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<p><a id="X7E210A267E85D052" name="X7E210A267E85D052"></a></p>
<div class="ChapSects"><a href="chap6.html#X7E210A267E85D052">6 <span class="Heading">Orbital Structures</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7A489A5D79DA9E5C">6.1 <span class="Heading">Examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7DE994FE7D595E0C">6.1-1 IsOrbitalStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86F56EAD87C74596">6.1-2 OrbitalStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87395C8C8592FF00">6.1-3 OS_OrbitRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X79BD0DBA7B72D4C9">6.1-4 OS_CanonisingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X815676077CFC8CF4">6.1-5 OS_CanonisingElementAndRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7CC301797B101011">6.1-6 OS_StabilizerOf</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7F2D94D9870E46B9">6.1-7 OrbitalRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8595D3EA85AA992F">6.1-8 AllOrbitalRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7CDB09DF87DD88C5">6.1-9 OrbitalCanonizingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X79D2C53084F773B2">6.1-10 OrbitalCanonizingElementInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X81A62A257FAC1BC7">6.1-11 OrbitalTransversalIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7FDDCFB078F55048">6.1-12 UnorderedOrbitalRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X79E001B18591F1AE">6.1-13 AllUnorderedOrbitalRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78ABE8E9844AC143">6.1-14 UnorderedOrbitalTransversalIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7FA8E4BE7F8F47DD">6.1-15 UnorderedOrbitalCanonizingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X82CAF97C7AE60FA4">6.1-16 UnorderedOrbitalCanonizingElementInverse</a></span>
</div></div>
</div>

<h3>6 <span class="Heading">Orbital Structures</span></h3>

<p>The functions for orbital structures are based on recent work in permutation group algorithms. An orbital structure contains information about orbits and stabilisers of a group acting on a set for the purposes of quickly determining representatives, canonising elements, and transversal elements (directed) orbitals (orbits of ordered pairs of elements of the domain), and undirected orbitals, i.e. orbits of sets of size two.</p>

<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>

<h4>6.1 <span class="Heading">Examples</span></h4>

<p>To create an orbital structure we need generators for a group, a set, and an action</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">os := OrbitalStructure([</span>
<span class="GAPprompt">></span> <span class="GAPinput">(1,13,4,14,5)(2,10,12,9,8)(3,7,15,6,11)(16,17,18,20,19),</span>
<span class="GAPprompt">></span> <span class="GAPinput">(1,2,3)(4,6,5)(7,10,13)(8,12,14)(9,11,15)(16,18,21)(17,19,20) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[1..21],</span>
<span class="GAPprompt">></span> <span class="GAPinput">OnPoints);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitalRepresentative(os, [16,15]);</span>
[ 16, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">c := OrbitalCanonizingElement(os, [16, 15]);</span>
(1,10,9,5,15)(2,7,6,8,4)(3,13,14,11,12)(17,20,18,19,21)
<span class="GAPprompt">gap></span> <span class="GAPinput">OnTuples(c, [16,15]);</span>
[ 16, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">UnorderedOrbitalRepresentative(os, [16,2]);</span>
[ 1, 16 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">c := UnorderedOrbitalCanonizingElement(os, [16,15]);</span>
(1,15)(2,4)(3,12)(5,10)(7,8)(11,13)(17,21)(19,20)
<span class="GAPprompt">gap></span> <span class="GAPinput">OnSets(c, Set([16,15]));</span>
[ 1, 16 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllOrbitalRepresentatives(os)</span>
[ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 16 ],
  [ 1, 18 ], [ 1, 20 ], [ 16, 1 ], [ 16, 2 ], [ 16, 3 ], [ 16, 16 ], [ 16, 17 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllUnorderedOrbitalRepresentatives(os)</span>
[ [ 1, 1 ], [ 1, 2 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 16 ], [ 1, 18 ],
  [ 1, 20 ], [ 16, 16 ], [ 16, 17 ] ]
</pre></div>

<p><a id="X7DE994FE7D595E0C" name="X7DE994FE7D595E0C"></a></p>

<h5>6.1-1 IsOrbitalStructure</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsOrbitalStructure</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p><a id="X86F56EAD87C74596" name="X86F56EAD87C74596"></a></p>

<h5>6.1-2 OrbitalStructure</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitalStructure</code>( <var class="Arg">gens</var>, <var class="Arg">domain</var>, <var class="Arg">act</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: An orbital structure</p>

<p>Given generators, a set, and an action function create an orbital structure. An orbital structure contains a list of orbits of the group generated by <var class="Arg">gens</var> on <var class="Arg">domain</var>, a hashmap that maps any element of <var class="Arg">domain</var> to the index of its orbit in the list of orbits. We choose the smallest element of each orbit as representative. For each orbit, the orbital structure also contains the stabilizer of the chosen orbit representative, together with all orbits of that stabilizer on <var class="Arg">domain</var> with chosen representatives.</p>

<p><a id="X87395C8C8592FF00" name="X87395C8C8592FF00"></a></p>

<h5>6.1-3 OS_OrbitRepresentative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OS_OrbitRepresentative</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><a id="X79BD0DBA7B72D4C9" name="X79BD0DBA7B72D4C9"></a></p>

<h5>6.1-4 OS_CanonisingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OS_CanonisingElement</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><a id="X815676077CFC8CF4" name="X815676077CFC8CF4"></a></p>

<h5>6.1-5 OS_CanonisingElementAndRepresentative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OS_CanonisingElementAndRepresentative</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><a id="X7CC301797B101011" name="X7CC301797B101011"></a></p>

<h5>6.1-6 OS_StabilizerOf</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OS_StabilizerOf</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><a id="X7F2D94D9870E46B9" name="X7F2D94D9870E46B9"></a></p>

<h5>6.1-7 OrbitalRepresentative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitalRepresentative</code>( <var class="Arg">os</var>, <var class="Arg">pair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: pair</p>

<p>Given an orbital structure <var class="Arg">os</var> and a pair <var class="Arg">pair</var> of elements of the domain that <var class="Arg">os</var> is defined on, returns a canonical representative of <var class="Arg">pair</var> in its orbit of ordered pairs.</p>

<p><a id="X8595D3EA85AA992F" name="X8595D3EA85AA992F"></a></p>

<h5>6.1-8 AllOrbitalRepresentatives</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllOrbitalRepresentatives</code>( <var class="Arg">os</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Return the set of canonical representatives of orbits of pairs under the action of the orbital structure.</p>

<p><a id="X7CDB09DF87DD88C5" name="X7CDB09DF87DD88C5"></a></p>

<h5>6.1-9 OrbitalCanonizingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitalCanonizingElement</code>( <var class="Arg">os</var>, <var class="Arg">pair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a group element</p>

<p>Given an orbital structure <var class="Arg">os</var> and the pair <var class="Arg">pair</var> returns an element <span class="SimpleMath">g</span> of the group that maps <var class="Arg">pair</var> to <code class="code">OrbitalRepresentative(os, pair)</code>.</p>

<p><a id="X79D2C53084F773B2" name="X79D2C53084F773B2"></a></p>

<h5>6.1-10 OrbitalCanonizingElementInverse</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitalCanonizingElementInverse</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><a id="X81A62A257FAC1BC7" name="X81A62A257FAC1BC7"></a></p>

<h5>6.1-11 OrbitalTransversalIterator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitalTransversalIterator</code>( <var class="Arg">os</var>, <var class="Arg">pair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an iterator</p>

<p>Given an orbital structure <var class="Arg">os</var> and a pair <var class="Arg">pair</var>, returns an iterator that produces an element <code class="code">g</code> for every element <code class="code">e</code> in the orbit such that <code class="code">OnTuples(OrbitalRepresentative(os, pair), g) = e</code>.</p>

<p><a id="X7FDDCFB078F55048" name="X7FDDCFB078F55048"></a></p>

<h5>6.1-12 UnorderedOrbitalRepresentative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnorderedOrbitalRepresentative</code>( <var class="Arg">os</var>, <var class="Arg">pair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: pair</p>

<p>Given an orbital structure <var class="Arg">os</var> and a pair <var class="Arg">pair</var> of elements of the domain that <var class="Arg">os</var> is defined on, returns a canonical representative of <var class="Arg">pair</var> in its orbit of sets.</p>

<p><a id="X79E001B18591F1AE" name="X79E001B18591F1AE"></a></p>

<h5>6.1-13 AllUnorderedOrbitalRepresentatives</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllUnorderedOrbitalRepresentatives</code>( <var class="Arg">os</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Return the set of canonical representatives of orbits of sets of size two under the action of the orbital structure.</p>

<p><a id="X78ABE8E9844AC143" name="X78ABE8E9844AC143"></a></p>

<h5>6.1-14 UnorderedOrbitalTransversalIterator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnorderedOrbitalTransversalIterator</code>( <var class="Arg">os</var>, <var class="Arg">pair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an iterator</p>

<p>Given an orbital structure <var class="Arg">os</var> and a pair <var class="Arg">pair</var>, returns an iterator that produces an element <code class="code">g</code> for every element <code class="code">e</code> in the orbit such that <code class="code">OnSets(UnorderedOrbitalRepresentative(os, pair), g) = e</code>.</p>

<p><a id="X7FA8E4BE7F8F47DD" name="X7FA8E4BE7F8F47DD"></a></p>

<h5>6.1-15 UnorderedOrbitalCanonizingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnorderedOrbitalCanonizingElement</code>( <var class="Arg">os</var>, <var class="Arg">pair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a group element</p>

<p>Given an orbital structure <var class="Arg">os</var> and the pair <var class="Arg">pair</var> returns an element <span class="SimpleMath">g</span> of the group that maps <var class="Arg">pair</var> to <code class="code">UnorderedOrbitalRepresentative(os, pair)</code>.</p>

<p><a id="X82CAF97C7AE60FA4" name="X82CAF97C7AE60FA4"></a></p>

<h5>6.1-16 UnorderedOrbitalCanonizingElementInverse</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnorderedOrbitalCanonizingElementInverse</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>

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