Version 4.5.1

March 13, 2017

**
Stefan Kohl
**

Email: stefan@gap-system.org

Homepage: https://stefan-kohl.github.io/

**RCWA** is a package for **GAP** 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:

Finite groups, and certain divisible torsion groups which they embed into.

Free groups of finite rank.

Free products of finitely many finite groups.

Direct products of the above groups.

Wreath products of the above groups with finite groups and with (ℤ,+).

Subgroups of any such groups.

With 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].

© 2003 - 2017 by Stefan Kohl.

**RCWA** 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.

**RCWA** 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.

For a copy of the GNU General Public License, see the file `GPL`

in the `etc`

directory of the **GAP** distribution or see http://www.gnu.org/licenses/gpl.html.

I am grateful to John P. McDermott for the discovery that the group discussed in Section 7.1 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.

2 Residue-Class-Wise Affine Mappings

2.8
On trajectories and cycles of residue-class-wise affine mappings

2.8-1 Trajectory (methods for rcwa mappings)

2.8-2 Trajectory (methods for rcwa mappings -- "accumulated coefficients")

2.8-3 IncreasingOn & DecreasingOn (for an rcwa mapping)

2.8-4 TransitionGraph

2.8-5 OrbitsModulo

2.8-6 FactorizationOnConnectedComponents

2.8-7 TransitionMatrix

2.8-8 Sources & Sinks (of an rcwa mapping)

2.8-9 Loops

2.8-10 GluckTaylorInvariant

2.8-11 LikelyContractionCentre

2.8-12 GuessedDivergence

2.8-1 Trajectory (methods for rcwa mappings)

2.8-2 Trajectory (methods for rcwa mappings -- "accumulated coefficients")

2.8-3 IncreasingOn & DecreasingOn (for an rcwa mapping)

2.8-4 TransitionGraph

2.8-5 OrbitsModulo

2.8-6 FactorizationOnConnectedComponents

2.8-7 TransitionMatrix

2.8-8 Sources & Sinks (of an rcwa mapping)

2.8-9 Loops

2.8-10 GluckTaylorInvariant

2.8-11 LikelyContractionCentre

2.8-12 GuessedDivergence

3 Residue-Class-Wise Affine Groups

3.1 Constructing residue-class-wise affine groups

3.1-1 IsomorphismRcwaGroup

3.1-2 DirectProduct

3.1-3 WreathProduct (for an rcwa group over Z, with a permutation group or (ℤ,+))

3.1-4 MergerExtension

3.1-5 GroupByResidueClasses

3.1-6 Restriction (of an rcwa mapping or -group, by an injective rcwa mapping)

3.1-7 Induction (of an rcwa mapping or -group, by an injective rcwa mapping)

3.1-8 RCWA

3.1-9 CT

3.1-1 IsomorphismRcwaGroup

3.1-2 DirectProduct

3.1-3 WreathProduct (for an rcwa group over Z, with a permutation group or (ℤ,+))

3.1-4 MergerExtension

3.1-5 GroupByResidueClasses

3.1-6 Restriction (of an rcwa mapping or -group, by an injective rcwa mapping)

3.1-7 Induction (of an rcwa mapping or -group, by an injective rcwa mapping)

3.1-8 RCWA

3.1-9 CT

3.3
The natural action of an rcwa group on the underlying ring

3.3-1 Orbit (for an rcwa group and either a point or a set)

3.3-2 GrowthFunctionOfOrbit

3.3-3 DrawOrbitPicture

3.3-4 ShortOrbits (for rcwa groups) & ShortCycles (for rcwa permutations)

3.3-5 ShortResidueClassOrbits & ShortResidueClassCycles

3.3-6 ComputeCycleLength

3.3-7 CycleRepresentativesAndLengths

3.3-8 FixedResidueClasses

3.3-9 Ball (for group, element and radius or group, point, radius and action)

3.3-10 RepresentativeAction

3.3-11 ProjectionsToInvariantUnionsOfResidueClasses

3.3-12 RepresentativeAction

3.3-13 CollatzLikeMappingByOrbitTree

3.3-1 Orbit (for an rcwa group and either a point or a set)

3.3-2 GrowthFunctionOfOrbit

3.3-3 DrawOrbitPicture

3.3-4 ShortOrbits (for rcwa groups) & ShortCycles (for rcwa permutations)

3.3-5 ShortResidueClassOrbits & ShortResidueClassCycles

3.3-6 ComputeCycleLength

3.3-7 CycleRepresentativesAndLengths

3.3-8 FixedResidueClasses

3.3-9 Ball (for group, element and radius or group, point, radius and action)

3.3-10 RepresentativeAction

3.3-11 ProjectionsToInvariantUnionsOfResidueClasses

3.3-12 RepresentativeAction

3.3-13 CollatzLikeMappingByOrbitTree

5
Residue-Class-Wise Affine Mappings, Groups and Monoids over ℤ^2

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