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# RCWA

## Residue Class-Wise Affine Groups

( Version 1.7.2 )

May 3, 2006

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

### Abstract

The RCWA package provides methods for investigating Residue Class-Wise Affine groups by means of computation. Residue class-wise affine groups are permutation groups acting on the integers, whose elements are bijective residue class-wise affine mappings. Typically they are infinite.

A mapping f: Z -> Z is called residue class-wise affine provided that there is a positive integer m such that the restrictions of f to the residue classes (mod m) are all affine. This means that for any residue class r(m) in Z/mZ there are coefficients a_r(m), b_r(m), c_r(m) in Z such that the restriction of the mapping f to the set r(m) = {r + km | k in Z} is given by

                                            a_r(m) * n + b_r(m)
               f|_r(m):  r(m) -> Z,  n |->  -------------------.
                                                  c_r(m)

Residue class-wise affine groups are countable. "Many" of them act multiply transitively on Z or on subsets thereof. Only relatively basic facts about their structure are known so far. This package is intended to serve as a tool for obtaining a better understanding of their rich and interesting group theoretical and combinatorial structure.

Residue class-wise affine groups can be generalized in a natural way to euclidean rings other than the ring of integers. While this package undoubtedly provides most functionality for residue class-wise affine groups over the integers, at least rudimentarily it also covers the cases that the underlying ring is a semilocalization of Z or a polynomial ring in one variable over a finite field.

The original motivation for investigating residue class-wise affine groups comes from the famous 3n+1 Conjecture, which is an assertion about a surjective, but not injective residue class-wise affine mapping.

Residue class-wise affine groups are introduced in the author's thesis Restklassenweise affine Gruppen. This thesis is published at http://deposit.ddb.de/cgi-bin/dokserv?idn=977164071 (Archivserver Deutsche Bibliothek) and at http://elib.uni-stuttgart.de/opus/volltexte/2005/2448/ (OPUS-Datenbank Universität Stuttgart). A copy of this thesis and an english translation thereof are distributed with this package (see `thesis/thesis.pdf` resp. `thesis/thesis_e.pdf`).

### Copyright

(C) 2003 - 2006 by Stefan Kohl

This package is distributed under the GNU General Public License.

### Contents

1. Preface
1.1 Motivation
1.2 Purpose of this package
1.3 Scope of this package
1.4 Acknowledgements
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 rcwa mappings
2.4 Factoring rcwa mappings
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
2.6-1 Multiplier
2.6-2 Divisor
2.6-3 PrimeSet
2.6-4 IsIntegral
2.6-5 IsClassWiseOrderPreserving
2.7 Functionality related to the affine partial mappings
2.7-1 LargestSourcesOfAffineMappings
2.7-2 Multpk
2.7-3 SetOnWhichMappingIsClassWiseOrderPreserving
2.7-4 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
2.10 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 Random
3.1-3 IsomorphismRcwaGroupOverZ
3.2 Attributes and properties of rcwa groups
3.2-1 Modulus
3.2-2 IsTame
3.2-3 PrimeSet
3.3 Membership testing, order computation, permutation- / matrix representations
3.3-1 \in
3.3-2 Size
3.3-3 IsomorphismPermGroup
3.3-4 IsomorphismMatrixGroup
3.4 Factoring elements into generators
3.4-1 PreImagesRepresentative
3.4-2 PreImagesRepresentatives
3.5 The action of an rcwa group on the underlying ring R
3.5-1 IsTransitive
3.5-2 RepresentativeAction
3.5-3 RepresentativeActionPreImage
3.5-4 RepresentativeAction
3.5-5 ShortOrbits
3.5-6 OrbitsModulo
3.5-7 Ball
3.6 Conjugacy in RCWA(R)
3.6-1 IsConjugate
3.6-2 RepresentativeAction
3.6-3 ShortCycles
3.6-4 NrConjugacyClassesOfRCWAZOfOrder
3.7 Restriction and induction
3.7-1 Restriction
3.7-2 Restriction
3.7-3 Induction
3.7-4 DirectProduct
3.7-5 WreathProduct
3.8 Special attributes for tame rcwa groups
3.8-1 RespectedPartition
3.8-2 ActionOnRespectedPartition
3.8-3 IntegralConjugate
3.8-4 IntegralizingConjugator
3.9 The categories of rcwa groups
3.9-1 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 An rcwa representation of a small group
4.9 An rcwa representation of the symmetric group on 10 points
4.10 Checking for solvability
4.11 Some examples over (semi)localizations of the integers
4.12 Twisting 257-cycles into an rcwa mapping with modulus 32
4.13 The behaviour of the moduli of powers
4.14 Images and preimages under the Collatz mapping
4.15 A group which acts 4-transitively on the positive integers
4.16 A group which acts 3-transitively, but not 4-transitively on Z
4.17 Grigorchuk groups
4.18 Forward orbits of a monoid with 2 generators
4.19 Representations of the free group of rank 2
4.20 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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