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<p><a id="X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5"></a></p>
<div class="pcenter">
<h1><strong class="pkg">RCWA</strong></h1>
<h2>Residue-Class-Wise Affine Groups</h2>
<p>
Version 2.5.4</p>
<p>September 26, 2007</p>
</div>
<p><b>
Stefan Kohl
</b>
<br />e-mail: <span class="URL"><a href="mailto:kohl@mathematik.uni-stuttgart.de">kohl@mathematik.uni-stuttgart.de</a></span>
<br />WWW: <span class="URL"><a href=" http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/ "> http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/ </a></span>
<br />Address: <br />Institut für Geometrie und Topologie <br /> Pfaffenwaldring 57 <br /> Universität Stuttgart <br /> 70550 Stuttgart <br /> Germany
</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. In principle, this package can deal at least with the following types of groups and their subgroups:</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, thus in particular the modular group PSL(2,Z).</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>
</ul>
<p>With substancial help of this package, the author has found a countable simple group which has an uncountable series of simple subgroups. This simple group is generated by involutions which interchange disjoint residue classes of the integers. All the above groups embed into it.</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2003 - 2007 by Stefan Kohl. This package is distributed under the GNU General Public License.</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>I am very grateful to Bettina Eick for communicating this package and for her kind help in improving its documentation. Further I would like to thank the two anonymous referees for their constructive criticism and their helpful suggestions.</p>
<p>I am also very grateful to Laurent Bartholdi for his hint on how to construct wreath products of residue-class-wise affine groups with (Z,+). Last but not least I would like to thank all the people who have invited me so far to give talks on the subject in their seminars and on their conferences.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents</h3>
<div class="ContChap"><a href="chap1.html#X83A8C2927FAE2C23">1. <span class="Heading">About the RCWA Package</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X816FA3667FFEDC3F">1.1 <span class="Heading">Motivation</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X78FE3F9D80DB633E">1.2 <span class="Heading">Purpose of this package</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X82190FB67F7F3325">1.3 <span class="Heading">Groups which this package can deal with</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X84BA84177E640F2B">1.4 <span class="Heading">Scope of this package</span></a>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X7FD73FCB8510050E">2. <span class="Heading">Residue-Class-Wise Affine Mappings</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7F0B50947EEA87F8">2.1 <span class="Heading">Basic definitions</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X86BC55648302D643">2.2 <span class="Heading">Entering residue-class-wise affine mappings</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X86B611BD7EED62A1">2.2-1 ClassShift</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7896C5417E3692B4">2.2-2 ClassReflection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8716A75F7DD1C46B">2.2-3 ClassTransposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X87EB8C1C87F78A17">2.2-4 ClassRotation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D87A85584BA7E48">2.2-5 <span class="Heading"> RcwaMapping (the general constructor) </span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F1A559387D0226E">2.2-6 LocalizedRcwaMapping</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X78E796B8824C4FC8">2.3 <span class="Heading">Basic arithmetic for residue-class-wise affine mappings</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X861C5DE6812425F1">2.4 <span class="Heading">
Attributes and properties of residue-class-wise affine mappings
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C21406085B69C30">2.4-1 LargestSourcesOfAffineMappings</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D6D0F2783AD02F4">2.4-2 FixedPointsOfAffinePartialMappings</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A2E308C860B46E3">2.4-3 Multpk</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B1E53127D9AE52F">2.4-4 Determinant</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8365EEEB82C946FD">2.4-5 Sign</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X8475F844869DD060">2.5 <span class="Heading">Factoring residue-class-wise affine permutations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X853885A182EC5104">2.5-1 FactorizationIntoCSCRCT</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X861C74E97AE5DA3B">2.5-2 PrimeSwitch</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X789CB69C7D97B0C4">2.5-3 mKnot</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X8141065381B0942B">2.6 <span class="Heading">
Extracting roots of residue-class-wise affine mappings
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X873692CE78433859">2.6-1 Root</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X8322C6848305EC4C">2.7 <span class="Heading">
Special functions for non-bijective mappings
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7AEFF16E86533633">2.7-1 RightInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X87C5B9CA7E319233">2.7-2 CommonRightInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X808D9EDF7BA27467">2.7-3 ImageDensity</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7A34724386A2E9F3">2.8 <span class="Heading">
On trajectories and cycles of residue-class-wise affine mappings
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.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.html#X7FFD09837E934853">2.8-2 <span class="Heading">
Trajectory (methods for rcwa mappings -- "accumulated coefficients")
</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.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.html#X780841E07CAE7543">2.8-4 TransitionGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F03CC4179424AA9">2.8-5 OrbitsModulo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F11051E866C197F">2.8-6 FactorizationOnConnectedComponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B6833D67D916EF9">2.8-7 TransitionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.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.html#X80221A4D81AF7453">2.8-9 Loops</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8773152E81A30123">2.8-10 GluckTaylorInvariant</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X84F6A29280E2F925">2.8-11 LikelyContractionCentre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81E0D8E3817B3D16">2.8-12 GuessedDivergence</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X83FA71DD842377F0">2.9 <span class="Heading">The categories and families of rcwa mappings</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7927C13782729CE9">2.9-1 IsRcwaMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X825DD365822934AF">2.9-2 RcwaMappingsFamily</a></span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X874A3BB684F0639A">3. <span class="Heading">Residue-Class-Wise Affine Groups</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7D7B075385435151">3.1 <span class="Heading">Constructing residue-class-wise affine groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C0DCF887D324CF9">3.1-1 RCWA</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BD42D8481300E25">3.1-2 CT</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7EB8A301790290C7">3.1-3 IsomorphismRcwaGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79CAE48981C11FE8">3.1-4 DirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7AEFAB7E7F81444B">3.1-5 <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.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.html#X8709A96C8640E4C2">3.1-7 <span class="Heading">
Induction (of an rcwa mapping or -group, by an injective rcwa mapping)
</span></a>
</span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X80C042BE82EE0F9A">3.2 <span class="Heading">
Basic routines for investigating residue-class-wise affine groups
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X864A7E3E87F366A8">3.2-1 StructureDescription</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83527DA37C5CB2C7">3.2-2 EpimorphismFromFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8463E34286344F06">3.2-3 PreImagesRepresentative</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X8151BE577FFDCE87">3.3 <span class="Heading">
The natural action of an rcwa group on the underlying ring
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.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.html#X7B30F7207817D859">3.3-2 DrawOrbitPicture</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78F145197F63A25D">3.3-3 <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.html#X805D691083F2FAFC">3.3-4 <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.html#X87A3462C82FD376E">3.3-5 RepresentativeAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F3CBDD2806ACECF">3.3-6 Projections</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X866843D08213067E">3.3-7 RepresentativeAction</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X781CBEFA7F39B58D">3.4 <span class="Heading">
Special attributes of tame residue-class-wise affine groups
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.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.html#X7A6786727EC1E4BB">3.4-2 <span class="Heading">
ActionOnRespectedPartition & KernelOfActionOnRespectedPartition
</span></a>
</span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X8302BA78810C1DEE">3.5 <span class="Heading">Generating pseudo-random elements of RCWA(R) and CT(R)</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X86327F6C83D09798">3.6 <span class="Heading">The categories of residue-class-wise affine groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84AFBB997B694A3D">3.6-1 IsRcwaGroup</a></span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X81C90F7C7BA25BDF">4. <span class="Heading">Residue-Class-Wise Affine Monoids</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7F95328B7C7E49EA">4.1 <span class="Heading">Constructing residue-class-wise affine monoids</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X788C654D8358E642">4.1-1 Rcwa</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X8759954F7EB1A658">4.2 <span class="Heading">Computing with residue-class-wise affine monoids</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87DB896687475084">4.2-1 ShortOrbits</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.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 class="ContChap"><a href="chap5.html#X7A489A5D79DA9E5C">5. <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X86C2BAE3876985A6">5.1 <span class="Heading">
Factoring Collatz' permutation of the integers
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X878499AF7889FD9E">5.2 <span class="Heading">
An rcwa mapping which seems to be contracting, but very slow
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X84A915BA833E0BDE">5.3 <span class="Heading">Checking a result by P. Andaloro</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7E8CD9B67ED78735">5.4 <span class="Heading">Two examples by Matthews and Leigh</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7A8F55CA87A16900">5.5 <span class="Heading">Exploring the structure of a wild rcwa group</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7D0928AE839F1C49">5.6 <span class="Heading">A wild rcwa mapping which has only finite cycles</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7A8605E680F664BF">5.7 <span class="Heading">An abelian rcwa group over a polynomial ring</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X87383623856ED81B">5.8 <span class="Heading">
A tame group generated by commutators of wild permutations
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X78DFE4B4821E07A6">5.9 <span class="Heading">Checking for solvability</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X783D54DC7A646273">5.10 <span class="Heading">Some examples over (semi)localizations of the integers</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X846D7D087861E0AC">5.11 <span class="Heading">
Twisting 257-cycles into an rcwa mapping with modulus 32
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X78D5DC93845CA6A0">5.12 <span class="Heading"> The behaviour of the moduli of powers </span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X855A3CD88459958B">5.13 <span class="Heading"> Images and preimages under the Collatz mapping </span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7968C1DF7EF0BD8E">5.14 <span class="Heading">
A group which acts 4-transitively on the positive integers
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X87E7FCE27EACDA38">5.15 <span class="Heading">
A group which acts 3-transitively, but not 4-transitively on Z
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X85DD5E9C7878CAE3">5.16 <span class="Heading">
Grigorchuk groups
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7DD9502F80364631">5.17 <span class="Heading">
Forward orbits of a monoid with 2 generators
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X81407BD58648D206">5.18 <span class="Heading">
Representations of the free group of rank 2
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X85B4B4DA83B09E81">5.19 <span class="Heading">
Representations of the modular group PSL(2,Z)
</span></a>
</div>
</div>
<div class="ContChap"><a href="chap6.html#X79EA0B717B045756">6. <span class="Heading">The Algorithms Implemented in RCWA</span></a>
</div>
<div class="ContChap"><a href="chap7.html#X859F6BF88754E5CC">7. <span class="Heading">Installation and auxiliary functions</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X85A08CF187A6D986">7.1 <span class="Heading">Requirements</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X8360C04082558A12">7.2 <span class="Heading">Installation</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X7A31FA44791E93C5">7.3 <span class="Heading">The Info class of the package</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7BAF5F4986288983">7.3-1 InfoRCWA</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X8667D5027AC3DE8E">7.4 <span class="Heading">The testing routine</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7A51ED5F839759C0">7.4-1 RCWATest</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X854E65D281B80D3B">7.5 <span class="Heading">Building the manual</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7AA556D17F61A44C">7.5-1 RCWABuildManual</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X7C7AB20486E56B83">7.6 <span class="Heading">Loading and saving bitmap pictures</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X81DE1E838615C214">7.6-1 SaveAsBitmapPicture</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X85E215DB85F610ED">7.7 <span class="Heading">Running demonstrations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X850524E47E3DB78D">7.7-1 RunDemonstration</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X8677C67F7AD4C9C7">7.8 <span class="Heading">Some general utility functions</span></a>
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