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### 1. Preface

#### 1.1 Motivation

The development of this package has originally been inspired by the famous 3n+1-Conjecture. This conjecture asserts that iterated application of the Collatz mapping

                                           /
                                          | n/2 if n even,
                   T:  Z -> Z,   n  |->  <
                                          | (3n+1)/2 if n odd
                                           \

to any given positive integer eventually yields 1.

The 3n+1-Conjecture has been made by Lothar Collatz in the 1930s, and is still lacking a proof today. Jeffrey C. Lagarias has written and maintains a comprehensive annotated bibliography [L06], which currently lists about 200 references to publications related to Collatz' conjecture. None of the articles mentioned there tries to attack the problem by means of group theory or investigates the structure of groups generated by bijective mappings which are "similar to the Collatz mapping", i.e. residue class-wise affine. In fact, residue class-wise affine groups apparently have not been treated anywhere in the literature before.

After having investigated these objects for a couple of years, the author feels that this is a gap which is worth to be filled.

#### 1.2 Groups which can be represented

This section lists some types of groups which are known to be representable as residue class-wise affine groups. Proofs of most of the results have not yet appeared in print, but can be found in the preprint A Simple Group Generated by Involutions Interchanging Residue Classes of the Integers on the author's homepage.

Obviously, the infinite dihedral group < n -> -n, n -> n+1 > can be represented. Further the class of representable groups is closed under forming direct products and under forming wreath products with finite groups and with the infinite cyclic group (Z,+). Free groups and free products of finite groups can be represented as well. Further there are divisible torsion groups and finitely generated groups with unsolvable membership problem which have faithful residue class-wise affine representations. In principle this package permits to construct and investigate groups of all mentioned types.

The group which is generated by all class transpositions -- these are involutions which interchange two disjoint residue classes, see `ClassTransposition` (2.2-3) -- is a simple group which contains all the above groups. It has uncountably many simple subgroups.

#### 1.3 Purpose of this package

So far, compared to classes of groups like for example matrix groups, finite permutation groups or polycyclic groups, only relatively basic facts about residue class-wise affine groups are known. This package is intended to serve as a tool for obtaining a better understanding of their rich and interesting group theoretical and combinatorial structure.

#### 1.4 Scope of this package

This package being a research tool which can be applied in various ways to various different problems, it is simply not possible to say what can be found out with its help about which mappings or groups. The best way to get an idea about this is likely to experiment with the examples discussed in this manual and included in the file `pkg/rcwa/examples/examples.g`.

Of course this package often does not provide an out-of-the-box solution for a given problem. At the beginning the user may perhaps notice long runtimes for seemingly easy things. But with some experience he will learn to estimate how long something will take and to see why raising some harmlessly-looking mapping to the 20th power would take terabytes of memory, while one can easily find out nontrivial things about groups which look much more complicate. Quite often it is possible to find an answer for a given question by using an interactive trial-and-error approach.

Among many other results, with substancial help of this package the author has found a proof that the group generated by all class transpositions is simple. Interactive sessions with this package have also lead to the development of a method for factoring residue class-wise affine permutations into involutions which have a particularly simple structure (see `FactorizationIntoCSCRCT` (2.4-1)).

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