&RCWA; is a package for &GAP; 4 which provides methods for investigating Residue Class-Wise Affine groups by means of computation.

Residue class-wise affine groups are countable permutation groups acting on the integers, whose elements are bijective residue class-wise affine mappings.

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) Many residue class-wise affine groups act multiply transitively on \Z or on subsets thereof. The class of groups which can faithfully be represented as residue class-wise affine groups is closed under forming wreath products with finite groups and with the infinite cyclic group (\Z,+). It includes free groups of any rank, all free products of finite groups and certain divisible torsion groups. It also contains finitely generated groups with unsolvable membership problem. The group which is generated by all class transpositions -- these are involutions which interchange two disjoint residue classes -- is a simple group which has subgroups of all mentioned types and whose class of isomorphism types of subgroups is closed under the above operations.

Despite of what has been said in the preceding paragraph, still not very much is known about the structure of residue class-wise affine groups. 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). A copy of this thesis and an english translation thereof are distributed with this package.

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