The &RCWA; package provides methods for investigating Residue Class-Wise Affine groups by means of computation. Residue class-wise affine groups are permutation groups acting on the integers, whose elements are bijective residue class-wise affine mappings. Typically they are infinite.

A mapping f: \Z \rightarrow \Z is called residue class-wise affine 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/m\Z 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 f|_{r(m)}: \ \ r(m) \rightarrow \Z, \ \ \ n \ \mapsto \ \frac{a_{r(m)} \cdot n + b_{r(m)}}{c_{r(m)}}. a_r(m) * n + b_r(m) f|_r(m): r(m) -> Z, n |-> -------------------. c_r(m) Residue class-wise affine groups are countable. Many of them act multiply transitively on \Z or on subsets thereof. Only relatively basic facts about their structure are known so far. 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 Restklassenweise affine Gruppen. This thesis is published at http://deposit.ddb.de/cgi-bin/dokserv?idn=977164071 (Archivserver Deutsche Bibliothek) and at http://elib.uni-stuttgart.de/opus/volltexte/2005/2448/ (OPUS-Datenbank Universität Stuttgart). A copy of this thesis and an english translation thereof are distributed with this package (see thesis/thesis.pdf resp. thesis/thesis&uscore;e.pdf).