About the RCWA Package
Motivation
Collatz conjecture
Collatz mapping
The development of this package has originally been inspired by the famous 3n+1-Conjecture, which asserts that iterated application of the Collatz mapping T: \ \ \Z \longrightarrow \Z, \ \ \ \ n \ \longmapsto \ \begin{cases} \frac{n}{2} & \text{if} \ \ n \ \ \text{is even}, \\ \frac{3n+1}{2} & \text{if} \ \ n \ \ \text{is odd} \end{cases} <center> <img src = "collatz.png" width = "342" height = "63" alt = "T: Z -> Z, n |-> (n/2 if n even, (3n+1)/2 if n odd)"/> </center> / | n/2 if n even, T: Z -> Z, n |-> < | (3n+1)/2 if n odd \ to any given positive integer eventually yields 1 (cp. ).

So far, no attempts have been made to investigate the structure of groups whose elements are permutations which are similar to the Collatz mapping, i.e. residue-class-wise affine.

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

Purpose of this package

The present scope of computational group theory essentially comprises finite permutation groups, matrix groups, finitely presented groups, polycyclically presented groups and automata groups. For details, we refer to .

The purpose of this package is twofold:

On the one hand, it introduces a new class of groups which are accessible to computational methods, and it therefore extends the current scope of computational group theory as outlined above.
On the other, residue-class-wise affine groups are interesting mathematical objects in their own right, and this package is intended to serve as a tool for obtaining a better understanding of their rich and often complicated group theoretical and combinatorial structure.
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Groups which this package can deal with

In principle this package permits to construct and investigate all groups which have faithful representations as residue-class-wise affine groups. Among many others, the following groups and their subgroups belong to this class:

Finite groups, and certain divisible torsion groups which they embed into.
Free groups of finite rank.
Free products of finitely many finite groups, thus in particular the modular group PSL(2,\Z).
Direct products of the above groups.
Wreath products of the above groups with finite groups and with (\Z,+).

This list permits already to conclude that there are finitely generated residue-class-wise affine groups which do not have finite presentations, and such with algorithmically unsolvable membership problem. However the list is certainly by far not exhaustive, and using this package it is easy to construct groups of types which are not mentioned there.

The group CT(\Z) which is generated by all class transpositions of \Z -- these are involutions which interchange two disjoint residue classes, see -- is a simple group which has subgroups of all types listed above. It is countable, but it has an uncountable series of simple subgroups which is parametrized by the sets of odd primes.

Proofs of most of the results mentioned here have not yet appeared in print. However they can be found in the preprint , which is available on the author's homepage. Descriptions of many of the algorithms and methods which are implemented in this package can be found in .

Scope of this package

This package can be applied in various ways to various different problems, and it is just not possible to say what can be found out with its help about which 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. Quite often it is possible to find an answer for a given question by using an interactive trial-and-error approach.

With substancial help of this package, the author has found the results mentioned in Section . Interactive sessions with this package have also lead to the development of most of the algorithms which are now implemented in it. Just to mention one example, developing the factorization method for residue-class-wise affine permutations (see ) solely by means of theory would likely have been very hard.