  
  
                                      [1m[4m[31m[1mRCWA[1m[4m[31m[0m
  
  
                        [1m[4m[31mResidue Class-Wise Affine Groups[0m
  
  
                                 Version 2.1.7
  
  
                               September 1, 2006
  
  
                                  Stefan Kohl
  
  
  
  Stefan Kohl
      Email:    [34mmailto:kohl@mathematik.uni-stuttgart.de[0m
      Homepage: [34mhttp://www.cip.mathematik.uni-stuttgart.de/~kohlsn[0m
      Address:  Institut für Geometrie und Topologie
                Universität Stuttgart
                70550 Stuttgart
                Germany
  
  
  
  -------------------------------------------------------
  [1m[4m[31mAbstract[0m
  [1mRCWA[0m is a package for [1mGAP[0m 4 which provides methods for investigating [22m[36mR[0mesidue
  [22m[36mC[0mlass-[22m[36mW[0mise  [22m[36mA[0mffine 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 [22m[36mresidue class-wise affine[0m 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 [22m[36mclass
  transpositions[0m  --  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
  [22m[36mRestklassenweise    affine    Gruppen[0m.   This   thesis   is   published   at
  [34mhttp://deposit.ddb.de/cgi-bin/dokserv?idn=977164071[0m  (Archivserver  Deutsche
  Bibliothek).  A  copy  of this thesis and an english translation thereof are
  distributed with this package.
  
  
  -------------------------------------------------------
  [1m[4m[31mCopyright[0m
  (C)  2003  -  2006 by Stefan Kohl. This package is distributed under the GNU
  General Public License.
  
  
  -------------------------------------------------------
  [1m[4m[31mAcknowledgements[0m
  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.
  
  
  -------------------------------------------------------
  
  
  [1m[4m[31mContent (RCWA)[0m
  
  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
  
  
  -------------------------------------------------------
