GAP (RCWA) - Chapter 1: About the RCWA Package

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### 1. About the RCWA Package

#### 1.1 Motivation

The development of this package has originally been inspired 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 (cp. [L06]).

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.

#### 1.2 Groups which can be dealt with

The following groups are known to be faithfully representable as residue class-wise affine groups:

- Free groups of finite rank.

- Free products of finitely many finite groups, thus in particular the modular group PSL(2,Z).

- Direct products of residue class-wise affine groups.

- Wreath products of residue class-wise affine groups with finite groups and with (Z,+).

Further every finite group embeds into some divisible residue class-wise affine torsion group. There are also finitely generated residue class-wise affine groups which are not finitely presented, and such with unsolvable membership problem. In principle this package permits to construct and investigate groups of all mentioned -- and most likely many more not yet known -- 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 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 above have not yet appeared in print. However they can be found in the preprint [K06], which is available on the author's homepage.

#### 1.3 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.

#### 1.4 Scope of this package

This package can be applied in various ways to various different problems, and it is just impossible 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 1.2. 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 `FactorizationIntoCSCRCT` (2.4-1)) solely by means of theory would likely have been very hard.

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