  
  [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
  
  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  [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 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 [K06],
  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 [K07].
  
  
  [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 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
  [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.
  
