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<h1>RCWA</h1>


<h2>Residue-Class-Wise Affine Groups</h2>

<p>
    4.8.0</p>

<p>
    22 September 2025
  </p>

</div>
<p><b>
    Stefan Kohl



  </b>
<br />Email: <span class="URL"><a href="mailto:sk239@st-andrews.ac.uk">sk239@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://stefan-kohl.github.io/">https://stefan-kohl.github.io/</a></span>
</p>

<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p><strong class="pkg">RCWA</strong> is a package for <strong class="pkg">GAP</strong> 4. It provides implementations of algorithms and methods for computing in certain infinite permutation groups acting on the set of integers. This package can be used to investigate the following types of groups and many more:</p>


<ul>
<li><p>Finite groups, and certain divisible torsion groups which they embed into.</p>

</li>
<li><p>Free groups of finite rank.</p>

</li>
<li><p>Free products of finitely many finite groups.</p>

</li>
<li><p>Direct products of the above groups.</p>

</li>
<li><p>Wreath products of the above groups with finite groups and with (ℤ,+).</p>

</li>
<li><p>Subgroups of any such groups.</p>

</li>
</ul>
<p>With the help of this package, the author has found a countable simple group which is generated by involutions interchanging disjoint residue classes of ℤ and which all the above groups embed into -- see <a href="chapBib_mj.html#biBKohl09">[Koh10]</a>.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2003 - 2018 by Stefan Kohl.</p>

<p><strong class="pkg">RCWA</strong> is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the License, or (at your option) any later version.</p>

<p><strong class="pkg">RCWA</strong> is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.</p>

<p>For a copy of the GNU General Public License, see the file <code class="file">GPL</code> in the <code class="file">etc</code> directory of the <strong class="pkg">GAP</strong> distribution or see <span class="URL"><a href="https://www.gnu.org/licenses/gpl.html">https://www.gnu.org/licenses/gpl.html</a></span>.</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>I am grateful to John P. McDermott for the discovery that the group discussed in Section <a href="chap7_mj.html#X84A058CF7C65A908"><span class="RefLink">7.1</span></a> is isomorphic to Thompson's Group V in July 2008, and to Laurent Bartholdi for his hint on how to construct wreath products of residue-class-wise affine groups with (ℤ,+) in April 2006. Further, I thank Bettina Eick for communicating this package and for her valuable suggestions on its manual in the time before its first public release in April 2005. Last but not least I thank the two anonymous referees for their constructive criticism and their helpful suggestions.



<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X83A8C2927FAE2C23">1 <span class="Heading">About the RCWA Package</span></a>
</div>
<div class="ContChap"><a href="chap2_mj.html#X7FD73FCB8510050E">2 <span class="Heading">Residue-Class-Wise Affine Mappings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X78ED07E37FC2BD46">2.1 <span class="Heading">Basic definitions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X86BC55648302D643">2.2 <span class="Heading">Entering residue-class-wise affine mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X86B611BD7EED62A1">2.2-1 ClassShift</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7896C5417E3692B4">2.2-2 ClassReflection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8716A75F7DD1C46B">2.2-3 ClassTransposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X87EB8C1C87F78A17">2.2-4 ClassRotation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8799551B83644B37">2.2-5 <span class="Heading"> RcwaMapping (the general constructor) </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7F1A559387D0226E">2.2-6 LocalizedRcwaMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X78E796B8824C4FC8">2.3 <span class="Heading">Basic arithmetic for residue-class-wise affine mappings</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7C16D22C7BD40FDC">2.4 <span class="Heading">
  Attributes and properties of residue-class-wise affine mappings
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7C21406085B69C30">2.4-1 LargestSourcesOfAffineMappings</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7D6D0F2783AD02F4">2.4-2 FixedPointsOfAffinePartialMappings</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7A2E308C860B46E3">2.4-3 Multpk</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7B1E53127D9AE52F">2.4-4 Determinant</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8365EEEB82C946FD">2.4-5 Sign</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8475F844869DD060">2.5 <span class="Heading">Factoring residue-class-wise affine permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X829BA0537F2372FF">2.5-1 CTCSCRSplit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X853885A182EC5104">2.5-2 FactorizationIntoCSCRCT</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X861C74E97AE5DA3B">2.5-3 PrimeSwitch</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X789CB69C7D97B0C4">2.5-4 mKnot</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8141065381B0942B">2.6 <span class="Heading">
  Extracting roots of residue-class-wise affine mappings
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X873692CE78433859">2.6-1 Root</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8322C6848305EC4C">2.7 <span class="Heading">
  Special functions for non-bijective mappings
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7AEFF16E86533633">2.7-1 RightInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X87C5B9CA7E319233">2.7-2 CommonRightInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X808D9EDF7BA27467">2.7-3 ImageDensity</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7A34724386A2E9F3">2.8 <span class="Heading">
  On trajectories and cycles of residue-class-wise affine mappings
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7C72174D7CCB6348">2.8-1 <span class="Heading"> Trajectory (methods for rcwa mappings) </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7FFD09837E934853">2.8-2 <span class="Heading">
    Trajectory (methods for rcwa mappings -- <q>accumulated coefficients</q>)
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7E0244A386744185">2.8-3 <span class="Heading"> IncreasingOn & DecreasingOn (for an rcwa mapping) </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X780841E07CAE7543">2.8-4 TransitionGraph</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7F03CC4179424AA9">2.8-5 OrbitsModulo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7F11051E866C197F">2.8-6 FactorizationOnConnectedComponents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7B6833D67D916EF9">2.8-7 TransitionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X81DBA2D58526BE7E">2.8-8 <span class="Heading"> Sources & Sinks (of an rcwa mapping) </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X80221A4D81AF7453">2.8-9 Loops</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8773152E81A30123">2.8-10 GluckTaylorInvariant</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X84F6A29280E2F925">2.8-11 LikelyContractionCentre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X81E0D8E3817B3D16">2.8-12 GuessedDivergence</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X86F0E0D17E6A9663">2.9 <span class="Heading">
  Saving memory -- the sparse representation of rcwa mappings
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X879451B17AD78B07">2.9-1 SparseRepresentation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X83FA71DD842377F0">2.10 <span class="Heading">The categories and families of rcwa mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7927C13782729CE9">2.10-1 IsRcwaMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X825DD365822934AF">2.10-2 RcwaMappingsFamily</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3_mj.html#X874A3BB684F0639A">3 <span class="Heading">Residue-Class-Wise Affine Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X81242A6586A604A3">3.1 <span class="Heading">Constructing residue-class-wise affine groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7EB8A301790290C7">3.1-1 IsomorphismRcwaGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79CAE48981C11FE8">3.1-2 DirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80D13D2A7AD73C2C">3.1-3 <span class="Heading">
    WreathProduct
    (for an rcwa group over Z, with a permutation group or (ℤ,+))
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8794913B878DD5C4">3.1-4 MergerExtension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8143AB647801F438">3.1-5 GroupByResidueClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X852EF2C079E4D7FF">3.1-6 <span class="Heading">
    Restriction (of an rcwa mapping or -group, by an injective rcwa mapping)
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X82171D7287CBED95">3.1-7 <span class="Heading">
    Induction (of an rcwa mapping or -group, by an injective rcwa mapping)
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79450C1C8756FEB3">3.1-8 RCWA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7BD42D8481300E25">3.1-9 CT</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X80C042BE82EE0F9A">3.2 <span class="Heading">
  Basic routines for investigating residue-class-wise affine groups
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X864A7E3E87F366A8">3.2-1 StructureDescription</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83527DA37C5CB2C7">3.2-2 EpimorphismFromFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8463E34286344F06">3.2-3 PreImagesRepresentative</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8151BE577FFDCE87">3.3 <span class="Heading">
  The natural action of an rcwa group on the underlying ring
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C046BE97EE53692">3.3-1 <span class="Heading"> Orbit (for an rcwa group and either a point or a set) </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B7A3AF97D195E33">3.3-2 GrowthFunctionOfOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D9DFAC97F9F0891">3.3-3 DrawOrbitPicture</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78F145197F63A25D">3.3-4 <span class="Heading">
    ShortOrbits (for rcwa groups) & ShortCycles (for rcwa permutations)
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80D18D0778A96C16">3.3-5 <span class="Heading">
    ShortResidueClassOrbits & ShortResidueClassCycles
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80C080287A355EFF">3.3-6 ComputeCycleLength</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F76B04E86C77B94">3.3-7 CycleRepresentativesAndLengths</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8777A62286597D53">3.3-8 FixedResidueClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8735855587CC029F">3.3-9 <span class="Heading">
    Ball (for group, element and radius or group, point, radius and action)
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87A3462C82FD376E">3.3-10 RepresentativeAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8587246A7F890849">3.3-11 ProjectionsToInvariantUnionsOfResidueClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X866843D08213067E">3.3-12 RepresentativeAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X82DBAF35788FA239">3.3-13 CollatzLikeMappingByOrbitTree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X781CBEFA7F39B58D">3.4 <span class="Heading">
  Special attributes of tame residue-class-wise affine groups
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F523A6B87825AB8">3.4-1 <span class="Heading">
    RespectedPartition (of a tame rcwa group or -permutation)
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X831ADC1584DE6113">3.4-2 <span class="Heading">
    ActionOnRespectedPartition & KernelOfActionOnRespectedPartition
  </span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X81941A247942FB99">3.5 <span class="Heading">Generating pseudo-random elements of RCWA(R) and CT(R)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X86327F6C83D09798">3.6 <span class="Heading">The categories of residue-class-wise affine groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84AFBB997B694A3D">3.6-1 IsRcwaGroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4_mj.html#X81C90F7C7BA25BDF">4 <span class="Heading">Residue-Class-Wise Affine Monoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X83D42E26849D5580">4.1 <span class="Heading">Constructing residue-class-wise affine monoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B95FCA279E0D6CC">4.1-1 Rcwa</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8759954F7EB1A658">4.2 <span class="Heading">Computing with residue-class-wise affine monoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X87DB896687475084">4.2-1 ShortOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X787848137DF1C245">4.2-2 <span class="Heading">
    Ball (for monoid, element and radius or monoid, point, radius and action)
  </span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap5_mj.html#X788EB00B82897762">5 <span class="Heading">
  Residue-Class-Wise Affine Mappings, Groups and Monoids over <span class="SimpleMath">\(ℤ^2\)</span>
</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X781907CA785CC7AC">5.1 <span class="Heading">
  The definition of residue-class-wise affine mappings of <span class="SimpleMath">\(ℤ^d\)</span>
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7A39FCF08030AB9B">5.2 <span class="Heading">
  Entering residue-class-wise affine mappings of <span class="SimpleMath">\(ℤ^2\)</span>
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X790649618012C606">5.2-1 <span class="Heading">
    RcwaMapping (the general constructor; methods for <span class="SimpleMath">\(ℤ^2\)</span>)
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B450EE17B465E02">5.2-2 <span class="Heading"> ClassTransposition (for <span class="SimpleMath">\(ℤ^2\)</span>) </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X828438127DDAEBB4">5.2-3 <span class="Heading"> ClassRotation (for <span class="SimpleMath">\(ℤ^2\)</span>) </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A14A8F48247E651">5.2-4 <span class="Heading"> ClassShift (for <span class="SimpleMath">\(ℤ^2\)</span>) </span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8531E39785FFF8A7">5.3 <span class="Heading">
  Methods for residue-class-wise affine mappings of <span class="SimpleMath">\(ℤ^2\)</span>
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8408B7837C9EED36">5.3-1 ProjectionsToCoordinates</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X83A1752F7BE9CE85">5.4 <span class="Heading">
  Methods for residue-class-wise affine groups and -monoids over <span class="SimpleMath">\(ℤ^2\)</span>
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X79A8F9AD7E839862">5.4-1 <span class="Heading">
    IsomorphismRcwaGroup (Embeddings of SL(2,ℤ) and GL(2,ℤ))
  </span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X812135EB87527F01">5.4-2 <span class="Heading"> DrawGrid </span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap6_mj.html#X81BA344979567342">6 <span class="Heading">
  Databases of Residue-Class-Wise Affine Groups and -Mappings
</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X86CCBF017A746F50">6.1 <span class="Heading">The collection of examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X8714254784AFD64B">6.1-1 LoadRCWAExamples</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X85DD85DF87DE47C9">6.2 <span class="Heading">Databases of rcwa groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X793E2C5C7FC935B8">6.2-1 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7A77F7D57B08E4A5">6.2-2 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X792C90B48692D0D7">6.2-3 LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X78A1A8E587C7FFD5">6.3 <span class="Heading">Databases of rcwa mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X843E94467A1BB86C">6.3-1 LoadDatabaseOfProductsOf2ClassTranspositions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X85B492697ACC4A54">6.3-2 LoadDatabaseOfNonbalancedProductsOfClassTranspositions</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap7_mj.html#X7A489A5D79DA9E5C">7 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X84A058CF7C65A908">7.1 <span class="Heading">
  Thompson's group V
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X86C2BAE3876985A6">7.2 <span class="Heading">
  Factoring Collatz' permutation of the integers
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X811919107D5DAAC1">7.3 <span class="Heading">
  The <span class="SimpleMath">\(3n+1\)</span> group
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7DCFDC797FF213C5">7.4 <span class="Heading">
  A group with huge finite orbits
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7968C1DF7EF0BD8E">7.5 <span class="Heading">
  A group which acts 4-transitively on the positive integers
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X85C529088050BEA3">7.6 <span class="Heading">
  A group which acts 3-transitively, but not 4-transitively on ℤ
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X878499AF7889FD9E">7.7 <span class="Heading">
  An rcwa mapping which seems to be contracting, but very slow
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X84A915BA833E0BDE">7.8 <span class="Heading">Checking a result by P. Andaloro</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7E8CD9B67ED78735">7.9 <span class="Heading">Two examples by Matthews and Leigh</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X854E9F65817E4F63">7.10 <span class="Heading">Orders of commutators</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7F085B867D799293">7.11 <span class="Heading">
  An infinite subgroup of CT(GF(2)[x]) with many torsion elements
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7A8605E680F664BF">7.12 <span class="Heading">An abelian rcwa group over a polynomial ring</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X78DFE4B4821E07A6">7.13 <span class="Heading">Checking for solvability</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X783D54DC7A646273">7.14 <span class="Heading">Some examples over (semi)localizations of the integers</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X846D7D087861E0AC">7.15 <span class="Heading">
  Twisting 257-cycles into an rcwa mapping with modulus 32
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X78D5DC93845CA6A0">7.16 <span class="Heading"> The behaviour of the moduli of powers </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X855A3CD88459958B">7.17 <span class="Heading"> Images and preimages under the Collatz mapping </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X84B6A498838A5509">7.18 <span class="Heading">
  An extension of the Collatz mapping T to a permutation of <span class="SimpleMath">\(ℤ^2\)</span>
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X81EB8D397898C6B2">7.19 <span class="Heading">
  Finite quotients of Grigorchuk groups
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7DD9502F80364631">7.20 <span class="Heading">
  Forward orbits of a monoid with 2 generators
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X815800ED820C6ECF">7.21 <span class="Heading">
  The free group of rank 2 and the modular group PSL(2,ℤ)
</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap8_mj.html#X79EA0B717B045756">8 <span class="Heading">The Algorithms Implemented in RCWA</span></a>
</div>
<div class="ContChap"><a href="chap9_mj.html#X859F6BF88754E5CC">9 <span class="Heading">Installation and Auxiliary Functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X85A08CF187A6D986">9.1 <span class="Heading">Requirements</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X8360C04082558A12">9.2 <span class="Heading">Installation</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X865D6A49826B92EC">9.3 <span class="Heading">The testing routines</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X8314E1597BF1555B">9.3-1 RCWATestInstall</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X877DDD787E4ABDC2">9.3-2 RCWATestAll</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X799793987AA3F34C">9.3-3 RCWATestExamples</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7A31FA44791E93C5">9.4 <span class="Heading">The Info class of the package</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X7BAF5F4986288983">9.4-1 InfoRCWA</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
<br />
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