Goto Chapter:

RCWA

Residue Class-Wise Affine Representations of Groups

( Version 1.0 )

April 24, 2002

Stefan Kohl
e-mail: kohl@mathematik.uni-stuttgart.de
WWW: http://www.cip.mathematik.uni-stuttgart.de/~kohlsn

Abstract

This package for GAP 4 (at least version 4.3) provides routines for computations with Residue Class-Wise Affine mappings of Z, its semilocalizations Z_pi and the polynomial rings GF( q )[ x ], hence it can be used for computations in certain types of infinite permutation groups. It is completely written in the GAP language and contains / requires no external binaries. For the documentation, the package GAPDoc [LN02] is needed. The package GRAPE[S02] is loaded for use by the function RcwaGraph, if present. RCWA must be installed in the pkg subdirectory of the GAP distribution.

Copyright

(C) 2002 by Stefan Kohl

After an official release, we adopt the copyright regulations of GAP as detailed in the copyright notice in the GAP manual.

Contents

1. Preface
2. Introduction
   2.1 Definitions
3. Semilocalizations of the integers
   3.1 Computing with semilocalizations of the integers
      3.1-1 Z_pi
      3.1-2 IsZ_pi
      3.1-3 NoninvertiblePrimes
      3.1-4 \in
      3.1-5 Intersection
      3.1-6 IsSubset
      3.1-7 StandardAssociate
      3.1-8 GcdOp
      3.1-9 LcmOp
      3.1-10 Factors
      3.1-11 IsUnit
4. Residue Class-Wise Affine Mappings
   4.1 The categories and families of rcwa mappings
      4.1-1 IsRcwaMapping
      4.1-2 IsRationalBasedRcwaMapping
      4.1-3 IsIntegralRcwaMapping
      4.1-4 IntegralRcwaMappingsFamily
      4.1-5 IsSemilocalIntegralRcwaMapping
      4.1-6 SemilocalIntegralRcwaMappingsFamily
      4.1-7 IsModularRcwaMapping
      4.1-8 ModularRcwaMappingsFamily
   4.2 Constructing rcwa mappings
      4.2-1 IntegralRcwaMapping
      4.2-2 SemilocalIntegralRcwaMapping
      4.2-3 ModularRcwaMapping
      4.2-4 AllGFqPolynomialsModDegree
   4.3 Extracting the components of rcwa mappings
      4.3-1 Coefficients
      4.3-2 Modulus
   4.4 Flat and order-preserving mappings, multiplier and divisor
      4.4-1 Multiplier
      4.4-2 Divisor
      4.4-3 IsFlat
      4.4-4 IsClassWiseOrderPreserving
   4.5 Checking for equality
      4.5-1 \=
   4.6 Printing and displaying rcwa mappings
      4.6-1 Print
      4.6-2 String
      4.6-3 Display
   4.7 Images and preimages under rcwa mappings
      4.7-1 ImageElm
      4.7-2 PreImageElm
      4.7-3 PreImagesElm
      4.7-4 PreImagesRepresentative
   4.8 Testing for injectivity, surjectivity and bijectivity
      4.8-1 IsInjective
   4.9 Arithmetical operations and neutral elements
      4.9-1 \+
      4.9-2 \*
      4.9-3 Inverse
      4.9-4 \^
      4.9-5 ZeroIntegralRcwaMapping
      4.9-6 IdentityIntegralRcwaMapping
   4.10 Computing the order of an rcwa mapping
      4.10-1 Order
      4.10-2 IsTame
   4.11 Graph, transition matrix and prime set
      4.11-1 RcwaGraph
      4.11-2 TransitionMatrix
      4.11-3 PrimeSet
   4.12 The normal form of an rcwa mapping
      4.12-1 ShortCycles
      4.12-2 CycleType
      4.12-3 StandardConjugate
      4.12-4 StandardizingConjugator
      4.12-5 IsConjugate
5. Residue Class-Wise Affine Groups
   5.1 The categories of rcwa groups
      5.1-1 IsRcwaGroup
      5.1-2 IsRationalBasedRcwaGroup
      5.1-3 IsIntegralRcwaGroup
      5.1-4 IntegralRcwaGroupsFamily
      5.1-5 TrivialIntegralRcwaGroup
      5.1-6 IsSemilocalIntegralRcwaGroup
      5.1-7 IsModularRcwaGroup
      5.1-8 RCWA
   5.2 Constructing rcwa groups
      5.2-1 IntegralRcwaGroupByPermGroup
      5.2-2 IsomorphismIntegralRcwaGroup
      5.2-3 Display
   5.3 Computing with rcwa groups
      5.3-1 \in
      5.3-2 Size
      5.3-3 IsomorphismPermGroup
      5.3-4 NiceMonomorphism
      5.3-5 NiceObject
      5.3-6 Modulus
      5.3-7 PrimeSet
      5.3-8 IsFlat
      5.3-9 IsClassWiseOrderPreserving
      5.3-10 IsTame
      5.3-11 ShortOrbits
   5.4 Properties of RCWA(Z)
      5.4-1 NrConjugacyClassesOfRCWAZOfOrder
   5.5 Predefined rcwa groups
6. Getting Information about Computations
   6.1 The Info class of the package
      6.1-1 RcwaInfo
      6.1-2 RCWAInfo
7. Auxiliary functions
   7.1 Building the manual
      7.1-1 BuildRCWAManual
   7.2 The testing routine
      7.2-1 RCWATest
   7.3 The color markup
8. Examples
   8.1 Replacing the Collatz mapping by conjugates
   8.2 An rcwa representation of a small group
   8.3 An rcwa representation of the symmetric group on 10 points
   8.4 Twisting 257-cycles into an rcwa mapping with modulus 32
   8.5 Two mappings with isomorphic graphs, but different orders
   8.6 A group with a free abelian normal subgroup of rank 12
   8.7 Behaviour of the moduli of powers

Goto Chapter:

generated by GAPDoc2HTML