  
  
                                      [1m[4m[31m[1mRCWA[1m[4m[31m[0m
  
  
                        [1m[4m[31mResidue Class-Wise Affine Groups[0m
  
  
                                 Version 2.2.1
  
  
                               September 29, 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
                Pfaffenwaldring 57
                Universität Stuttgart
                70550 Stuttgart
                Germany
  
  
  
  -------------------------------------------------------
  [1m[4m[31mAbstract[0m
  [1mRCWA[0m  is  a  package  for  [1mGAP[0m  4, which provides methods for computing with
  [22m[36mR[0mesidue  [22m[36mC[0mlass-[22m[36mW[0mise  [22m[36mA[0mffine  groups.  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.
  
  
  -------------------------------------------------------
  [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. 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 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
  
  
  -------------------------------------------------------
