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<!-- ##  about.xml          RCWA documentation            Stefan Kohl  ## -->
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<Chapter Label="ch:AboutRCWA"><Heading>About the RCWA Package</Heading>

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<Section Label="sec:motivation">
<Heading>Motivation</Heading>

<Index Key="Collatz conjecture">Collatz conjecture</Index>
<Index Key="Collatz conjecture">Collatz mapping</Index>
The development of this package has originally been inspired by the famous
<M>3n+1</M>-Conjecture, which asserts that iterated application of the
<E>Collatz mapping</E>
<Alt Only="LaTeX">
  <Display>
  <![CDATA[
    T: \ \ \Z \longrightarrow \Z, \ \ \ \
    n \ \longmapsto \
    \begin{cases}
      \frac{n}{2}    & \text{if} \ \ n \ \ \text{is even}, \\
      \frac{3n+1}{2} & \text{if} \ \ n \ \ \text{is odd}
    \end{cases}
  ]]>
  </Display>
</Alt>
<Alt Only="Text,HTML"><Verb><![CDATA[
                                       /
                                      | n/2 if n even,
               T:  Z -> Z,   n  |->  <
                                      | (3n+1)/2 if n odd
                                       \
]]></Verb></Alt>
to any given positive integer eventually yields&nbsp;1
(cp.&nbsp;<Cite Key="Lagarias06"/>). <P/>

So far, no attempts have been made to investigate the structure of groups
whose elements are permutations which are <Q>similar to the Collatz
mapping</Q>, i.e. <E>residue class-wise affine</E>. <P/>

After having investigated these groups for a couple of years,
the author feels that this is a gap which is worth to be filled.

</Section>

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<Section Label="sec:GroupsWhichEmbed">
<Heading>Groups which can be dealt with</Heading>

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

<List>

  <Item>
    Free groups of finite rank.
  </Item>

  <Item>
    Free products of finitely many finite groups, thus in particular
    the modular group PSL(2,<M>\Z</M>).
  </Item>

  <Item>
    Direct products of residue class-wise affine groups.
  </Item>

  <Item>
    Wreath products of residue class-wise affine groups with finite groups
    and with&nbsp;<M>(\Z,+)</M>.
  </Item>

</List>
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. <P/>

The group which is generated by all <E>class transpositions</E>
-- these are involutions which interchange two disjoint residue classes,
see <Ref Func="ClassTransposition" Label="r1, m1, r2, m2"/> -- 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. <P/>

Proofs of most of the results mentioned above have not yet appeared in print.
However they can be found in the preprint&nbsp;<Cite Key="Kohl06b"/>, which
is available on the author's homepage.

</Section>

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<Section Label="sec:Purpose">
<Heading>Purpose of this package</Heading>

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.

</Section>

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<Section Label="sec:Scope">
<Heading>Scope of this package</Heading>

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 <File>pkg/rcwa/examples/examples.g</File>. <P/>

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. <P/>

With substancial help of this package, the author has found the results
mentioned in Section&nbsp;<Ref Label="sec:GroupsWhichEmbed"/>.
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&nbsp;<Ref Attr="FactorizationIntoCSCRCT"
Label="g"/>) solely by means of theory would likely have been very hard.

</Section>

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</Chapter>

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