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# RCWA

## Residue Class-Wise Affine Groups

Version 2.1.7

September 1, 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

RCWA is a package for GAP 4 which provides methods for investigating Residue Class-Wise Affine groups by means of computation. Residue class-wise affine groups are countable permutation groups acting on the integers, whose elements are bijective residue class-wise affine mappings.

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)

"Many" residue class-wise affine groups act multiply transitively on Z or on subsets thereof. The class of groups which can faithfully be represented as residue class-wise affine groups is closed under forming wreath products with finite groups and with the infinite cyclic group (Z,+). It includes free groups of any rank, all free products of finite groups and certain divisible torsion groups. It also contains finitely generated groups with unsolvable membership problem. The group which is generated by all class transpositions -- these are involutions which interchange two disjoint residue classes -- is a simple group which has subgroups of all mentioned types and whose class of isomorphism types of subgroups is closed under the above operations.

Despite of what has been said in the preceding paragraph, still not very much is known about the structure of residue class-wise affine groups. 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). A copy of this thesis and an english translation thereof are distributed with this package.

### Copyright

(C) 2003 - 2006 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 inviting me to give a talk on the subject in Lausanne in April 2006, and for his hint on how to construct wreath products of residue class-wise affine groups with (Z,+). I would like to thank Otto H. Kegel, Katrin Tent and Oliver Röndigs for their related invitations to Freiburg resp. Bielefeld in February and March 2006.

### Contents

1. Preface
1.1 Motivation
1.2 Groups which can be represented
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 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
2.7 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 Some general utility functions
3.13 The categories of rcwa groups
3.13-1 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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