  
  [1m[4m[31m1. About the RCWA Package[0m
  
  
  [1m[4m[31m1.1 Motivation[0m
  
  The  development  of this package has originally been inspired by the famous
  3n+1-Conjecture,  which  asserts  that  iterated  application of the [22m[36mCollatz
  mapping[0m
  
  
                                         /
                                        | 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. [22m[36mresidue class-wise affine[0m.
  
  After  having  investigated  these  groups for a couple of years, the author
  feels that this is a gap which is worth to be filled.
  
  
  [1m[4m[31m1.2 Groups which can be dealt with[0m
  
  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  [22m[36mclass transpositions[0m -- these are
  involutions   which   interchange   two   disjoint   residue   classes,  see
  [1m[34mClassTransposition[0m ([1m2.2-3[0m) -- 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.
  
  
  [1m[4m[31m1.3 Purpose of this package[0m
  
  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.
  
  
  [1m[4m[31m1.4 Scope of this package[0m
  
  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
  [1mpkg/rcwa/examples/examples.g[0m.
  
  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 [1m1.2[0m. 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 [1m[34mFactorizationIntoCSCRCT[0m ([1m2.4-1[0m))
  solely by means of theory would likely have been very hard.
  
