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# RCWA

## Residue-Class-Wise Affine Groups

Version 2.3.3

January 26, 2007

Stefan Kohl
e-mail: kohl@mathematik.uni-stuttgart.de
WWW: http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/
Address:
Institut für Geometrie und Topologie
Pfaffenwaldring 57
Universität Stuttgart
70550 Stuttgart
Germany

### Abstract

RCWA is a package for GAP 4. It introduces a new class of groups which are accessible to computational methods. In principle, this package can deal at least with the following types of groups and their subgroups:

- Finite groups, and certain divisible torsion groups which they embed into.

- Free groups of finite rank.

- Free products of finitely many finite groups, thus in particular the modular group PSL(2,Z).

- Direct products of the above groups.

- Wreath products of the above groups with finite groups and with (Z,+).

With substancial help of this package, the author has found a countable simple group which is generated by involutions interchanging residue classes of the integers and into which all the above groups embed. This simple group has an uncountable series of simple subgroups, which is parametrized by the sets of odd primes.

### Copyright

(C) 2003 - 2007 by Stefan Kohl. This package is distributed under the GNU General Public License.

### Acknowledgements

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. 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.

### Contents

1. About the RCWA Package
1.1 Motivation
1.2 Groups which can be dealt with
1.3 Purpose of this package
1.4 Scope of this package
2. Residue-Class-Wise Affine Mappings
2.1 Basic definitions
2.2 Entering residue-class-wise affine mappings
2.2-1 ClassShift
2.2-2 ClassReflection
2.2-3 ClassTransposition
2.2-4 PrimeSwitch
2.2-5 RcwaMapping
2.2-6 LaTeXObj
2.3 Basic functionality for residue-class-wise affine mappings
2.4 Factoring residue-class-wise affine permutations
2.4-1 FactorizationIntoCSCRCT
2.4-2 mKnot
2.5 Determinant and sign
2.5-1 Determinant
2.5-2 Sign
2.6 Attributes and properties derived from the coefficients
2.7 Functionality related to the affine partial mappings
2.7-1 LargestSourcesOfAffineMappings
2.7-2 Multpk
2.7-3 FixedPointsOfAffinePartialMappings
2.8 Transition graphs and transition matrices
2.8-1 TransitionGraph
2.8-2 OrbitsModulo
2.8-3 FactorizationOnConnectedComponents
2.8-4 TransitionMatrix
2.8-5 Sources
2.8-6 Sinks
2.8-7 Loops
2.9 Trajectories
2.9-1 Trajectory
2.9-2 Trajectory
2.9-3 IncreasingOn
2.9-4 GluckTaylorInvariant
2.10 Localizations of rcwa mappings of the integers
2.10-1 LocalizedRcwaMapping
2.11 Extracting roots of rcwa mappings
2.11-1 Root
2.12 Special functions for non-bijective mappings
2.12-1 RightInverse
2.12-2 CommonRightInverse
2.12-3 ImageDensity
2.13 Probabilistic guesses on the behaviour of trajectories
2.13-1 LikelyContractionCentre
2.13-2 GuessedDivergence
2.14 The categories and families of rcwa mappings
2.14-1 IsRcwaMapping
2.14-2 RcwaMappingsFamily
3. Residue-Class-Wise Affine Groups
3.1 Constructing residue-class-wise affine groups
3.1-1 RCWA
3.1-2 IsomorphismRcwaGroupOverZ
3.1-3 StructureDescription
3.2 Direct products and wreath products
3.2-1 DirectProduct
3.2-2 WreathProduct
3.3 The membership test
3.4 Basic attributes and properties of rcwa groups
3.5 Permutation- and matrix representations
3.5-1 IsomorphismPermGroup
3.5-2 IsomorphismMatrixGroup
3.6 Factoring elements into generators
3.6-1 PreImagesRepresentative
3.6-2 PreImagesRepresentatives
3.7 The action of an rcwa group on the underlying ring
3.7-1 IsTransitive
3.7-2 RepresentativeAction
3.7-3 RepresentativeActionPreImage
3.7-4 RepresentativeAction
3.7-5 ShortOrbits
3.7-6 Projections
3.7-7 Ball
3.8 Conjugacy in RCWA(Z)
3.8-1 IsConjugate
3.8-2 RepresentativeAction
3.8-3 NrConjugacyClassesOfRCWAZOfOrder
3.9 Restriction and induction
3.9-1 Restriction
3.9-2 Induction
3.10 Getting pseudo-random elements of RCWA(Z)
3.11 Special attributes for tame rcwa groups
3.11-1 RespectedPartition
3.11-2 ActionOnRespectedPartition
3.11-3 KernelOfActionOnRespectedPartition
3.11-4 IntegralConjugate
3.12 Drawing pictures of orbits on Z^2
3.12-1 DrawOrbitPicture
3.12-2 SaveAsBitmapPicture
3.13 Some general utility functions
3.14 The categories of rcwa groups
3.14-1 IsRcwaGroup
4. Examples
4.1 Factoring Collatz' permutation of the integers
4.2 An rcwa mapping which seems to be contracting, but very slow
4.3 Checking a result by P. Andaloro
4.4 Two examples by Matthews and Leigh
4.5 Exploring the structure of a wild rcwa group
4.6 A wild rcwa mapping which has only finite cycles
4.7 An abelian rcwa group over a polynomial ring
4.8 A tame group generated by commutators of wild permutations
4.9 Checking for solvability
4.10 Some examples over (semi)localizations of the integers
4.11 Twisting 257-cycles into an rcwa mapping with modulus 32
4.12 The behaviour of the moduli of powers
4.13 Images and preimages under the Collatz mapping
4.14 A group which acts 4-transitively on the positive integers
4.15 A group which acts 3-transitively, but not 4-transitively on Z
4.16 Grigorchuk groups
4.17 Forward orbits of a monoid with 2 generators
4.18 Representations of the free group of rank 2
4.19 Representations of the modular group PSL(2,Z)
5. The Algorithms Implemented in RCWA
6. Installation and auxiliary functions
6.1 Requirements
6.2 Installation
6.3 The Info class of the package
6.3-1 InfoRCWA
6.4 The testing routine
6.4-1 RCWATest
6.5 Building the manual
6.5-1 RCWABuildManual

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