  
  
                                     [1X RCWA [101X
  
  
                       [1X Residue-Class-Wise Affine Groups [101X
  
  
                                     4.7.1
  
  
                                6 December 2022
  
  
                                  Stefan Kohl
  
  
  
  Stefan Kohl
      Email:    [7Xmailto:sk239@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://stefan-kohl.github.io/[107X
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0Y[5XRCWA[105X  is  a package for [5XGAP[105X 4. It provides implementations of algorithms and
  methods  for  computing in certain infinite permutation groups acting on the
  set of integers. This package can be used to investigate the following types
  of groups and many more:[133X
  
  [30X    [33X[0;6YFinite  groups,  and certain divisible torsion groups which they embed
        into.[133X
  
  [30X    [33X[0;6YFree groups of finite rank.[133X
  
  [30X    [33X[0;6YFree products of finitely many finite groups.[133X
  
  [30X    [33X[0;6YDirect products of the above groups.[133X
  
  [30X    [33X[0;6YWreath products of the above groups with finite groups and with (ℤ,+).[133X
  
  [30X    [33X[0;6YSubgroups of any such groups.[133X
  
  [33X[0;0YWith the help of this package, the author has found a countable simple group
  which  is generated by involutions interchanging disjoint residue classes of
  ℤ and which all the above groups embed into -- see [Koh10].[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2003 - 2018 by Stefan Kohl.[133X
  
  [33X[0;0Y[5XRCWA[105X  is  free  software: you can redistribute it and/or modify it under the
  terms  of  the  GNU General Public License as published by the Free Software
  Foundation,  either  version 2 of the License, or (at your option) any later
  version.[133X
  
  [33X[0;0Y[5XRCWA[105X  is  distributed  in  the  hope that it will be useful, but WITHOUT ANY
  WARRANTY;  without  even  the implied warranty of MERCHANTABILITY or FITNESS
  FOR  A  PARTICULAR  PURPOSE.  See  the  GNU  General Public License for more
  details.[133X
  
  [33X[0;0YFor  a  copy  of the GNU General Public License, see the file [11XGPL[111X in the [11Xetc[111X
  directory       of       the       [5XGAP[105X       distribution       or       see
  [7Xhttps://www.gnu.org/licenses/gpl.html[107X.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YI  am  grateful  to  John  P.  McDermott  for  the  discovery that the group
  discussed  in  Section [14X7.1[114X is isomorphic to Thompson's Group V in July 2008,
  and to Laurent Bartholdi for his hint on how to construct wreath products of
  residue-class-wise  affine groups with (ℤ,+) in April 2006. Further, I thank
  Bettina Eick for communicating this package and for her valuable suggestions
  on  its  manual  in  the time before its first public release in April 2005.
  Last but not least I thank the two anonymous referees for their constructive
  criticism and their helpful suggestions.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (RCWA)[101X
  
  1 [33X[0;0YAbout the RCWA Package[133X
  2 [33X[0;0YResidue-Class-Wise Affine Mappings[133X
    2.1 [33X[0;0YBasic definitions[133X
    2.2 [33X[0;0YEntering residue-class-wise affine mappings[133X
      2.2-1 ClassShift
      2.2-2 ClassReflection
      2.2-3 ClassTransposition
      2.2-4 ClassRotation
      2.2-5 [33X[0;0YRcwaMapping (the general constructor)[133X
      2.2-6 LocalizedRcwaMapping
    2.3 [33X[0;0YBasic arithmetic for residue-class-wise affine mappings[133X
    2.4 [33X[0;0YAttributes and properties of residue-class-wise affine mappings[133X
      2.4-1 LargestSourcesOfAffineMappings
      2.4-2 FixedPointsOfAffinePartialMappings
      2.4-3 Multpk
      2.4-4 Determinant
      2.4-5 Sign
    2.5 [33X[0;0YFactoring residue-class-wise affine permutations[133X
      2.5-1 CTCSCRSplit
      2.5-2 FactorizationIntoCSCRCT
      2.5-3 PrimeSwitch
      2.5-4 mKnot
    2.6 [33X[0;0YExtracting roots of residue-class-wise affine mappings[133X
      2.6-1 Root
    2.7 [33X[0;0YSpecial functions for non-bijective mappings[133X
      2.7-1 RightInverse
      2.7-2 CommonRightInverse
      2.7-3 ImageDensity
    2.8 [33X[0;0YOn trajectories and cycles of residue-class-wise affine mappings[133X
      2.8-1 [33X[0;0YTrajectory (methods for rcwa mappings)[133X
      2.8-2 [33X[0;0YTrajectory (methods for rcwa mappings -- [21Xaccumulated coefficients[121X)[133X
      2.8-3 [33X[0;0YIncreasingOn & DecreasingOn (for an rcwa mapping)[133X
      2.8-4 TransitionGraph
      2.8-5 OrbitsModulo
      2.8-6 FactorizationOnConnectedComponents
      2.8-7 TransitionMatrix
      2.8-8 [33X[0;0YSources & Sinks (of an rcwa mapping)[133X
      2.8-9 Loops
      2.8-10 GluckTaylorInvariant
      2.8-11 LikelyContractionCentre
      2.8-12 GuessedDivergence
    2.9 [33X[0;0YSaving memory -- the sparse representation of rcwa mappings[133X
      2.9-1 SparseRepresentation
    2.10 [33X[0;0YThe categories and families of rcwa mappings[133X
      2.10-1 IsRcwaMapping
      2.10-2 RcwaMappingsFamily
  3 [33X[0;0YResidue-Class-Wise Affine Groups[133X
    3.1 [33X[0;0YConstructing residue-class-wise affine groups[133X
      3.1-1 IsomorphismRcwaGroup
      3.1-2 DirectProduct
      3.1-3 [33X[0;0YWreathProduct (for an rcwa group over Z, with a permutation group
      or (ℤ,+))[133X
      3.1-4 MergerExtension
      3.1-5 GroupByResidueClasses
      3.1-6 [33X[0;0YRestriction (of an rcwa mapping or -group, by an injective rcwa
      mapping)[133X
      3.1-7 [33X[0;0YInduction (of an rcwa mapping or -group, by an injective rcwa
      mapping)[133X
      3.1-8 RCWA
      3.1-9 CT
    3.2 [33X[0;0YBasic routines for investigating residue-class-wise affine groups[133X
      3.2-1 StructureDescription
      3.2-2 EpimorphismFromFpGroup
      3.2-3 PreImagesRepresentative
    3.3 [33X[0;0YThe natural action of an rcwa group on the underlying ring[133X
      3.3-1 [33X[0;0YOrbit (for an rcwa group and either a point or a set)[133X
      3.3-2 GrowthFunctionOfOrbit
      3.3-3 DrawOrbitPicture
      3.3-4 [33X[0;0YShortOrbits (for rcwa groups) & ShortCycles (for rcwa
      permutations)[133X
      3.3-5 [33X[0;0YShortResidueClassOrbits & ShortResidueClassCycles[133X
      3.3-6 ComputeCycleLength
      3.3-7 CycleRepresentativesAndLengths
      3.3-8 FixedResidueClasses
      3.3-9 [33X[0;0YBall (for group, element and radius or group, point, radius and
      action)[133X
      3.3-10 RepresentativeAction
      3.3-11 ProjectionsToInvariantUnionsOfResidueClasses
      3.3-12 RepresentativeAction
      3.3-13 CollatzLikeMappingByOrbitTree
    3.4 [33X[0;0YSpecial attributes of tame residue-class-wise affine groups[133X
      3.4-1 [33X[0;0YRespectedPartition (of a tame rcwa group or -permutation)[133X
      3.4-2 [33X[0;0YActionOnRespectedPartition & KernelOfActionOnRespectedPartition[133X
    3.5 [33X[0;0YGenerating pseudo-random elements of RCWA(R) and CT(R)[133X
    3.6 [33X[0;0YThe categories of residue-class-wise affine groups[133X
      3.6-1 IsRcwaGroup
  4 [33X[0;0YResidue-Class-Wise Affine Monoids[133X
    4.1 [33X[0;0YConstructing residue-class-wise affine monoids[133X
      4.1-1 Rcwa
    4.2 [33X[0;0YComputing with residue-class-wise affine monoids[133X
      4.2-1 ShortOrbits
      4.2-2 [33X[0;0YBall (for monoid, element and radius or monoid, point, radius and
      action)[133X
  5 [33X[0;0YResidue-Class-Wise Affine Mappings, Groups and Monoids over [22Xℤ^2[122X[133X
    5.1 [33X[0;0YThe definition of residue-class-wise affine mappings of [22Xℤ^d[122X[133X
    5.2 [33X[0;0YEntering residue-class-wise affine mappings of [22Xℤ^2[122X[133X
      5.2-1 [33X[0;0YRcwaMapping (the general constructor; methods for [22Xℤ^2[122X)[133X
      5.2-2 [33X[0;0YClassTransposition (for [22Xℤ^2[122X)[133X
      5.2-3 [33X[0;0YClassRotation (for [22Xℤ^2[122X)[133X
      5.2-4 [33X[0;0YClassShift (for [22Xℤ^2[122X)[133X
    5.3 [33X[0;0YMethods for residue-class-wise affine mappings of [22Xℤ^2[122X[133X
      5.3-1 ProjectionsToCoordinates
    5.4 [33X[0;0YMethods for residue-class-wise affine groups and -monoids over [22Xℤ^2[122X[133X
      5.4-1 [33X[0;0YIsomorphismRcwaGroup (Embeddings of SL(2,ℤ) and GL(2,ℤ))[133X
      5.4-2 [33X[0;0YDrawGrid[133X
  6 [33X[0;0YDatabases of Residue-Class-Wise Affine Groups and -Mappings[133X
    6.1 [33X[0;0YThe collection of examples[133X
      6.1-1 LoadRCWAExamples
    6.2 [33X[0;0YDatabases of rcwa groups[133X
      6.2-1 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions
      6.2-2 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions
      6.2-3 LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions
    6.3 [33X[0;0YDatabases of rcwa mappings[133X
      6.3-1 LoadDatabaseOfProductsOf2ClassTranspositions
      6.3-2 LoadDatabaseOfNonbalancedProductsOfClassTranspositions
  7 [33X[0;0YExamples[133X
    7.1 [33X[0;0YThompson's group V[133X
    7.2 [33X[0;0YFactoring Collatz' permutation of the integers[133X
    7.3 [33X[0;0YThe [22X3n+1[122X group[133X
    7.4 [33X[0;0YA group with huge finite orbits[133X
    7.5 [33X[0;0YA group which acts 4-transitively on the positive integers[133X
    7.6 [33X[0;0YA group which acts 3-transitively, but not 4-transitively on ℤ[133X
    7.7 [33X[0;0YAn rcwa mapping which seems to be contracting, but very slow[133X
    7.8 [33X[0;0YChecking a result by P. Andaloro[133X
    7.9 [33X[0;0YTwo examples by Matthews and Leigh[133X
    7.10 [33X[0;0YOrders of commutators[133X
    7.11 [33X[0;0YAn infinite subgroup of CT(GF(2)[x]) with many torsion elements[133X
    7.12 [33X[0;0YAn abelian rcwa group over a polynomial ring[133X
    7.13 [33X[0;0YChecking for solvability[133X
    7.14 [33X[0;0YSome examples over (semi)localizations of the integers[133X
    7.15 [33X[0;0YTwisting 257-cycles into an rcwa mapping with modulus 32[133X
    7.16 [33X[0;0YThe behaviour of the moduli of powers[133X
    7.17 [33X[0;0YImages and preimages under the Collatz mapping[133X
    7.18 [33X[0;0YAn extension of the Collatz mapping T to a permutation of [22Xℤ^2[122X[133X
    7.19 [33X[0;0YFinite quotients of Grigorchuk groups[133X
    7.20 [33X[0;0YForward orbits of a monoid with 2 generators[133X
    7.21 [33X[0;0YThe free group of rank 2 and the modular group PSL(2,ℤ)[133X
  8 [33X[0;0YThe Algorithms Implemented in RCWA[133X
  9 [33X[0;0YInstallation and Auxiliary Functions[133X
    9.1 [33X[0;0YRequirements[133X
    9.2 [33X[0;0YInstallation[133X
    9.3 [33X[0;0YThe testing routines[133X
      9.3-1 RCWATestInstall
      9.3-2 RCWATestAll
      9.3-3 RCWATestExamples
    9.4 [33X[0;0YThe Info class of the package[133X
      9.4-1 InfoRCWA
  
  
  [32X
