35">41.12-9 ExternalOrbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7BAFF02B7D6DF9F2">41.12-10 StabilizerOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X867262FA82FDD592">41.12-11 <span class="Heading">ExternalOrbits</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7A64EF807CE8893E">41.12-12 <span class="Heading">ExternalOrbitsStabilizers</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X8048AE727A7F1A2F">41.12-13 CanonicalRepresentativeOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X8071A8D784DC8325">41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X85E9A6A77B8D00B8">41.12-15 ActorOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X8190A8247F29A5C7">41.12-16 UnderlyingExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap41.html#X7A3D87DE809FBFD4">41.12-17 SurjectiveActionHomomorphismAttr</a></span>
</div></div>
</div>
<h3>41 <span class="Heading">Group Actions</span></h3>
<p>A <em>group action</em> is a triple <span class="SimpleMath">(G, Ω, μ)</span>, where <span class="SimpleMath">G</span> is a group, <span class="SimpleMath">Ω</span> a set and <span class="SimpleMath">μ : Ω × G → Ω</span> a function that is compatible with the group arithmetic. We call <span class="SimpleMath">Ω</span> the <em>domain</em> of the action.</p>
<p>In <strong class="pkg">GAP</strong>, <span class="SimpleMath">Ω</span> can be a duplicate-free collection (an object that permits access to its elements via the <span class="SimpleMath">Ω[n]</span> operation, for example a list), it does not need to be sorted (see <code class="func">IsSet</code> (<a href="chap21.html#X80CDAF45782E8DCB"><span class="RefLink">21.17-4</span></a>)).</p>
<p>The acting function <span class="SimpleMath">μ</span> is a binary <strong class="pkg">GAP</strong> function that returns the image <span class="SimpleMath">μ( x, g )</span> for a point <span class="SimpleMath">x ∈ Ω</span> and a group element <span class="SimpleMath">g ∈ G</span>.</p>
<p>In <strong class="pkg">GAP</strong>, groups always act from the right, that is <span class="SimpleMath">μ( μ( x, g ), h ) = μ( x, gh )</span>.</p>
<p><strong class="pkg">GAP</strong> does not test whether the acting function <span class="SimpleMath">μ</span> satisfies the conditions for a group operation but silently assumes that is does. (If it does not, results are unpredictable.)</p>
<p>The first section of this chapter, <a href="chap41.html#X83661AFD7B7BD1D9"><span class="RefLink">41.1</span></a>, describes the various ways how operations for group actions can be called.</p>
<p>Functions for several commonly used action are already built into <strong class="pkg">GAP</strong>. These are listed in section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a>.</p>
<p>The sections <a href="chap41.html#X87F73CCA7921DE65"><span class="RefLink">41.7</span></a> and <a href="chap41.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 other sections then describe operations to compute orbits, stabilizers, as well as properties of actions.</p>
<p>Finally section <a href="chap41.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a> describes the concept of "external sets" which represent the concept of a <em><span class="SimpleMath">G</span>-set</em> and underly the actions mechanism.</p>
<p><a id="X83661AFD7B7BD1D9" name="X83661AFD7B7BD1D9"></a></p>
<h4>41.1 <span class="Heading">About Group Actions</span></h4>
<p>The syntax which is 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.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>
<dl>
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">( G, Ω[, μ] )</span></strong></dt>
<dd><p>The acting function <span class="SimpleMath">μ</span> is optional. If it is not given, the built-in action <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) (which defines an action via the caret operator <code class="code">^</code>) is used as a default.</p>
</dd>
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">( G, Ω, gens, acts[, μ] )</span></strong></dt>
<dd><p>This second version of <code class="func">OrbitsDomain</code> (<a href="chap41.html#X86BC8B958123F953"><span class="RefLink">41.4-3</span></a>) permits one to implement an action induced by a homomorphism: If the group <span class="SimpleMath">H</span> acts on <span class="SimpleMath">Ω</span> via <span class="SimpleMath">μ</span> and <span class="SimpleMath">φ : G → H</span> is a homomorphism, <span class="SimpleMath">G</span> acts on <span class="SimpleMath">Ω</span> via the induced action <span class="SimpleMath">μ'( x, g ) = μ( x, g^φ )</span>.</p>
<p>Here <span class="SimpleMath">gens</span> must be a set of generators of <span class="SimpleMath">G</span> and <span class="SimpleMath">acts</span> the images of <span class="SimpleMath">gens</span> under <span class="SimpleMath">φ</span>. <span class="SimpleMath">μ</span> is the acting function for <span class="SimpleMath">H</span>. Again, the function <span class="SimpleMath">μ</span> is optional and <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) is used as a default.</p>
<p>The advantage of this notation is that <strong class="pkg">GAP</strong> does not need to construct this homomorphism <span class="SimpleMath">φ</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>
<p><strong class="pkg">GAP</strong> does not test whether the mapping <span class="SimpleMath">gens ↦ acts</span> actually induces a homomorphism and the results are unpredictable if this is not the case.</p>
</dd>
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">( extset )</span></strong></dt>
<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">Ω</span> and <span class="SimpleMath">μ</span>. You will find more about external sets in Section <a href="chap41.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a>.</p>
</dd>
</dl>
<p>For operations like <code class="func">Stabilizer</code> (<a href="chap41.html#X86FB962786397E02"><span class="RefLink">41.5-2</span></a>) of course the domain must be replaced by an element of the domain of the action.</p>
<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.html#X86FB962786397E02"><span class="RefLink">41.5-2</span></a>) special methods are available for many of these actions.</p>
<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.html#X7FA394D27E721E2B"><span class="RefLink">41.2-13</span></a>):</p>
<div class="example"><pre>
OnIndeterminatesPairs:= function( polypair, g )
return [ OnIndeterminates( polypair[1], g ),
OnIndeterminates( polypair[2], g ) ];
end;
</pre></div>
<p>Note that this function <em>must</em> implement a group action from the <em>right</em>. This is not verified by <strong class="pkg">GAP</strong> and results are unpredictable otherwise.</p>
<p><a id="X7FE417DD837987B4" name="X7FE417DD837987B4"></a></p>
<h5>41.2-1 OnPoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="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>
<p>returns <code class="code"><var class="Arg">pnt</var> ^ <var class="Arg">g</var></code>. This is for example the action of a permutation group on points, or the action of a group on its elements via conjugation. The action of a matrix group on vectors from the right is described by both <code class="func">OnPoints</code> and <code class="func">OnRight</code> (<a href="chap41.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>).</p>
<p><a id="X7960924D84B5B18F" name="X7960924D84B5B18F"></a></p>
<h5>41.2-2 OnRight</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnRight</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</var> * <var class="Arg">g</var></code>. 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.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) and <code class="func">OnRight</code>.</p>
<p><a id="X832DF5327ECA0E44" name="X832DF5327ECA0E44"></a></p>
<h5>41.2-3 OnLeftInverse</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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"><var class="Arg">g</var>^{-1}</span> <code class="code">* <var class="Arg">pnt</var></code>. Forming the inverse is necessary to make this a proper action, as in <strong class="pkg">GAP</strong> groups always act from the right.</p>
<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.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a>), that is, a right coset <span class="SimpleMath">Ug</span> is an external set for the group <span class="SimpleMath">U</span> acting on it via <code class="func">OnLeftInverse</code>.)</p>
<p><a id="X85AA04347CD117F9" name="X85AA04347CD117F9"></a></p>
<h5>41.2-4 OnSets</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">set</var> be a proper set (see <a href="chap21.html#X80ABC25582343910"><span class="RefLink">21.19</span></a>). <code class="func">OnSets</code> returns the proper set formed by the images of all points <span class="SimpleMath">x</span> of <var class="Arg">set</var> via the action function <code class="func">OnPoints</code> (<a href="chap41.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> is for example used to compute the action of a permutation group on blocks.</p>
<p>(<code class="func">OnTuples</code> (<a href="chap41.html#X832CC5F87EEA4A7E"><span class="RefLink">41.2-5</span></a>) is an action on lists that preserves the ordering of entries.)</p>
<p><a id="X832CC5F87EEA4A7E" name="X832CC5F87EEA4A7E"></a></p>
<h5>41.2-5 OnTuples</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnTuples</code>( <var class="Arg">tup</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">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.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.html#X85AA04347CD117F9"><span class="RefLink">41.2-4</span></a>) is an action on lists that additionally sorts the entries of the result.)</p>
<p><a id="X80DAA1D2855B1456" name="X80DAA1D2855B1456"></a></p>
<h5>41.2-6 OnPairs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnPairs</code>( <var class="Arg">tup</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a special case of <code class="func">OnTuples</code> (<a href="chap41.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="tdleft"><code class="func">‣ OnSetsSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the action on sets of sets. For the special case that the sets are pairwise disjoint, it is possible to use <code class="func">OnSetsDisjointSets</code> (<a href="chap41.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 <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<p><a id="X7E23686E7A9D3A20" name="X7E23686E7A9D3A20"></a></p>
<h5>41.2-8 OnSetsDisjointSets</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnSetsDisjointSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the action on sets of pairwise disjoint sets (see also <code class="func">OnSetsSets</code> (<a href="chap41.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="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<p><a id="X7ADE244E819035FF" name="X7ADE244E819035FF"></a></p>
<h5>41.2-9 OnSetsTuples</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 on sets of tuples. <var class="Arg">set</var> must be a sorted list, otherwise an error is triggered (see <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<p><a id="X7FF556CD7E6739A9" name="X7FF556CD7E6739A9"></a></p>
<h5>41.2-10 OnTuplesSets</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnTuplesSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<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 href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<p><a id="X844E902382EB4151" name="X844E902382EB4151"></a></p>
<h5>41.2-11 OnTuplesTuples</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnTuplesTuples</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the action on tuples of tuples.</p>
<div class="example"><pre>
<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,OnPoints);</span>
[ 1, 2, 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(g,(),OnRight);</span>
[ (), (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) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(g,[1,2],OnPairs);</span>
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 1 ], [ 3, 4 ], [ 2, 1 ],
[ 1, 4 ], [ 4, 1 ], [ 4, 2 ], [ 3, 2 ], [ 2, 4 ], [ 4, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(g,[1,2],OnSets);</span>
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(g,[[1,2],[3,4]],OnSetsSets);</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ],
[ [ 1, 3 ], [ 2, 4 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(g,[[1,2],[3,4]],OnTuplesSets);</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 ] ] ]
<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 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(g,[[1,2],[3,4]],OnTuplesTuples);</span>
[ [ [ 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 ] ] ]
</pre></div>
<p><a id="X86DC2DD5829CAD9A" name="X86DC2DD5829CAD9A"></a></p>
<h5>41.2-12 OnLines</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnLines</code>( <var class="Arg">vec</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">vec</var> be a <em>normed</em> row vector, that is, its first nonzero entry is normed to the identity of the relevant field, see <code class="func">NormedRowVector</code> (<a href="chap23.html#X785DC60D8482695D"><span class="RefLink">23.2-1</span></a>). The function <code class="func">OnLines</code> returns the row vector obtained from first multiplying <var class="Arg">vec</var> from the right with <var class="Arg">g</var> (via <code class="func">OnRight</code> (<a href="chap41.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.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">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:=Action(gl,Orbit(gl,v,OnLines),OnLines);</span>
Group([ (2,3,5,6), (1,2,4)(3,6,5) ])
</pre></div>
<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">‣ 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="GAPinput">Stabilizer(Group((1,2,3,4),(1,2)),x*y,OnIndeterminates);</span>
Group([ (1,2), (3,4) ])
</pre></div>
<p><a id="X7BA8D76586F1F06E" name="X7BA8D76586F1F06E"></a></p>
<h5>41.2-14 Permuted</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Permuted</code>( <var class="Arg">list</var>, <var class="Arg">perm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The following example demonstrates <code class="func">Permuted</code> (<a href="chap21.html#X7B5A19098406347A"><span class="RefLink">21.20-18</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 id="X85124D197F0F9C4D" name="X85124D197F0F9C4D"></a></p>
<h5>41.2-15 OnSubspacesByCanonicalBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnSubspacesByCanonicalBasis</code>( <var class="Arg">bas</var>, <var class="Arg">mat</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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</var> must be a list of (linearly independent) vectors which forms a basis of the subspace in Hermite normal form. <var class="Arg">mat</var> is an element of the acting matrix group. The function returns a mutable matrix which gives the basis of the image of the subspace in Hermite normal form. (In other words: it triangulizes the product of <var class="Arg">bas</var> with <var class="Arg">mat</var>.)</p>
<p><var class="Arg">bas</var> must be given in Hermite normal form, otherwise an error is triggered (see <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>
<p><a id="X82181CA07A5B2056" name="X82181CA07A5B2056"></a></p>
<h4>41.3 <span class="Heading">Action on canonical representatives</span></h4>
<p>A variety of action functions assumes that the objects on which it acts are given in a particular form, for example canonical representatives. Affected actions are for example <code class="func">OnSetsSets</code> (<a href="chap41.html#X7C10492081D72376"><span class="RefLink">41.2-7</span></a>), <code class="func">OnSetsDisjointSets</code> (<a href="chap41.html#X7E23686E7A9D3A20"><span class="RefLink">41.2-8</span></a>), <code class="func">OnSetsTuples</code> (<a href="chap41.html#X7ADE244E819035FF"><span class="RefLink">41.2-9</span></a>), <code class="func">OnTuplesSets</code> (<a href="chap41.html#X7FF556CD7E6739A9"><span class="RefLink">41.2-10</span></a>), <code class="func">OnLines</code> (<a href="chap41.html#X86DC2DD5829CAD9A"><span class="RefLink">41.2-12</span></a>) and <code class="func">OnSubspacesByCanonicalBasis</code> (<a href="chap41.html#X85124D197F0F9C4D"><span class="RefLink">41.2-15</span></a>).</p>
<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(SymmetricGroup(5),[[2,4],[1,3]],OnSetsSets);</span>
Error, Action not well-defined. See the manual section
``Action on canonical representatives''.
</pre></div>
<p>In this case the affected domain elements have to be brought in canonical form, as documented for the respective action function. For interactive use this is most easily done by acting with the identity element of the group.</p>
<p>(A similar error could arise if a user-defined action function is used which actually does not implement an action from the right.)</p>
<p><a id="X81E0FF0587C54543" name="X81E0FF0587C54543"></a></p>
<h4>41.4 <span class="Heading">Orbits</span></h4>
<p>If a group <span class="SimpleMath">G</span> acts on a set <span class="SimpleMath">Ω</span>, the set of all images of <span class="SimpleMath">x ∈ Ω</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">Ω</span>.</p>
<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">‣ Orbit</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></table></div>
<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.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) if no action function 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 class="GAPinput">Orbit(g,1);</span>
[ 1, 3, 2, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(g,[1,2],OnSets);</span>
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ], [ 2, 4 ] ]
</pre></div>
<p>(See Section <a href="chap41.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="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Orbits</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 )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Orbits</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) if no action function is given.</p>
<p>(Note that the arrangement of orbits or of points within one orbit is not defined by the operation.)</p>
<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">‣ 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><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitsDomain</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the orbits of <var class="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.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) if no action function is given.</p>
<p>This operation is often faster than <code class="func">Orbits</code> (<a href="chap41.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>(Note that the arrangement of orbits or of points within one orbit is not defined by the operation.)</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">Orbits(g,[1..5]);</span>
[ [ 1, 3, 2, 4 ], [ 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitsDomain(g,Arrangements([1..4],3),OnTuples);</span>
[ [ [ 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 ] ] ]
<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.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">‣ 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></table></div>
<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.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>41.4-5 <span class="Heading">OrbitLengths</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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><a id="X8520E2487F7E98AF" name="X8520E2487F7E98AF"></a></p>
<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">‣ OrbitLengthsDomain</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="func">OrbitLengths</code> (<a href="chap41.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>
<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,Arrangements([1..4],4),OnTuples);</span>
[ 12, 12 ]
</pre></div>
<p><a id="X797BD60E7ACEF1B1" name="X797BD60E7ACEF1B1"></a></p>
<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 fix <span class="SimpleMath">x</span>.</p>
<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">‣ OrbitStabilizer</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></table></div>
<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">‣ 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>
<p>computes the stabilizer in <var class="Arg">G</var> of the point <var class="Arg">pnt</var>, that is the subgroup of those 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>
<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">Stabilizer(g,4);</span>
Group([ (1,3,2) ])
</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 class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(g,[1,2],OnTuples);</span>
Group(())
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitStabilizer(g,[1,2],OnSets);</span>
rec(
orbit := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ],
[ 2, 4 ] ], stabilizer := Group([ (1,2)(3,4) ]) )
</pre></div>
<p>(See Section <a href="chap41.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 for solvable groups, which uses the fact that the orbits of a normal subgroup form a block system (see <a href="chapBib.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">‣ 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">( function )</td></tr></table></div>
<p>This operation should not be called by a user. It is documented however for purposes to extend or maintain the group actions package (the word "package" here refers to the <strong class="pkg">GAP</strong> functionality for group actions, not to a <strong class="pkg">GAP</strong> package).</p>
<p><code class="func">OrbitStabilizerAlgorithm</code> 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 given, this must be a subgroup stabilizing <em>all</em> points in the domain and can be used to abbreviate stabilizer calculations.</p>
<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> is given it must be a bit list corresponding to <var class="Arg">Omega</var> in which elements which have been found already will be "ticked off" with <code class="keyw">true</code>. (In particular, the entries for the orbit of <var class="Arg">pnt</var> still must be all set to <code class="keyw">false</code>). Again the remaining action domain (the bits set initially to <code class="keyw">false</code>) can be used to stop if the orbit cannot grow any longer. Another use of the bit list is if <var class="Arg">Omega</var> is an enumerator which can determine <code class="func">PositionCanonical</code> (<a href="chap21.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>) values very quickly. In this situation it can be worth to search images not in the orbit found so far, but via their position in <var class="Arg">Omega</var> and use a the bit list to keep track whether the element is in the orbit found so far.</p>
<p><a id="X7A9389097BAF670D" name="X7A9389097BAF670D"></a></p>
<h4>41.6 <span class="Heading">Elements with Prescribed Images</span></h4>
<p><a id="X857DC7B085EB0539" name="X857DC7B085EB0539"></a></p>
<h5>41.6-1 RepresentativeAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeAction</code>( <var class="Arg">G</var>[, <var class="Arg">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">( operation )</td></tr></table></div>
<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>
<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">RepresentativeAction(g,1,3);</span>
(1,3)(2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(g,1,3,OnPoints);</span>
(1,3)(2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(g,(1,2,3),(2,4,3));</span>
(1,2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(g,(1,2,3),(2,3,4));</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></span> <span class="GAPinput"> RepresentativeAction(g,[1,2,3],[1,2,4],OnSets);</span>
(2,4,3)
<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.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>
<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="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 <span class="SimpleMath">Ω</span>, an enumeration of <span class="SimpleMath">Omega</span> yields a homomorphism from <span class="SimpleMath">G</span> into the symmetric group on <span class="SimpleMath">{ 1, ..., |Ω| }</span>. In <strong class="pkg">GAP</strong>, the enumeration of <span class="SimpleMath">Ω</span> is provided by the <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) value of <span class="SimpleMath">Ω</span> which of course is <span class="SimpleMath">Ω</span> itself if it is a list.</p>
<p>For an action homomorphism, the operation <code class="func">UnderlyingExternalSet</code> (<a href="chap41.html#X8190A8247F29A5C7"><span class="RefLink">41.12-16</span></a>) will return the external set on <span class="SimpleMath">Ω</span> which affords the action.</p>
<p><a id="X78E6A002835288A4" name="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">‣ 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">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActionHomomorphism</code>( <var class="Arg">xset</var>[, <var class="Arg">"surjective"</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">‣ ActionHomomorphism</code>( <var class="Arg">action</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a homomorphism from <var class="Arg">G</var> into the symmetric group on <span class="SimpleMath">|<var class="Arg">Omega</var>|</span> points that gives the permutation action of <var class="Arg">G</var> on <var class="Arg">Omega</var>.</p>
<p>By default the homomorphism returned by <code class="func">ActionHomomorphism</code> is not necessarily surjective (its <code class="func">Range</code> (<a href="chap32.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 is supported only for <strong class="pkg">GAP</strong>3 compatibility) returns the action homomorphism that belongs to the image obtained via <code class="func">Action</code> (<a href="chap41.html#X85A8E93D786C3C9C"><span class="RefLink">41.7-2</span></a>).</p>
<p>(See Section <a href="chap41.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 class="GAPprompt">gap></span> <span class="GAPinput">hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples);</span>
<action homomorphism>
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(hom);</span>
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></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));</span>
6
</pre></div>
<p>When acting on a domain, the operation <code class="func">PositionCanonical</code> (<a href="chap21.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 by a list of representatives for which <code class="func">PositionCanonical</code> (<a href="chap21.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>) is implemented, for example the return value of <code class="func">RightTransversal</code> (<a href="chap39.html#X85C65D06822E716F"><span class="RefLink">39.8-1</span></a>).</p>
<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"><code class="func">‣ Action</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Action</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the image group of <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>) called with the same parameters.</p>
<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.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 class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Action(g,AsList(g),OnRight);</span>
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
</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="func">‣ 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="100%"><tr><td class="tdleft"><code class="func">‣ SortedSparseActionHomomorphism</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>
<p><code class="func">SparseActionHomomorphism</code> computes the action homomorphism (see <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>)) with arguments <var class="Arg">G</var>, <span class="SimpleMath">D</span>, and the optional arguments given, where <span class="SimpleMath">D</span> is the union of the <var class="Arg">G</var>-orbits of all points in <var class="Arg">start</var>. In the <code class="func">Orbit</code> (<a href="chap41.html#X80E0234E7BD79409"><span class="RefLink">41.4-1</span></a>) calls that are used to create <span class="SimpleMath">D</span>, again the optional arguments given are entered.)</p>
<p>If <var class="Arg">G</var> acts on a very large domain not surjectively this may yield a permutation image of substantially smaller degree than by action on the whole domain.</p>
<p>The operation <code class="func">SparseActionHomomorphism</code> will only use <code class="func">\=</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>) comparisons of points in the orbit. Therefore it can be used even if no good <code class="func">\<</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>) comparison method for these points is available. However the image group will depend on the generators <var class="Arg">gens</var> of <var class="Arg">G</var>.</p>
<p>The operation <code class="func">SortedSparseActionHomomorphism</code> in contrast will sort the orbit and thus produce an image group which does not depend on these generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group(Z(3)*[[[1,1],[0,1]]]);</span>
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">Action of a group on itself</span></h4>
<p>Of particular importance is the action of a group on its elements or cosets of a subgroup. These actions can be obtained by using <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>) for a suitable domain (for example a list of subgroups). For the following (frequently used) types of actions however special (often particularly efficient) functions are provided. A special case is the regular action on all elements.</p>
<p><a id="X78C37C4C7B2BDC44" name="X78C37C4C7B2BDC44"></a></p>
<h5>41.8-1 FactorCosetAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="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>
<p>This command computes the action of the group <var class="Arg">G</var> on the right cosets of the subgroup <var class="Arg">U</var>. If a normal subgroup <var class="Arg">N</var> of <var class="Arg">G</var> is given, it is stored as kernel of this action.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4,5),(1,2));;u:=SylowSubgroup(g,2);;Index(g,u);</span>
15
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorCosetAction(g,u);</span>
<action epimorphism>
<span class="GAPprompt">gap></span> <span class="GAPinput">Range(last);</span>
Group([ (1,9,13,10,4)(2,8,14,11,5)(3,7,15,12,6),
(1,7)(2,8)(3,9)(5,6)(10,11)(14,15) ])
</pre></div>
<p><a id="X8561DEBA79E01ABD" name="X8561DEBA79E01ABD"></a></p>
<h5>41.8-2 RegularActionHomomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RegularActionHomomorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an isomorphism from <var class="Arg">G</var> onto the regular permutation representation of <var class="Arg">G</var>.</p>
<p><a id="X835317A7847477D4" name="X835317A7847477D4"></a></p>
<h5>41.8-3 AbelianSubfactorAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianSubfactorAction</code>( <var class="Arg">G</var>, <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a group and <span class="SimpleMath"><var class="Arg">M</var> ≥ <var class="Arg">N</var></span> be subgroups of a common parent that are normal under <var class="Arg">G</var>, such that the subfactor <span class="SimpleMath"><var class="Arg">M</var>/<var class="Arg">N</var></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"><var class="Arg">M</var>/<var class="Arg">N</var></span>, <var class="Arg">alpha</var> is a map from <var class="Arg">M</var> into a <span class="SimpleMath">n</span>-dimensional row space over <span class="SimpleMath">GF(p)</span> where <span class="SimpleMath">[<var class="Arg">M</var>:<var class="Arg">N</var>] = p^n</span> that is the natural homomorphism of <var class="Arg">M</var> by <var class="Arg">N</var> with the quotient represented as an additive group. Finally <var class="Arg">phi</var> is a homomorphism from <var class="Arg">G</var> into <span class="SimpleMath">GL_n(p)</span> that represents the action of <var class="Arg">G</var> on the factor <span class="SimpleMath"><var class="Arg">M</var>/<var class="Arg">N</var></span>.</p>
<p>Note: If only matrices for the action are needed, <code class="func">LinearActionLayer</code> (<a href="chap45.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),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (1,9,5,6,3,10)(2,11,12,8,4,7));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:=ChiefSeries(g);;List(c,Size);</span>
[ 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>
<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( 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>
Group([ <an immutable 2x2 matrix over GF2>,
<an immutable 2x2 matrix over GF2> ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(mat);</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=PreImagesRepresentative(a[2],[Z(2),0*Z(2)]);</span>
(2,9,3,8)(4,11,5,10)
<span class="GAPprompt">gap></span> <span class="GAPinput">e in c[3];e in c[4];</span>
true
false
</pre></div>
<p><a id="X807AA91E841D132B" name="X807AA91E841D132B"></a></p>
<h4>41.9 <span class="Heading">Permutations Induced by Elements and Cycles</span></h4>
<p>If only the permutation image of a single element is needed, 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><a id="X7807A33381DCAB26" name="X7807A33381DCAB26"></a></p>
<h5>41.9-1 <span class="Heading">Permutation</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Permutation</code>( <var class="Arg">g</var>, <var class="Arg">xset</var> )</td><td class="tdright">( function )</td></tr></table></div>
<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 of 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.html#X86A0CC1479A5932A"><span class="RefLink">41.12-5</span></a>) value of this external set (see Section <a href="chap41.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a>). Note that the points of the returned permutation refer to the positions in <var class="Arg">Omega</var>, even if <var class="Arg">Omega</var> itself consists of integers.</p>
<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>
<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"><code class="func">‣ PermutationCycle</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>, <var class="Arg">pnt</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<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>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Permutation([[Z(3),-Z(3)],[Z(3),0*Z(3)]],AsList(GF(3)^2));</span>
(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="GAPinput">PermutationCycle((1,2,3)(4,5)(6,7),[4..7],4);</span>
(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">‣ Cycle</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>, <var class="Arg">pnt</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of the points in the cycle of <var class="Arg">pnt</var> under the action of the element <var class="Arg">g</var>.</p>
<p><a id="X7F559E897B333758" name="X7F559E897B333758"></a></p>
<h5>41.9-4 CycleLength</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CycleLength</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>, <var class="Arg">pnt</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the length of the cycle of <var class="Arg">pnt</var> under the action of the element <var class="Arg">g</var>.</p>
<p><a id="X7F3B387A7FD8AE5E" name="X7F3B387A7FD8AE5E"></a></p>
<h5>41.9-5 Cycles</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cycles</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<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="100%"><tr><td class="tdleft"><code class="func">‣ CycleLengths</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the lengths of all the cycles under the action of the element <var class="Arg">g</var> on <var class="Arg">Omega</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cycle((1,2,3)(4,5)(6,7),[4..7],4);</span>
[ 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">gap></span> <span class="GAPinput">Cycles((1,2,3)(4,5)(6,7),[4..7]);</span>
[ [ 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>
<h5>41.9-7 <span class="Heading">CycleIndex</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CycleIndex</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CycleIndex</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>The <em>cycle index</em> of a permutation <var class="Arg">g</var> acting on <var class="Arg">Omega</var> is defined as</p>
<p class="pcenter">z(<var class="Arg">g</var>) = s_1^{c_1} s_2^{c_2} ⋯ s_n^{c_n}</p>
<p>where <span class="SimpleMath">c_k</span> is the number of <span class="SimpleMath">k</span>-cycles in the cycle decomposition of <var class="Arg">g</var> and the <span class="SimpleMath">s_i</span> are indeterminates.</p>
<p>The <em>cycle index</em> of a group <var class="Arg">G</var> is defined as</p>
<p class="pcenter">Z(<var class="Arg">G</var>) = ( ∑_{g ∈ <var class="Arg">G</var>} z(g) ) / |<var class="Arg">G</var>| .</p>
<p>The indeterminates used by <code class="func">CycleIndex</code> are the indeterminates <span class="SimpleMath">1</span> to <span class="SimpleMath">n</span> over the rationals (see <code class="func">Indeterminate</code> (<a href="chap66.html#X79D0380D7FA39F7D"><span class="RefLink">66.1-1</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=TransitiveGroup(6,8);</span>
S_4(6c) = 1/2[2^3]S(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">CycleIndex(g);</span>
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>
<h4>41.10 <span class="Heading">Tests for Actions</span></h4>
<p><a id="X79B15750851828CB" name="X79B15750851828CB"></a></p>
<h5>41.10-1 <span class="Heading">IsTransitive</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransitive</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransitive</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransitive</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 transitive, or <code class="keyw">false</code> otherwise.</p>
<p>We say that a group <var class="Arg">G</var> acts <em>transitively</em> on a domain <span class="SimpleMath">D</span> if and only if for every pair of points <span class="SimpleMath">d, e ∈ D</span> there is an element <span class="SimpleMath">g</span> in <var class="Arg">G</var> such that <span class="SimpleMath">d^g = e</span>.</p>
<p>For permutation groups, the syntax <code class="code">IsTransitive(<var class="Arg">G</var>)</code> is also permitted and tests whether the group is transitive on the points moved by it, that is the group <span class="SimpleMath">⟨ (2,3,4),(2,3) ⟩</span> is transitive (on 3 points).</p>
<p><a id="X8295D733796B7A37" name="X8295D733796B7A37"></a></p>
<h5>41.10-2 <span class="Heading">Transitivity</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Transitivity</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Transitivity</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the degree <span class="SimpleMath">k</span> (a non-negative integer) of transitivity of the action implied by the arguments, i.e. the largest integer <span class="SimpleMath">k</span> such that the action is <span class="SimpleMath">k</span>-transitive. If the action is not transitive <code class="code">0</code> is returned.</p>
<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>
<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
</pre></div>
<p><a id="X8166A6A17C8D6E73" name="X8166A6A17C8D6E73"></a></p>
<h5>41.10-3 <span class="Heading">RankAction</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankAction</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankAction</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the rank of a transitive action, i.e. the number of orbits of the point stabilizer.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RankAction(g,Combinations([1..4],2),OnSets);</span>
4
</pre></div>
<p><a id="X7B77040F8543CD6E" name="X7B77040F8543CD6E"></a></p>
<h5>41.10-4 <span class="Heading">IsSemiRegular</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemiRegular</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemiRegular</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 semiregular, or <code class="keyw">false</code> otherwise.</p>
<p>An action is <em>semiregular</em> is the stabilizer of each point is the identity.</p>
<p><a id="X7CF02C4785F0EAB5" name="X7CF02C4785F0EAB5"></a></p>
<h5>41.10-5 <span class="Heading">IsRegular</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRegular</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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.html#X7B77040F8543CD6E"><span class="RefLink">41.10-4</span></a>)) and transitive (see <code class="func">IsTransitive</code> (<a href="chap41.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>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemiRegular(g,Arrangements([1..4],3),OnTuples);</span>
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>
<h5>41.10-6 <span class="Heading">Earns</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Earns</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 methods for a primitive group <var class="Arg">G</var> are implemented.</p>
<p><a id="X84C19AD68247B760" name="X84C19AD68247B760"></a></p>
<h5>41.10-7 <span class="Heading">IsPrimitive</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimitive</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimitive</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 primitive, or <code class="keyw">false</code> otherwise.</p>
<p>An action is <em>primitive</em> if it is transitive and the action admits no nontrivial block systems. See <a href="chap41.html#X7E9D3D0B7A9A8572"><span class="RefLink">41.11</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimitive(g,Orbit(g,(1,2)(3,4)));</span>
true
</pre></div>
<p><a id="X7E9D3D0B7A9A8572" name="X7E9D3D0B7A9A8572"></a></p>
<h4>41.11 <span class="Heading">Block Systems</span></h4>
<p>A <em>block system</em> (system of imprimitivity) for the action of a group <span class="SimpleMath">G</span> on an action domain <span class="SimpleMath">Ω</span> is a partition of <span class="SimpleMath">Ω</span> which –as a partition– remains invariant under the action of <span class="SimpleMath">G</span>.</p>
<p><a id="X84FE699F85371643" name="X84FE699F85371643"></a></p>
<h5>41.11-1 <span class="Heading">Blocks</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 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">‣ 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 action. If <var class="Arg">seed</var> is not given and the action is imprimitive, a minimal nontrivial block system will be found. If <var class="Arg">seed</var> is given, a block system in which <var class="Arg">seed</var> is the subset of one block is computed. The action must be transitive.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=TransitiveGroup(8,3);</span>
E(8)=2[x]2[x]2
<span class="GAPprompt">gap></span> <span class="GAPinput">Blocks(g,[1..8]);</span>
[ [ 1, 8 ], [ 2, 3 ], [ 4, 5 ], [ 6, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Blocks(g,[1..8],[1,4]);</span>
[ [ 1, 4 ], [ 2, 7 ], [ 3, 6 ], [ 5, 8 ] ]
</pre></div>
<p>(See Section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>
<p><a id="X79936EB97AAD1144" name="X79936EB97AAD1144"></a></p>
<h5>41.11-2 <span class="Heading">MaximalBlocks</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>][, <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">‣ MaximalBlocks</code>( <var class="Arg">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 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>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalBlocks(g,[1..8]);</span>
[ [ 1, 2, 3, 8 ], [ 4, 5, 6, 7 ] ]
</pre></div>
<p><a id="X7941DB6380B74510" name="X7941DB6380B74510"></a></p>
<h5>41.11-3 <span class="Heading">RepresentativesMinimalBlocks</span></h5>
<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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesMinimalBlocks</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a list of block representatives for all minimal (i.e blocks are minimal with respect to inclusion) nontrivial block systems for the action.</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">‣ AllBlocks</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</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>
<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>
<p>The stabilizer of a block can be computed via the action <code class="func">OnSets</code> (<a href="chap41.html#X85AA04347CD117F9"><span class="RefLink">41.2-4</span></a>):</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(g,[1,8],OnSets);</span>
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.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((1,2,3,4,5,6,7,8),(1,2));;</span>
<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>
<p><a id="X7FD3D2D2788709B7" name="X7FD3D2D2788709B7"></a></p>
<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 set <span class="SimpleMath">Ω</span> endowed with an action of <span class="SimpleMath">G</span>. The elements of the <span class="SimpleMath">G</span>-set are the same as those of <span class="SimpleMath">Ω</span>, however concepts like equality and equivalence of <span class="SimpleMath">G</span>-sets do not only consider the underlying domain <span class="SimpleMath">Ω</span> but the group action as well.</p>
<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.html#X7B9DB15D80CE28B4"><span class="RefLink">41.12-3</span></a>), <code class="func">FunctionAction</code> (<a href="chap41.html#X86153CB087394DC1"><span class="RefLink">41.12-4</span></a>) and <code class="func">HomeEnumerator</code> (<a href="chap41.html#X86A0CC1479A5932A"><span class="RefLink">41.12-5</span></a>).</p>
<p>Most operations for actions are applicable as an attribute for an external set.</p>
<p>The most prominent external subsets are orbits, see <code class="func">ExternalOrbit</code> (<a href="chap41.html#X7FB656AE7A066C35"><span class="RefLink">41.12-9</span></a>).</p>
<p>Many subsets of a group, such as conjugacy classes or cosets (see <code class="func">ConjugacyClass</code> (<a href="chap39.html#X7B2F207F7F85F5B8"><span class="RefLink">39.10-1</span></a>) and <code class="func">RightCoset</code> (<a href="chap39.html#X8412ABD57986B9FC"><span class="RefLink">39.7-1</span></a>)) are implemented as external orbits.</p>
<p>External sets also are implicitly underlying action homomorphisms, see <code class="func">UnderlyingExternalSet</code> (<a href="chap41.html#X8190A8247F29A5C7"><span class="RefLink">41.12-16</span></a>) and <code class="func">SurjectiveActionHomomorphismAttr</code> (<a href="chap41.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">‣ IsExternalSet</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>An <em>external set</em> specifies a group action <span class="SimpleMath">μ: Ω × G ↦ Ω</span> of a group <span class="SimpleMath">G</span> on a domain <span class="SimpleMath">Ω</span>. The external set knows the group, the domain and the actual acting function. Mathematically, an external set is the set <span class="SimpleMath">Ω</span>, which is endowed with the action of a group <span class="SimpleMath">G</span> via the group action <span class="SimpleMath">μ</span>. For this reason <strong class="pkg">GAP</strong> treats an external set as a domain whose elements are the elements of <span class="SimpleMath">Ω</span>. An external set is always a union of orbits. Currently the domain <span class="SimpleMath">Ω</span> must always be finite. If <span class="SimpleMath">Ω</span> is not a list, an enumerator for <span class="SimpleMath">Ω</span> is automatically chosen, see <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>).</p>
<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">‣ ExternalSet</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></table></div>
<p>creates the external set for the action <var class="Arg">act</var> of <var class="Arg">G</var> on <var class="Arg">Omega</var>. <var class="Arg">Omega</var> can be either a proper set, or a domain which is represented as described in <a href="chap12.html#X7BAF69417BB925F6"><span class="RefLink">12.4</span></a> and <a href="chap30.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.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) ) of this domain.</p>
<div class="example"><pre>
<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">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), (2,3,4), (1,3,2), (1,4,3), (1,2,4) ] ]
</pre></div>
<p><a id="X7B9DB15D80CE28B4" name="X7B9DB15D80CE28B4"></a></p>
<h5>41.12-3 ActingDomain</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActingDomain</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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">‣ FunctionAction</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the acting function with which the external set <var class="Arg">xset</var> was defined.</p>
<p><a id="X86A0CC1479A5932A" name="X86A0CC1479A5932A"></a></p>
<h5>41.12-5 HomeEnumerator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) value of <var class="Arg">xset</var>, which enumerates only the subset.</p>
<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)=OnRight;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">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">‣ 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" width="100%"><tr><td class="tdleft"><code class="func">‣ ExternalSubset</code>( <var class="Arg">G</var>, <var class="Arg">xset</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>
<p>constructs the external subset of <var class="Arg">xset</var> on the union of orbits of the points in <var class="Arg">start</var>.</p>
<p><a id="X7E081F568407317F" name="X7E081F568407317F"></a></p>
<h5>41.12-8 IsExternalOrbit</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsExternalOrbit</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>An external orbit is an external subset consisting of one orbit.</p>
<p><a id="X7FB656AE7A066C35" name="X7FB656AE7A066C35"></a></p>
<h5>41.12-9 ExternalOrbit</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExternalOrbit</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></table></div>
<p>constructs the external subset on the orbit of <var class="Arg">pnt</var>. The <code class="func">Representative</code> (<a href="chap30.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>) value of this external set is <var class="Arg">pnt</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StabilizerOfExternalSet</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the stabilizer of the <code class="func">Representative</code> (<a href="chap30.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></span> <span class="GAPinput">StabilizerOfExternalSet(e);</span>
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">‣ ExternalOrbits</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExternalOrbits</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a list of external orbits that give the orbits of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExternalOrbits(g,AsList(g));</span>
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
</pre></div>
<p><a id="X7A64EF807CE8893E" name="X7A64EF807CE8893E"></a></p>
<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">‣ ExternalOrbitsStabilizers</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></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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.html#X867262FA82FDD592"><span class="RefLink">41.12-11</span></a>), this operation also computes the stabilizers of the representatives of the external orbits at the same time. (This can be quicker than computing the <code class="func">ExternalOrbits</code> (<a href="chap41.html#X867262FA82FDD592"><span class="RefLink">41.12-11</span></a>) value first and the stabilizers afterwards.)</p>
<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">‣ CanonicalRepresentativeOfExternalSet</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The canonical representative of an external set <var class="Arg">xset</var> may only depend on the defining attributes <var class="Arg">G</var>, <var class="Arg">Omega</var>, <var class="Arg">act</var> of <var class="Arg">xset</var> and (in the case of external subsets) <code class="code">Enumerator( <var class="Arg">xset</var> )</code>. It must <em>not</em> depend, e.g., on the representative of an external orbit. <strong class="pkg">GAP</strong> does not know methods for arbitrary external sets to compute a canonical representative, see <code class="func">CanonicalRepresentativeDeterminatorOfExternalSet</code> (<a href="chap41.html#X8071A8D784DC8325"><span class="RefLink">41.12-14</span></a>).</p>
<p><a id="X8071A8D784DC8325" name="X8071A8D784DC8325"></a></p>
<h5>41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanonicalRepresentativeDeterminatorOfExternalSet</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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">‣ ActorOfExternalSet</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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>
<span class="GAPprompt">gap></span> <span class="GAPinput">CanonicalRepresentativeOfExternalSet(e);</span>
(2,4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">ActorOfExternalSet(e);</span>
(1,3,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">FunctionAction(e)((1,2)(3,4),last);</span>
(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">‣ 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">‣ 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.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.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>).)</p>
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