  
  [1m[4m[31m1. Preface[0m
  
  
  [1m[4m[31m1.1 Motivation[0m
  
  The  development  of this package has originally been inspired by the famous
  3n+1-Conjecture.  This  conjecture  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.
  
  The  3n+1-Conjecture  has  been  made by Lothar Collatz in the 1930s, and is
  still lacking a proof today. Jeffrey C. Lagarias has written and maintains a
  comprehensive  annotated  bibliography  [L06],  which  currently lists about
  200 references  to  publications related to Collatz' conjecture. None of the
  articles  mentioned  there  tries  to  attack  the problem by means of group
  theory  or  investigates  the  structure  of  groups  generated by bijective
  mappings which are "similar to the Collatz mapping", i.e. [22m[36mresidue class-wise
  affine[0m.  In  fact, [22m[36mresidue class-wise affine groups[0m apparently have not been
  treated anywhere in the literature before.
  
  After  having  investigated  these objects 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 represented[0m
  
  This  section lists some types of groups which are known to be representable
  as  residue class-wise affine groups. Proofs of most of the results have not
  yet  appeared  in  print,  but  can  be found in the preprint [22m[36mA Simple Group
  Generated  by  Involutions  Interchanging Residue Classes of the Integers[0m on
  the author's homepage.
  
  Obviously,  the  infinite  dihedral  group  <  n  ->  -n,  n -> n+1 > can be
  represented.  Further  the  class  of  representable  groups is closed under
  forming direct products and under forming wreath products with finite groups
  and  with  the infinite cyclic group (Z,+). Free groups and free products of
  finite  groups  can  be  represented  as  well.  Further there are divisible
  torsion  groups  and  finitely  generated  groups with unsolvable membership
  problem  which  have  faithful residue class-wise affine representations. In
  principle  this  package  permits to construct and investigate groups of all
  mentioned 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 has uncountably many simple subgroups.
  
  
  [1m[4m[31m1.3 Purpose of this package[0m
  
  So far, compared to classes of groups like for example matrix groups, finite
  permutation  groups  or polycyclic groups, only relatively basic facts about
  residue  class-wise  affine  groups  are  known. 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  being a research tool which can be applied in various ways to
  various  different  problems,  it  is simply not possible to say what can be
  found  out with its help about which mappings or 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. At the beginning the user may perhaps notice long runtimes
  for  seemingly  easy  things.  But  with  some  experience  he will learn to
  estimate  how  long  something  will  take  and  to  see  why  raising  some
  harmlessly-looking mapping to the 20th power would take terabytes of memory,
  while one can easily find out nontrivial things about groups which look much
  more  complicate.  Quite  often it is possible to find an answer for a given
  question by using an interactive trial-and-error approach.
  
  Among  many  other results, with substancial help of this package the author
  has  found  a  proof that the group generated by all class transpositions is
  simple.  Interactive  sessions  with  this  package  have  also  lead to the
  development of a method for factoring residue class-wise affine permutations
  into    involutions    which    have   a   particularly   simple   structure
  (see [1m[34mFactorizationIntoCSCRCT[0m ([1m2.4-1[0m)).
  
