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} / | 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.

Groups which can be dealt with

In principle this package permits to construct and investigate all groups which have faithful representations as residue-class-wise affine groups. A couple of types of such groups are listed in the abstract. From the list given there, one can already conclude that there are finitely generated residue-class-wise affine groups which do not have finite presentations, and such with unsolvable membership problem. However the list is by far not exhaustive.

The group which is generated by all class transpositions -- these are involutions which interchange two disjoint residue classes, see -- is a simple group which has subgroups of all types listed in the abstract. 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 and in the abstract 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 .

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Purpose of this package

So far, compared to classes of groups like for example finite groups, matrix groups, finitely presented groups or polycyclic groups, not very much is known about 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.

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.