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### 1. Preface

#### 1.1 Motivation

The development of this package has originally been motivated by the famous 3n+1 - Conjecture, which 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.

This has been conjectured by Lothar Collatz in the 1930s, and is still an unsolved problem today. Jeffrey C. Lagarias has written and maintains a commented 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 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.

This manual is pure software documentation, and as such it does not contain any theorems or proofs. In a few places, where this is absolutely necessary for understanding what some function is good for, corresponding mathematical assertions are made. Proofs of all of them as well as a detailed introduction into the subject can be found in the author's PhD thesis [K05]. A copy of this thesis and an english translation thereof are distributed with this package (see `thesis/thesis.pdf` resp. `thesis/thesis_e.pdf`).

#### 1.3 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 it 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`. Another source of examples is the `Random` (3.1-2) - function. If you have LaTeX and xdvi installed, you can nicely display examples of residue class-wise affine mappings by repeatedly issueing `Display(Random(RCWA(Integers)):xdvi);`.

Often the package does not provide an out-of-the-box solution for a given problem. At the beginning you will perhaps notice extremely long runtimes for seemingly trivial things. But with some experience you will learn to estimate in advance 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 some group which looks 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 non-trivial normal subgroup of the group of all residue class-wise affine permutations of the integers. 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)).

#### 1.4 Acknowledgements

I would like to thank Bettina Eick for her kind help in trying to make this package and in particular its documentation more useful and more interesting for a larger number of people. Furthermore I would like to thank the two anonymous referees for their constructive criticism and helpful suggestions.

If you use RCWA in some work leading to a publication, I ask you to cite it just as you would cite a journal article. I would be grateful for any bug reports, comments or suggestions and of course for reports of results found with the help of this package.

Stefan Kohl

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