  
  [1X1. About the RCWA Package[0X
  
  
  [1X1.1 Motivation[0X
  
  The  development  of this package has originally been inspired by the famous
  3n+1-Conjecture,  which  asserts  that  iterated  application of the [13XCollatz
  mapping[0X
  
  
                                         /
                                        | n/2 if n even,
                 T:  Z -> Z,   n  |->  <
                                        | (3n+1)/2 if n odd
                                         \
  to any given positive integer eventually yields 1 (cf. [Lag06]).
  
  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. [13Xresidue-class-wise affine[0X.
  
  After  having  investigated  these  groups for a couple of years, the author
  feels that this is a gap which is worth to be filled.
  
  
  [1X1.2 Purpose of this package[0X
  
  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 [HEO05].
  
  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.
  
  
  [1X1.3 Groups which this package can deal with[0X
  
  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 [13Xclass transpositions[0X of Z -- these
  are   involutions  which  interchange  two  disjoint  residue  classes,  see
  [2XClassTransposition[0X  ([14X2.2-3[0X)  -- 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 [Koh06a], 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 [Koh07b].
  
  
  [1X1.4 Scope of this package[0X
  
  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
  [11Xpkg/rcwa/examples/examples.g[0X.
  
  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 [14X1.3[0X. 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 [2XFactorizationIntoCSCRCT[0X ([14X2.5-1[0X))
  solely by means of theory would likely have been very hard.
  
