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\begin{titlepage}
\begin{center}{\Huge \textbf{RCWA}}\\[1cm]
\hypersetup{pdftitle=RCWA}
\markright{\scriptsize \mbox{}\hfill RCWA \hfill\mbox{}}
{\Large \textbf{Residue Class-Wise Affine Groups}}\\[1cm]
{ ( Version 1.7.2 ) }\\[1cm]
{May 3, 2006}\\[1cm]
\mbox{}\\[2cm]
{\large \textbf{ Stefan Kohl    }}\\
\hypersetup{pdfauthor= Stefan Kohl    }
\end{center}\vfill

\mbox{}\\
{\mbox{}\\
\small \noindent \textbf{ Stefan Kohl    } --- Email: \href{mailto://kohl@mathematik.uni-stuttgart.de}{\texttt{kohl@mathematik.uni-stuttgart.de}}\\
 --- Homepage: \href{http://www.cip.mathematik.uni-stuttgart.de/~kohlsn}{\texttt{http://www.cip.mathematik.uni-stuttgart.de/\~{}kohlsn}}\\
 --- Address: \begin{minipage}[t]{8cm}\noindent
 Institut für Geometrie und Topologie \\
 Universität Stuttgart \\
 70550 Stuttgart \\
 Germany \end{minipage}
}\\
\end{titlepage}

\newpage\setcounter{page}{2}
{\small 
\section*{Abstract}
\logpage{[ 0, 0, 2 ]}
 The \textsf{RCWA} package provides methods for investigating \emph{R}esidue \emph{C}lass-\emph{W}ise \emph{A}ffine groups by means of computation. Residue class-wise affine groups are
permutation groups acting on the integers, whose elements are bijective
residue class-wise affine mappings. Typically they are infinite. 

 A mapping $f: \Z \rightarrow \Z$ is called \emph{residue class-wise affine} provided that there is a positive integer $m$ such that the restrictions of $f$ to the residue classes (mod~$m$) are all affine. This means that for any residue class $r(m) \in \Z/m\Z$ there are coefficients $a_{r(m)}, b_{r(m)}, c_{r(m)} \in \Z$ such that the restriction of the mapping~$f$ to the set $r(m) = \{{r + km | k \in \Z\}}$ is given~by  
\[ f|_{r(m)}: \ \ r(m) \rightarrow \Z, \ \ \ n \ \mapsto \ \frac{a_{r(m)} \cdot n
+ b_{r(m)}}{c_{r(m)}}. \]
   Residue class-wise affine groups are countable. ``Many'' of them act multiply transitively on~$\Z$ or on subsets thereof. Only relatively basic facts about their structure are
known so far. This package is intended to serve as a tool for obtaining a
better understanding of their rich and interesting group theoretical and
combinatorial structure. 

 Residue class-wise affine groups can be generalized in a natural way to
euclidean rings other than the ring of integers. While this package
undoubtedly provides most functionality for residue class-wise affine groups
over the integers, at least rudimentarily it also covers the cases that the
underlying ring is a semilocalization of~$\Z$ or a polynomial ring in one variable over a finite field. 

 The original motivation for investigating residue class-wise affine groups
comes from the famous $3n+1$ Conjecture, which is an assertion about a surjective, but not injective
residue class-wise affine mapping. 

 Residue class-wise affine groups are introduced in the author's thesis \emph{Restklassenweise affine Gruppen}. This thesis is published at \href{http://deposit.ddb.de/cgi-bin/dokserv?idn=977164071}{\texttt{http://deposit.ddb.de/cgi-bin/dokserv?idn=977164071}} (Archivserver Deutsche Bibliothek) and at \href{http://elib.uni-stuttgart.de/opus/volltexte/2005/2448/}{\texttt{http://elib.uni-stuttgart.de/opus/volltexte/2005/2448/}} (OPUS-Datenbank Universität Stuttgart). A copy of this thesis and an english
translation thereof are distributed with this package (see \texttt{thesis/thesis.pdf} resp. \texttt{thesis/thesis{\textunderscore}e.pdf}). }\\[1cm]
{\small 
\section*{Copyright}
\logpage{[ 0, 0, 1 ]}
 {\copyright} 2003 - 2006 by Stefan Kohl 

 This package is distributed under the GNU General Public License. }\\[1cm]
\newpage

\def\contentsname{Contents\logpage{[ 0, 0, 3 ]}}

\tableofcontents
\newpage

        
\chapter{\textcolor{Chapter }{Preface}}\label{ch:preface}
\logpage{[ 1, 0, 0 ]}
{
  
\section{\textcolor{Chapter }{Motivation}}\label{sec:motivation}
\logpage{[ 1, 1, 0 ]}
{
  \index{Collatz conjecture@Collatz conjecture} \index{Collatz conjecture@Collatz mapping} The development of this package has originally been motivated by the famous $3n+1$ - Conjecture, which asserts that iterated application of the Collatz mapping  
\[ T: \ \mathbb{Z} \longrightarrow \mathbb{Z}, \ \ \ \ n \ \longmapsto \
\begin{cases} \frac{n}{2} & \text{if} \ n \ \text{even}, \\ \frac{3n+1}{2} &
\text{if} \ n \ \text{odd} \end{cases} \]
   to any given positive integer eventually yields~1. 

 This has been conjectured by Lothar Collatz in the 1930s, and is still an
unsolved problem today. Jeffrey C. Lagarias has written and maintains a
commented bibliography \cite{Lagarias06}, 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. \emph{residue class-wise affine}. In fact, \emph{residue class-wise affine groups} 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. }

  
\section{\textcolor{Chapter }{Purpose of this package}}\label{sec:purpose}
\logpage{[ 1, 2, 0 ]}
{
  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. 

 This manual is pure software documentation, and as such it does not contain
any theorems or proofs. In a few places, where this is absolutely necessary
for understanding what some function is good for, corresponding mathematical
assertions are made. Proofs of all of them as well as a detailed introduction
into the subject can be found in the author's PhD thesis~\cite{Kohl05}. A copy of this thesis and an english translation thereof are distributed
with this package (see \texttt{thesis/thesis.pdf} resp. \texttt{thesis/thesis{\textunderscore}e.pdf}). }

 \pagebreak[4]  
\section{\textcolor{Chapter }{Scope of this package}}\label{sec:noteaboutcontents}
\logpage{[ 1, 3, 0 ]}
{
  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 it 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 \texttt{pkg/rcwa/examples/examples.g}. Another source of examples is the \texttt{Random} (\ref{Random:RCWA( Integers )}) - function. If you have {\LaTeX} and \textsf{xdvi} installed, you can nicely display examples of residue class-wise affine
mappings by repeatedly issueing \texttt{Display(Random(RCWA(Integers)):xdvi);}. 

 Often the package does not provide an out-of-the-box solution for a given
problem. At the beginning you will perhaps notice extremely long runtimes for
seemingly trivial things. But with some experience you will learn to estimate
in advance 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 some group which looks
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 non-trivial normal subgroup of the group of all residue class-wise
affine permutations of the integers. 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 \texttt{FactorizationIntoCSCRCT} (\ref{FactorizationIntoCSCRCT:g})). }

  
\section{\textcolor{Chapter }{Acknowledgements}}\label{sec:acknowledgements}
\logpage{[ 1, 4, 0 ]}
{
  I would like to thank Bettina~Eick for her kind help in trying to make this package and in particular its
documentation more useful and more interesting for a larger number of people.
Furthermore I would like to thank the two anonymous referees for their
constructive criticism and helpful suggestions. 

 If you use \textsf{RCWA} in some work leading to a publication, I ask you to cite it just as you would
cite a journal article. I would be grateful for any bug reports, comments or
suggestions and of course for reports of results found with the help of this
package.    \vfill \noindent Stuttgart, \today \hfill Stefan Kohl  }

  }

         
\chapter{\textcolor{Chapter }{Residue Class-Wise Affine Mappings}}\label{ch:RcwaMappings}
\logpage{[ 2, 0, 0 ]}
{
  This chapter describes the functionality provided by this package for
computing with residue class-wise affine mappings.  
\section{\textcolor{Chapter }{Basic definitions}}\label{sec:basicdefinitions}
\logpage{[ 2, 1, 0 ]}
{
  The abstract already gave a brief definition of residue class-wise affine
groups over the ring of integers. In the sequel, a slightly generalized and
more formal version of this definition is given. In the same time, some useful
notation is introduced. 

 \index{rcwa mapping@!definition} Let $R$ be an infinite euclidean domain which is not a field and all of whose proper
residue class rings are finite. A mapping $f: R \rightarrow R$ is called \emph{residue class-wise affine}, or for short an \emph{rcwa} mapping, if there is an $m \in R \setminus \{0\}$ such that the restrictions of $f$ to the residue classes $r(m) \in R/mR$ are all affine. This means that for any residue class $r(m)$ there are coefficients $a_{r(m)}, b_{r(m)}, c_{r(m)} \in R$ such that the restriction of the mapping $f$ to the set $r(m) = \{{ r+km | k \in R\}}$ is given by  
\[ f|_{r(m)}: \ r(m) \rightarrow R, \ \ \ n \ \mapsto \ \frac{a_{r(m)} \cdot n +
b_{r(m)}}{c_{r(m)}}. \]
   The value $m$ is called the \emph{modulus} of~$f$. \index{rcwa mapping@!modulus} It is understood that all fractions are reduced, i.e. that $\gcd( a_{r(m)}, b_{r(m)}, c_{r(m)} ) = 1$, and that $m$ is chosen multiplicatively minimal. 

 Apart from the restrictions imposed by the condition that the image of any
residue class $r(m)$ under $f$ must be a subset of $R$ and that one cannot divide by~0, the coefficients $a_{r(m)}$, $b_{r(m)}$ and $c_{r(m)}$ can be any ring elements. 

 When talking about the \emph{product} $f \cdot g$ of some rcwa mappings $f$ and~$g$ it is always meant their composition as mappings, where $f$ is applied first. By the inverse of a bijective rcwa mapping it is meant its
inverse mapping. 

 The set RCWA($R$) $\ := \ $ { $\{ \ \sigma \in$ Sym($R$) $\ | \ \sigma$ is residue class-wise affine $\}$ } is closed under multiplication and taking inverses (this can be verified
easily), hence forms a subgroup of Sym($R$). \index{rcwa group@!definition} A subgroup of RCWA($R$) is called a \emph{residue class-wise affine} group, or for short an \emph{rcwa} group. 

 There are two entirely different classes of rcwa mappings and -groups. One of
these classes comprises what could be called the ``trivial cases''. The members of the other have typically a quite complicate structure and are
in often very difficult to investigate. Accordingly, the former are called \emph{tame} and the latter are called \emph{wild}. By definition, an rcwa mapping is \emph{tame} if the set of moduli of its powers is bounded, and an rcwa group is \emph{tame} if the set of moduli of its elements is bounded. }

 \pagebreak[4]  
\section{\textcolor{Chapter }{Entering residue class-wise affine mappings}}\label{sec:EnteringRcwaMappings}
\logpage{[ 2, 2, 0 ]}
{
  Entering an rcwa mapping into \textsf{RCWA} in general requires specifying the underlying ring~$R$, the modulus~$m$ and the coefficients $a_{r(m)}$, $b_{r(m)}$ and~$c_{r(m)}$ for $r(m)$ running over the residue classes (mod~$m$). For the sake of simplicity, in this section we describe how to enter rcwa
mappings of $R = \Z$. This is likely the most prominent and certainly the best-supported case. For
the general constructor for rcwa mappings, see \texttt{RcwaMapping} (\ref{RcwaMapping:coeffs}). 

 The easiest way to enter an rcwa mapping of $\Z$ is by \texttt{RcwaMapping( \mbox{\texttt{coeffs}} )}. Here \mbox{\texttt{coeffs}} is a list of $m$ coefficient triples \texttt{coeffs[}$r+1$\texttt{]~= [}$a_{r(m)}$, $b_{r(m)}$, $c_{r(m)}$\texttt{]}, where $r$ runs from 0 to~$m-1$. 

 If some coefficient $c_{r(m)}$ is zero or if images of some integers under the mapping to be defined would
not be integers, an error message is printed and a break loop is entered. For
example, the coefficient triple \texttt{[1,1,3]} is not allowed at the first position. The reason for this is that not all
integers congruent to 0~+~1~=~1~mod~$m$ are divisible by~3. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> T := RcwaMapping([[1,0,2],[3,1,2]]); # The Collatz mapping.
  <rcwa mapping of Z with modulus 2>
  gap> [ IsSurjective(T), IsInjective(T) ];
  [ true, false ]
  gap> SetName(T,"T"); Display(T);
  
  Surjective rcwa mapping of Z with modulus 2
  
                  n mod 2                |                 n^T
  ---------------------------------------+--------------------------------------
    0                                    | n/2
    1                                    | (3n + 1)/2
  
  gap> a := RcwaMapping([[3,0,2],[3,1,4],[3,0,2],[3,-1,4]]); SetName(a,"a");
  <rcwa mapping of Z with modulus 4>
  gap> IsBijective(a); # Check whether this is a permutation.
  true
  gap> Display(a);
  
  Bijective rcwa mapping of Z with modulus 4
  
                  n mod 4                |                 n^a
  ---------------------------------------+--------------------------------------
    0 2                                  | 3n/2
    1                                    | (3n + 1)/4
    3                                    | (3n - 1)/4
  
  gap> MovedPoints(a);
  Z \ [ -1, 0, 1 ]
  gap> Cycle(a,44);
  [ 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59 ]
  
\end{Verbatim}
 \pagebreak[4] \noindent There is computational evidence for the conjecture that any residue class-wise
affine permutation of~$\Z$ can be factored into members of the following three series of rcwa mappings of
particularly simple structure (cp. \texttt{FactorizationIntoCSCRCT} (\ref{FactorizationIntoCSCRCT:g})): 

\subsection{\textcolor{Chapter }{ClassShift (r, m)}}
\logpage{[ 2, 2, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{ClassShift( r, m )\index{ClassShift@\texttt{ClassShift}!r, m}
\label{ClassShift:r, m}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 The \emph{class shift} $\nu_{r(m)}$. 



 The \emph{class shift} $\nu_{r(m)}$ is the rcwa mapping of $\mathbb{Z}$ which maps $n \in r(m)$ to $n + m$ and fixes $\mathbb{Z} \setminus r(m)$ pointwisely. The residue class \texttt{ResidueClass(\mbox{\texttt{r}},\mbox{\texttt{m}})} itself can be given in place of the arguments \mbox{\texttt{r}} and~\mbox{\texttt{m}}. Enclosing the argument list in list brackets is permitted. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(ClassShift(5,12));    
  
  Tame bijective rcwa mapping of Z with modulus 12, of order infinity
  
                 n mod 12                |          n^ClassShift(5,12)
  ---------------------------------------+--------------------------------------
     0  1  2  3  4  6  7  8  9 10 11     | n
     5                                   | n + 12
  
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{ClassReflection (r, m)}}
\logpage{[ 2, 2, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{ClassReflection( r, m )\index{ClassReflection@\texttt{ClassReflection}!r, m}
\label{ClassReflection:r, m}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 The \emph{class reflection} $\varsigma_{r(m)}$. 



 The \emph{class reflection} $\varsigma_{r(m)}$ is the rcwa mapping of $\mathbb{Z}$ which maps $n \in r(m)$ to $-n + 2r$ and fixes $\mathbb{Z}~\setminus~r(m)$ pointwisely. The residue class \texttt{ResidueClass(\mbox{\texttt{r}},\mbox{\texttt{m}})} itself can be given in place of the arguments \mbox{\texttt{r}} and~\mbox{\texttt{m}}. Enclosing the argument list in list brackets is permitted. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(ClassReflection(5,9));
  
  Bijective rcwa mapping of Z with modulus 9, of order 2
  
                  n mod 9                |        n^ClassReflection(5,9)
  ---------------------------------------+--------------------------------------
    0 1 2 3 4 6 7 8                      | n
    5                                    | -n + 10
  
  
\end{Verbatim}
 }

 \pagebreak[4] 

\subsection{\textcolor{Chapter }{ClassTransposition (r1, m1, r2, m2)}}
\logpage{[ 2, 2, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{ClassTransposition( r1, m1, r2, m2 )\index{ClassTransposition@\texttt{ClassTransposition}!r1, m1, r2, m2}
\label{ClassTransposition:r1, m1, r2, m2}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 The \emph{class transposition} $\tau_{r_1(m_1),r_2(m_2)}$. 



 The \emph{class transposition} $\tau_{r_1(m_1),r_2(m_2)}$ is an rcwa mapping of $\mathbb{Z}$ of order~2 which interchanges the disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$ of $\mathbb{\Z}$ and fixes the complement of their union pointwisely. The residue classes \texttt{ResidueClass(\mbox{\texttt{r1}},\mbox{\texttt{m1}})} and \texttt{ResidueClass(\mbox{\texttt{r2}},\mbox{\texttt{m2}})} themselves can be given in place of the arguments \mbox{\texttt{r1}}, \mbox{\texttt{m1}}, \mbox{\texttt{r2}} and \mbox{\texttt{m2}}. Enclosing the argument list in list brackets is permitted. The residue
classes $r_1(m_1)$ and $r_2(m_2)$ are stored as an attribute \texttt{TransposedClasses}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(ClassTransposition(1,2,8,10));
  
  Bijective rcwa mapping of Z with modulus 10, of order 2
  
                 n mod 10                |    n^ClassTransposition(1,2,8,10)
  ---------------------------------------+--------------------------------------
     0  2  4  6                          | n
     1  3  5  7  9                       | 5n + 3
     8                                   | (n - 3)/5
  
  
\end{Verbatim}
 }

 \noindent It can be shown that the group which is generated by all class transpositions
is simple. 

 The permutations of the following kind play an important role in factoring
bijective rcwa mappings into class shifts, class reflections and class
transpositions (cp. \texttt{FactorizationIntoCSCRCT} (\ref{FactorizationIntoCSCRCT:g})): 

\subsection{\textcolor{Chapter }{PrimeSwitch (p)}}
\logpage{[ 2, 2, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{PrimeSwitch( p )\index{PrimeSwitch@\texttt{PrimeSwitch}!p}
\label{PrimeSwitch:p}
}\hfill{\scriptsize (function)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{PrimeSwitch( p, k )\index{PrimeSwitch@\texttt{PrimeSwitch}!p, k}
\label{PrimeSwitch:p, k}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 In the one-argument form the \emph{prime switch} $\sigma_p := \tau_{0(8),1(2p)} \cdot \tau_{4(8),-1(2p)} \cdot \tau_{0(4),1(2p)}
\cdot \tau_{2(4),-1(2p)} \cdot \tau_{2(2p),1(4p)} \cdot \tau_{4(2p),2p+1(4p)}$, and in the two-argument form the restriction of $\sigma_p$ by $n \mapsto kn$. 



 For an odd prime $p$, the \emph{prime switch} $\sigma_p$ is a bijective rcwa mapping of $\Z$ with modulus~$4p$, multiplier~$p$ (see \texttt{Multiplier} (\ref{Multiplier:f})) and divisor~2 (see \texttt{Divisor} (\ref{Divisor:f})). 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(PrimeSwitch(3));
  
  Wild bijective rcwa mapping of Z with modulus 12
  
                 n mod 12                |           n^PrimeSwitch(3)
  ---------------------------------------+--------------------------------------
     0                                   | n/2
     1  7                                | n + 1
     2  6 10                             | (3n + 4)/2
     3  9                                | n
     4                                   | n - 3
     5  8 11                             | n - 1
  
  
\end{Verbatim}
 }

 There are properties \texttt{IsClassShift}, \texttt{IsClassReflection}, \texttt{IsClassTransposition} and \texttt{IsPrimeSwitch} which indicate whether a given rcwa mapping belongs to the corresponding
series. 

 In the sequel, a description of the general-purpose constructor for rcwa
mappings is given. This might look a bit technical on a first glance, but
knowing all possible ways of entering an rcwa mapping is by no means necessary
for understanding this manual or for using this package. 

\subsection{\textcolor{Chapter }{RcwaMapping (R, m, coeffs)}}
\logpage{[ 2, 2, 5 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( R, m, coeffs )\index{RcwaMapping@\texttt{RcwaMapping}!R, m, coeffs}
\label{RcwaMapping:R, m, coeffs}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( R, coeffs )\index{RcwaMapping@\texttt{RcwaMapping}!R, coeffs}
\label{RcwaMapping:R, coeffs}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( coeffs )\index{RcwaMapping@\texttt{RcwaMapping}!coeffs}
\label{RcwaMapping:coeffs}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( perm, range )\index{RcwaMapping@\texttt{RcwaMapping}!perm, range}
\label{RcwaMapping:perm, range}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( m, values )\index{RcwaMapping@\texttt{RcwaMapping}!m, values}
\label{RcwaMapping:m, values}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( pi, coeffs )\index{RcwaMapping@\texttt{RcwaMapping}!pi, coeffs}
\label{RcwaMapping:pi, coeffs}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( q, m, coeffs )\index{RcwaMapping@\texttt{RcwaMapping}!q, m, coeffs}
\label{RcwaMapping:q, m, coeffs}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( P1, P2 )\index{RcwaMapping@\texttt{RcwaMapping}!P1, P2}
\label{RcwaMapping:P1, P2}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMapping( cycles )\index{RcwaMapping@\texttt{RcwaMapping}!cycles}
\label{RcwaMapping:cycles}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 An rcwa mapping. 



 In all cases the argument~\mbox{\texttt{R}} is the underlying ring, \mbox{\texttt{m}} is the modulus and \mbox{\texttt{coeffs}} is the coefficient list. A coefficient list for an rcwa mapping with modulus $m$ consists of $|R/mR|$ coefficient triples \texttt{[}$a_{r(m)}$, $b_{r(m)}$, $c_{r(m)}$\texttt{]}. Their ordering is determined by the ordering of the representatives of the
residue classes (mod~$m$) in the sorted list returned by \texttt{AllResidues(\mbox{\texttt{R}}, \mbox{\texttt{m}})}. In case $R = \mathbb{Z}$ this means that the coefficient triple for the residue class $0(m)$ comes first and is followed by the one for $1(m)$, the one for $2(m)$ and so on. In case one or several of the arguments \mbox{\texttt{R}}, \mbox{\texttt{m}} and \mbox{\texttt{coeffs}} are omitted or replaced by other arguments, they are either derived from the
latter or default values are taken. The meaning of the other arguments is
defined in the detailed description of the particular methods given in the
sequel. The above methods return the rcwa mapping 
\begin{description}
\item[{(a)}]  of \mbox{\texttt{R}} with modulus \mbox{\texttt{modulus}} and coefficients \mbox{\texttt{coeffs}}, resp. 
\item[{(b)}]  of \mbox{\texttt{R}} = $\mathbb{Z}$ or \mbox{\texttt{R}} = $\mathbb{Z}_{(\pi)}$ with modulus \texttt{Length(\mbox{\texttt{coeffs}})} and coefficients \mbox{\texttt{coeffs}}, resp. 
\item[{(c)}]  of \mbox{\texttt{R}} = $\mathbb{Z}$ with modulus \texttt{Length(\mbox{\texttt{coeffs}})} and coefficients \mbox{\texttt{coeffs}}, resp. 
\item[{(d)}]  of \mbox{\texttt{R}} = $\mathbb{Z}$, acting on any set \texttt{\mbox{\texttt{range}}+k*Length(\mbox{\texttt{range}})} like the permutation \mbox{\texttt{perm}} on \mbox{\texttt{range}}, resp. 
\item[{(e)}]  of \mbox{\texttt{R}} = $\mathbb{Z}$ with modulus \mbox{\texttt{modulus}} and values prescribed by the list \mbox{\texttt{val}}, which consists of 2$\cdot$\mbox{\texttt{modulus}} pairs giving preimage and image for 2 points per residue class (mod \mbox{\texttt{modulus}}), resp. 
\item[{(f)}]  of \mbox{\texttt{R}} = $\Z_{(\pi)}$ with modulus \texttt{Length(\mbox{\texttt{coeffs}})} and coefficients \mbox{\texttt{coeffs}} (the set of primes $\pi$ denoting the underlying ring is given as argument \mbox{\texttt{pi}}), resp. 
\item[{(g)}]  of \mbox{\texttt{R}} = GF(\mbox{\texttt{q}})[\mbox{\texttt{x}}] with modulus \mbox{\texttt{modulus}} and coefficients \mbox{\texttt{coeffs}}, resp. 
\item[{(h)}]  an arbitrary rcwa mapping which induces a bijection between the partitions \mbox{\texttt{P1}} and \mbox{\texttt{P2}} of \mbox{\texttt{R}} into disjoint single residue classes and which is affine on the elements of \mbox{\texttt{P1}}, resp. 
\item[{(i)}]  an arbitrary rcwa mapping with ``residue class cycles'' as given by \mbox{\texttt{cycles}}. The latter is a list of lists of disjoint residue classes which the mapping
should permute cyclically, each. 
\end{description}
 The methods for the operation \texttt{RcwaMapping} perform a number of argument checks, which can be skipped by using \texttt{RcwaMappingNC} instead. \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> f := RcwaMapping([[1,1,1],[1,-1,1],[1,1,1],[1,-1,1]]);
  <rcwa mapping of Z with modulus 2>
  gap> f = RcwaMapping((2,3),[2..3]);
  true
  gap> g := RcwaMapping((1,2,3)(8,9),[4..20]);
  <bijective rcwa mapping of Z with modulus 17, of order 2>
  gap> Action(Group(g),[4..20]);
  Group([ (5,6) ])
  gap> T = RcwaMapping(2,[[1,2],[2,1],[3,5],[4,2]]);
  true
  gap> t := RcwaMapping(1,[[-1,1],[1,-1]]); # The involution n -> -n.
  Rcwa mapping of Z: n -> -n
  gap> d := RcwaMapping([2],[[1/3,0,1]]);
  Rcwa mapping of Z_( 2 ): n -> 1/3 n
  gap> RcwaMapping([2,3],ShallowCopy(Coefficients(T)));
  <rcwa mapping of Z_( 2, 3 ) with modulus 2>
  gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);
  <rcwa mapping of Z with modulus 5>
  gap> x := Indeterminate(GF(2),1);; SetName(x,"x");
  gap> R := PolynomialRing(GF(2),1); z := Zero(R);; e := One(R);;
  GF(2)[x]
  gap> r := RcwaMapping( R, x^2 + e,
  >                      [ [ x^2 + x + e, z      , x^2 + e ],
  >                        [ x^2 + x + e, x      , x^2 + e ],
  >                        [ x^2 + x + e, x^2    , x^2 + e ],
  >                        [ x^2 + x + e, x^2 + x, x^2 + e ] ] );
  <rcwa mapping of GF(2)[x] with modulus x^2+Z(2)^0>
  gap> rc := function(r,m) return ResidueClass(DefaultRing(m),m,r); end;;
  gap> f1 := RcwaMapping([[rc(1,6),rc(0, 8)],[rc(5,6),rc(4, 8)]]);;
  gap> f2 := RcwaMapping([[rc(1,6),rc(0, 4)],[rc(5,6),rc(2, 4)]]);;
  gap> f3 := RcwaMapping([[rc(2,6),rc(1,12)],[rc(4,6),rc(7,12)]]);;
  gap> List([f1,f2,f3],Order);
  [ 2, 2, 2 ]
  gap> f := f1*f2*f3;
  <bijective rcwa mapping of Z with modulus 12>
  gap> Order(f);
  infinity
  gap> a = RcwaMapping([rc(0,2),rc(1,4),rc(3,4)],[rc(0,3),rc(1,3),rc(2,3)]);
  true
  gap> [rc(0,2),rc(1,4),rc(3,4)]^a;
  [ 0(3), 1(3), 2(3) ]
  
\end{Verbatim}
 }

 \pagebreak[4] \noindent In most cases an rcwa mapping is not determined uniquely by the output of the \texttt{ViewObj} method. In these cases the output is enclosed in brackets. There are methods
installed for \texttt{Display}, \texttt{Print} and \texttt{String}. The \texttt{Print}ed representation of an rcwa mapping is \textsf{GAP} - readable if and only if the \texttt{Print}ed representation of the elements of the underlying ring is so. There is also
a method for \texttt{LaTeXObj}: 

\subsection{\textcolor{Chapter }{LaTeXObj (f)}}
\logpage{[ 2, 2, 6 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{LaTeXObj( f )\index{LaTeXObj@\texttt{LaTeXObj}!f}
\label{LaTeXObj:f}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 A {\LaTeX} representation of the rcwa mapping~\mbox{\texttt{f}}. 



 The output makes use of the {\LaTeX} macro package \textsf{amsmath}. If the option \mbox{\texttt{Factorization}} is set, a factorization of \mbox{\texttt{f}} into class shifts, class reflections, class transpositions and prime switches
is printed (cp. \texttt{FactorizationIntoCSCRCT} (\ref{FactorizationIntoCSCRCT:g})). For rcwa mappings with modulus larger than~1, an indentation by \mbox{\texttt{Indentation}} characters can be specified by setting this option value accordingly. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Print(LaTeXObj(a));
  n \ \longmapsto \
  \begin{cases}
    \frac{3n}{2}     & \text{if} \ n \in 0(2), \\
    \frac{3n + 1}{4} & \text{if} \ n \in 1(4), \\
    \frac{3n - 1}{4} & \text{if} \ n \in 3(4).
  \end{cases}
  gap> Print(LaTeXObj(Comm(a,ClassShift(0,4)):Factorization));
        &\nu_{8(12)} \cdot \nu_{0(12)}^{-1}
   \cdot \tau_{0(12),6(12)} \cdot \tau_{0(12),4(12)}
   \cdot \tau_{0(12),8(12)}
  
\end{Verbatim}
 }

 The \texttt{Display} method recognizes the option \mbox{\texttt{xdvi}}. If this option is set, the given rcwa mapping is displayed in an \textsf{xdvi} window. For this purpose, the string returned by the \texttt{LaTeXObj} - method described above is inserted into a {\LaTeX} template file. This file is {\LaTeX}'ed, and the result is shown with \textsf{xdvi}. This works only on UNIX systems, and requires suitable installations of {\LaTeX} and \textsf{xdvi}. }

  
\section{\textcolor{Chapter }{Basic functionality for rcwa mappings}}\label{sec:BasicFunctionalityForRcwaMappings}
\logpage{[ 2, 3, 0 ]}
{
  Checking whether two rcwa mappings are equal is cheap. Rcwa mappings can be
multiplied, thus there is a method for \texttt{*}. Bijective rcwa mappings can also be inverted, thus there is a method for \texttt{Inverse}. The latter method is usually accessed by raising a mapping to some power
with negative exponent. Multiplying, inverting and computing powers of tame
rcwa mappings is cheap. Computing powers of wild mappings is usually expensive
-- runtime and memory requirements normally grow approximately exponentially
with the exponent. How expensive multiplying a couple of wild mappings is,
varies very much. In any case, the amount of memory required for storing an
rcwa mapping is proportional to its modulus. Whether a given mapping is tame
or wild can be determined by the operation \texttt{IsTame}. There are methods for \texttt{Order}, which can not only compute a finite order, but can also detect infinite
order. 

 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List([-6..6],k->Modulus(f^k)); Order(f);
  [ 324, 108, 108, 36, 36, 12, 1, 12, 24, 48, 96, 192, 384 ]
  infinity
  gap> List( [ a, u, f ], IsTame );
  [ false, false, false ]
  gap> f^2*u;         
  <bijective rcwa mapping of Z with modulus 120>
  gap> f^2*u*a^2*f^-1;
  <bijective rcwa mapping of Z with modulus 3840>
  gap> Comm(f,ClassShift(6,12)*f)^1000;
  <bijective rcwa mapping of Z with modulus 18>
  
\end{Verbatim}
 There are methods installed for \texttt{IsInjective}, \texttt{IsSurjective}, \texttt{IsBijective} and \texttt{Image}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> [ IsInjective(T), IsSurjective(T), IsBijective(u) ];
  [ false, true, true ]
  gap> Image(RcwaMapping([[-4,-8,1]]));
  0(4)
  
\end{Verbatim}
 Images of elements, of finite sets of elements and of unions of finitely many
residue classes of the source of an rcwa mapping can be computed with \texttt{{\textasciicircum}} (the same symbol as used for exponentiation and conjugation). The same works
for partitions of the source into a finite number of residue classes. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> [ 15^T, 7^d, (x^3+x^2+x+One(R))^r ];
  [ 23, 7/3, x^3+Z(2)^0 ]
  gap> A := ResidueClass(Integers,3,2);;
  gap> [ A^T, A^u ];
  [ 1(3) U 8(9), 1(9) U 3(9) U 14(27) U 20(27) U 26(27) ]
  gap> [rc(0,2),rc(1,4),rc(3,4)]^f;
  [ 0(6) U 1(6) U 5(6), 2(12) U 4(12) U 9(12), 3(12) U 8(12) U 10(12) ]
  
\end{Verbatim}
 For computing preimages of elements under rcwa mappings, there are methods for \texttt{PreImageElm} and \texttt{PreImagesElm}. The preimage of a finite set of ring elements or of a union of finitely many
residue classes under an rcwa mapping can be computed using \texttt{PreImage}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> [ PreImageElm(d,37/17), PreImagesElm(T,8), PreImagesElm(Zero(T),0) ];
  [ 111/17, [ 5, 16 ], Integers ]
  gap> PreImage(T,ResidueClass(Integers,3,2));
  Z \ 0(6) U 2(6)
  gap> M := [1];; l := [1];;
  gap> while Length(M) < 10000 do M := PreImage(T,M); Add(l,Length(M)); od; l;
  [ 1, 1, 2, 2, 4, 5, 8, 10, 14, 18, 26, 36, 50, 67, 89, 117, 157, 208, 277, 
    367, 488, 649, 869, 1154, 1534, 2039, 2721, 3629, 4843, 6458, 8608, 11472 ]
  
\end{Verbatim}
 There is a method for the operation \texttt{MovedPoints} for computing the support of a bijective rcwa mapping, and there is a method
for \texttt{RestrictedPerm} for computing the restriction of a bijective rcwa mapping to a union of
residue classes it fixes setwisely. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> [ MovedPoints(u), MovedPoints(u^2) ];
  [ Z \ [ -1, 0 ], Z \ [ -10, -6, -1, 0, 1, 2, 3, 5 ] ]
  gap> MovedPoints(r);
  GF(2)[x] \ [ 0*Z(2), Z(2)^0, x, x+Z(2)^0 ]
  gap> RestrictedPerm(f,ResidueClassUnion(Integers,36,[7,8]));
  <rcwa mapping of Z with modulus 36>
  
\end{Verbatim}
 Rcwa mappings can be added and subtracted pointwisely. However, please note
that the set of rcwa mappings of a ring does not form a ring under \texttt{+} and \texttt{*}. 

 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> a  := RcwaMapping([[3,0,2],[3,1,4],[3,0,2],[3,-1,4]]);;
  gap> b  := ClassShift(1,4) * a;;
  gap> [ Image((a + b)), Image((a - b)) ];
  [ 0(6) U 4(6) U 5(6), [ -3, 0 ] ]
  gap> d+d+d;
  IdentityMapping( Z_( 2 ) )
  
\end{Verbatim}
 There are operations \texttt{Modulus} (abbreviated \texttt{Mod}) and \texttt{Coefficients} for extracting the modulus resp. the coefficient list of a given rcwa mapping.
The meaning of the return values is as described in the previous section.
General documentation for most operations mentioned in this section can be
found in the \textsf{GAP} reference manual. For rcwa mappings of rings other than $\mathbb{Z}$, not for all operations applicable methods are available. 

 }

  
\section{\textcolor{Chapter }{Factoring rcwa mappings}}\label{sec:FactoringRcwaMappings}
\logpage{[ 2, 4, 0 ]}
{
  Factoring group elements into elements of some ``nice'' set of generators is often helpful. The following can be seen as an attempt
towards getting a satisfactory solution of this problem for the group RCWA($\Z$): 

\subsection{\textcolor{Chapter }{FactorizationIntoCSCRCT (g)}}
\logpage{[ 2, 4, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{FactorizationIntoCSCRCT( g )\index{FactorizationIntoCSCRCT@\texttt{FactorizationIntoCSCRCT}!g}
\label{FactorizationIntoCSCRCT:g}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Factorization( g )\index{Factorization@\texttt{Factorization}!g}
\label{Factorization:g}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 A factorization of the bijective rcwa mapping~\mbox{\texttt{g}} into class shifts, class reflections and class transpositions, provided that
such a factorization exists and the method finds it. 



 The method may return \texttt{fail}, stop with an error message or run into an infinite loop. If it returns a
result, this result is always correct. By default, prime switches are taken as
one factor. If the option \mbox{\texttt{ExpandPrimeSwitches}} is set, they are each decomposed into the 6 class transpositions given in the
definition (see \texttt{PrimeSwitch} (\ref{PrimeSwitch:p})). By default, the factoring process begins with splitting off factors from
the right. This can be changed by setting the option \mbox{\texttt{Direction}} to \texttt{"from the left"}. By default, a reasonably coarse respected partition of the integral mapping
occuring in the final stage of the algorithm is computed. This can be
suppressed by setting the option \mbox{\texttt{ShortenPartition}} equal to \texttt{false}. By default, at the end it is checked whether the product of the determined
factors indeed equals \mbox{\texttt{g}}. This check can be suppressed by setting the option \mbox{\texttt{NC}}.

 The problem of obtaining a factorization as desired is algorithmically
difficult, and this factorization routine is currently perhaps the most
sophisticated part of the \textsf{RCWA} package. Information about the progress of the factorization process can be
obtained by setting the info level of the Info class \texttt{InfoRCWA} (\ref{InfoRCWA}) to~2. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Factorization(Comm(a,b));
  [ ClassShift(7,9), ClassShift(1,9)^-1, ClassTransposition(1,9,4,9), 
    ClassTransposition(1,9,7,9), ClassTransposition(6,18,15,18), 
    ClassTransposition(5,9,15,18), ClassTransposition(4,9,15,18), 
    ClassTransposition(5,9,6,18), ClassTransposition(4,9,6,18) ]
  
\end{Verbatim}
 }

 For purposes of demonstrating the capabilities of the factorization routine,
in Section~\ref{sec:FactoringTheCollatzPermutation} a permutation is factored which has already been mentioned by Lothar Collatz
in~1932, and whose cycle structure is unknown so far. 

 Obtaining a factorization of a bijective rcwa mapping into class shifts, class
reflections and class transpositions is particularly difficult if multiplier
and divisor are coprime. A prototype of permutations which have this property
has been introduced in a different context in~\cite{Keller99}: 

\subsection{\textcolor{Chapter }{mKnot (m)}}
\logpage{[ 2, 4, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{mKnot( m )\index{mKnot@\texttt{mKnot}!m}
\label{mKnot:m}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 The permutation $g_m$ as introduced in~\cite{Keller99}. 



 The argument \mbox{\texttt{m}} must be an odd integer~$\geq~3$. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(mKnot(5));
  
  Wild bijective rcwa mapping of Z with modulus 5
  
                  n mod 5                |              n^mKnot(5)
  ---------------------------------------+--------------------------------------
    0                                    | 6n/5
    1                                    | (4n + 1)/5
    2                                    | (6n - 2)/5
    3                                    | (4n + 3)/5
    4                                    | (6n - 4)/5
  
  
\end{Verbatim}
 }

 \noindent In his article, Timothy P. Keller shows that a permutation of this type cannot
have infinitely many cycles of any given finite length. }

  
\section{\textcolor{Chapter }{Determinant and sign}}\label{sec:DeterminantAndSign}
\logpage{[ 2, 5, 0 ]}
{
  

\subsection{\textcolor{Chapter }{Determinant (sigma)}}
\logpage{[ 2, 5, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Determinant( sigma )\index{Determinant@\texttt{Determinant}!sigma}
\label{Determinant:sigma}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Determinant( sigma, S )\index{Determinant@\texttt{Determinant}!sigma, S}
\label{Determinant:sigma, S}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 The determinant of the bijective rcwa mapping~\mbox{\texttt{sigma}}. 



 The \emph{determinant} of an affine mapping $n \mapsto (an+b)/c$ whose source is a residue class $r(m)$ is defined by $b/|a|m$. This definition is extended additively to determinants of rcwa mappings and
their restrictions to unions of residue classes. 

 Using the notation from the definition of an rcwa mapping, the \emph{determinant} det($\sigma$) of an rcwa mapping $\sigma$ is given by   
\[ \frac{1}{m} \left( \sum_{r(m) \in R/mR} \frac{b_{r(m)}}{|a_{r(m)}|} \right). \]
  In the author's thesis it is shown that the determinant mapping is an
epimorphism from the group of all class-wise order-preserving bijective rcwa
mappings of $\Z$ onto $(\Z,+)$ (see Theorem~2.11.9). 

 If a residue class union~\mbox{\texttt{S}} is given as an additional argument, the method returns the determinant of the
restriction of \mbox{\texttt{sigma}} to~\mbox{\texttt{S}}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> nu := ClassShift(0,1);;
  gap> List( [ nu, a, b, u ], Determinant );
  [ 1, 0, 1, 0 ]
  gap> [ Determinant(u^2*b^-3), Determinant(nu^7*a^2*nu^-1*b^-1*a^-3) ];
  [ -3, 5 ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Sign (sigma)}}
\logpage{[ 2, 5, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Sign( sigma )\index{Sign@\texttt{Sign}!sigma}
\label{Sign:sigma}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The sign of the bijective rcwa mapping~\mbox{\texttt{sigma}}. 



 Using the notation from the definition of an rcwa mapping, the \emph{sign} of a bijective rcwa mapping~$\sigma$ of~$\Z$ is defined by   
\[ (-1)^{\displaystyle{{\rm det}(\sigma) + \frac{1}{m} \left( \sum_{r(m): \
a_{r(m)} < 0} (m - 2r) \right)}}. \]
  In the author's thesis it is shown that the sign mapping is an epimorphism
from RCWA($\Z$) to the group $\Z^\times$ of units of~$\Z$ (see Theorem~2.12.8). This means that the kernel of the sign mapping is a normal subgroup
of RCWA($\Z$) of index~2. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List( [ nu, nu^2, nu^3 ], Sign );
  [ -1, 1, -1 ]
  gap> List( [ t, nu^3*t ], Sign );
  [ -1, 1 ]
  gap> List( [ a, a*b, (a*b)^2, Comm(a,b) ], Sign );
  [ 1, -1, 1, 1 ]
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{Attributes and properties derived from the coefficients}}\label{sec:PropertiesDerivedFromCoefficients}
\logpage{[ 2, 6, 0 ]}
{
  

\subsection{\textcolor{Chapter }{Multiplier (f)}}
\logpage{[ 2, 6, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Multiplier( f )\index{Multiplier@\texttt{Multiplier}!f}
\label{Multiplier:f}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Mult( f )\index{Mult@\texttt{Mult}!f}
\label{Mult:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The multiplier of the rcwa mapping~\mbox{\texttt{f}}. 



 In the notation used in the definition of an rcwa mapping, the \emph{multiplier} is the lcm of the coefficients $a_{r(m)}$ in the numerators. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List( [ g, u, T, d, r ], Multiplier );
  [ 1, 9, 3, 1, x^2+x+Z(2)^0 ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Divisor (f)}}
\logpage{[ 2, 6, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Divisor( f )\index{Divisor@\texttt{Divisor}!f}
\label{Divisor:f}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Div( f )\index{Div@\texttt{Div}!f}
\label{Div:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The divisor of the rcwa mapping~\mbox{\texttt{f}}. 



 In the notation used in the definition of an rcwa mapping, the \emph{divisor} is the lcm of the coefficients $c_{r(m)}$ in the denominators. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List( [ g, u, T, d, r ], Divisor );
  [ 1, 5, 2, 1, x^2+Z(2)^0 ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{PrimeSet (f)}}
\logpage{[ 2, 6, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{PrimeSet( f )\index{PrimeSet@\texttt{PrimeSet}!f}
\label{PrimeSet:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The prime set of the rcwa mapping~\mbox{\texttt{f}}. 



 The \emph{prime set} of an rcwa mapping is the set of prime divisors of the product of its modulus,
its multiplier and its divisor. See also~\texttt{PrimeSet} (\ref{PrimeSet:G}) for rcwa groups. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> PrimeSet(T);
  [ 2, 3 ]
  gap> List( [ u, T^u, T^(u^-1) ], PrimeSet );
  [ [ 3, 5 ], [ 2, 3 ], [ 2, 3, 5 ] ]
  gap> PrimeSet(r);
  [ x+Z(2)^0, x^2+x+Z(2)^0 ]
  
\end{Verbatim}
 }

 \pagebreak[4] 

\subsection{\textcolor{Chapter }{IsIntegral (f)}}
\logpage{[ 2, 6, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsIntegral( f )\index{IsIntegral@\texttt{IsIntegral}!f}
\label{IsIntegral:f}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
 \texttt{true} if the rcwa mapping~\mbox{\texttt{f}} is integral and \texttt{false} otherwise. 



 An rcwa mapping is called \emph{integral} if its divisor equals~1, thus ``if no proper divisions occur''. Computing with such mappings is particularly easy. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List( [ u, t, RcwaMapping([[2,0,1],[3,5,1]]) ], IsIntegral );
  [ false, true, true ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IsClassWiseOrderPreserving (f)}}
\logpage{[ 2, 6, 5 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsClassWiseOrderPreserving( f )\index{IsClassWiseOrderPreserving@\texttt{IsClassWiseOrderPreserving}!f}
\label{IsClassWiseOrderPreserving:f}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
 \texttt{true} if the rcwa mapping~\mbox{\texttt{f}} is class-wise order-preserving and \texttt{false} otherwise. 



 The term \emph{class-wise order-preserving} is defined only for rcwa mappings of ordered rings, e.g.~$\mathbb{Z}$. In the notation introduced in the definition of an rcwa mapping, \mbox{\texttt{f}} is class-wise order-preserving if and only if all coefficients $a_{r(m)}$ in the numerators of the affine partial mappings are positive. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List( [ g, u, T, t, d ], IsClassWiseOrderPreserving );
  [ true, true, true, false, true ]
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{Functionality related to the affine partial mappings}}\label{sec:AffinePartialMappings}
\logpage{[ 2, 7, 0 ]}
{
  

\subsection{\textcolor{Chapter }{LargestSourcesOfAffineMappings (f)}}
\logpage{[ 2, 7, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{LargestSourcesOfAffineMappings( f )\index{LargestSourcesOfAffineMappings@\texttt{LargestSourcesOfAffineMappings}!f}
\label{LargestSourcesOfAffineMappings:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The coarsest partition of \texttt{Source(\mbox{\texttt{f}})} on whose elements the rcwa mapping~\mbox{\texttt{f}} is affine. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> LargestSourcesOfAffineMappings(T);
  [ 0(2), 1(2) ]
  gap> List( [ u, u^-1 ], LargestSourcesOfAffineMappings );
  [ [ 0(5), 1(5), 2(5), 3(5), 4(5) ], [ 0(3), 1(3), 2(9), 5(9), 8(9) ] ]
  gap> LargestSourcesOfAffineMappings(t);
  [ Integers ]
  gap> kappa := RcwaMapping([[1,0,1],[1,0,1],[3,2,2],[1,-1,1],
  >                          [2,0,1],[1,0,1],[3,2,2],[1,-1,1],
  >                          [1,1,3],[1,0,1],[3,2,2],[2,-2,1]]);;
  gap> SetName(kappa,"kappa");
  gap> LargestSourcesOfAffineMappings(kappa);
  [ 2(4), 1(4) U 0(12), 3(12) U 7(12), 4(12), 8(12), 11(12) ]
  gap> LargestSourcesOfAffineMappings(r);
  [ 0*Z(2) ( mod x^2+Z(2)^0 ), Z(2)^0 ( mod x^2+Z(2)^0 ), x ( mod x^2+Z(2)^0 ), 
    x+Z(2)^0 ( mod x^2+Z(2)^0 ) ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Multpk (f, p, k)}}
\logpage{[ 2, 7, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Multpk( f, p, k )\index{Multpk@\texttt{Multpk}!f, p, k}
\label{Multpk:f, p, k}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The union of the residue classes $r(m)$ such that $p^k||a_{r(m)}$ if $k \geq 0$, and the union of the residue classes $r(m)$ such that $p^k||c_{r(m)}$ if $k \leq 0$. In this context, $m$ denotes the modulus and $a_{r(m)}$ and $c_{r(m)}$ denote the coefficients of~\mbox{\texttt{f}} as introduced in the definition of an rcwa mapping. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> [ Multpk(T,2,-1), Multpk(T,3,1) ];
  [ Integers, 1(2) ]
  gap> [ Multpk(u,3,0), Multpk(u,3,1), Multpk(u,3,2), Multpk(u,5,-1) ];
  [ [  ], 0(5) U 2(5), Z \ 0(5) U 2(5), Integers ]
  gap> [ Multpk(kappa,2,1), Multpk(kappa,2,-1), Multpk(kappa,3,1),
  >      Multpk(kappa,3,-1) ];
  [ 4(12) U 11(12), 2(4), 2(4), 8(12) ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{SetOnWhichMappingIsClassWiseOrderPreserving (f)}}
\logpage{[ 2, 7, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{SetOnWhichMappingIsClassWiseOrderPreserving( f )\index{SetOnWhichMappingIsClassWiseOrderPreserving@\texttt{SetOnWhichMappingIsClassWiseOrderPreserving}!f}
\label{SetOnWhichMappingIsClassWiseOrderPreserving:f}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{SetOnWhichMappingIsClassWiseConstant( f )\index{SetOnWhichMappingIsClassWiseConstant@\texttt{SetOnWhichMappingIsClassWiseConstant}!f}
\label{SetOnWhichMappingIsClassWiseConstant:f}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{SetOnWhichMappingIsClassWiseOrderReversing( f )\index{SetOnWhichMappingIsClassWiseOrderReversing@\texttt{SetOnWhichMappingIsClassWiseOrderReversing}!f}
\label{SetOnWhichMappingIsClassWiseOrderReversing:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The union of the residue classes (mod~\texttt{Modulus(\mbox{\texttt{f}})}) on which the rcwa mapping~\mbox{\texttt{f}} is class-wise order-preserving, class-wise constant resp. class-wise
order-reversing. 



 The source of the rcwa mapping~\mbox{\texttt{f}} must be ordered. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List( [ T, u, t ], SetOnWhichMappingIsClassWiseOrderPreserving );
  [ Integers, Integers, [  ] ]
  gap> SetOnWhichMappingIsClassWiseConstant(RcwaMapping([[2,0,1],[0,4,1]]));
  1(2)
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{FixedPointsOfAffinePartialMappings (f)}}
\logpage{[ 2, 7, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{FixedPointsOfAffinePartialMappings( f )\index{FixedPointsOfAffinePartialMappings@\texttt{FixedPointsOfAffinePartialMappings}!f}
\label{FixedPointsOfAffinePartialMappings:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 A list of the sets of fixed points of the affine partial mappings of the rcwa
mapping~\mbox{\texttt{f}} in the quotient field of its source. 



 The returned list contains entries for the restrictions of~\mbox{\texttt{f}} to all residue classes modulo \texttt{Mod(\mbox{\texttt{f}})}. A list entry can either be an empty set, the source of~\mbox{\texttt{f}} or a set of cardinality~1. The ordering of the entries is the same as the one which is used in
coefficient lists. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> FixedPointsOfAffinePartialMappings(ClassShift(0,2));
  [ [  ], Rationals ]
  gap> List([1..3],k->FixedPointsOfAffinePartialMappings(T^k));
  [ [ [ 0 ], [ -1 ] ], [ [ 0 ], [ 1 ], [ 2 ], [ -1 ] ], 
    [ [ 0 ], [ -7 ], [ 2/5 ], [ -5 ], [ 4/5 ], [ 1/5 ], [ -10 ], [ -1 ] ] ]
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{Transition graphs and transition matrices}}\label{sec:TransGraphMat}
\logpage{[ 2, 8, 0 ]}
{
  

\subsection{\textcolor{Chapter }{TransitionGraph (f, m)}}
\logpage{[ 2, 8, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{TransitionGraph( f, m )\index{TransitionGraph@\texttt{TransitionGraph}!f, m}
\label{TransitionGraph:f, m}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The transition graph of the rcwa mapping~\mbox{\texttt{f}} for modulus~\mbox{\texttt{m}}. 



 \index{rcwa mapping@!transition graph} \noindent The \emph{transition graph} $\Gamma_{f,m}$ of~$f$ for modulus~$m$ is defined as follows: 
\begin{enumerate}
\item  The vertices are the residue classes (mod~$m$). 
\item  There is an edge from $r_1(m)$ to $r_2(m)$ if and only if there is some $n \in r_1(m)$ such that $n^f \in r_2(m)$. 
\end{enumerate}
 The assignment of the residue classes (mod~$m$) to the vertices of the graph is given by the ordering of the residues in \texttt{AllResidues(Source(\mbox{\texttt{f}}),\mbox{\texttt{m}})}. The result is returned in the format used by the package \textsf{GRAPE}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> TransitionGraph(a,Modulus(a));
  rec( isGraph := true, order := 4, group := Group(()), 
    schreierVector := [ -1, -2, -3, -4 ], 
    adjacencies := [ [ 1, 3 ], [ 1, 2, 3, 4 ], [ 2, 4 ], [ 1, 2, 3, 4 ] ], 
    representatives := [ 1, 2, 3, 4 ], names := [ 1, 2, 3, 4 ] )
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{OrbitsModulo (f, m)}}
\logpage{[ 2, 8, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{OrbitsModulo( f, m )\index{OrbitsModulo@\texttt{OrbitsModulo}!f, m}
\label{OrbitsModulo:f, m}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The partition of \texttt{AllResidues(Source(\mbox{\texttt{f}}),\mbox{\texttt{m}})} corresponding to the weakly-connected components of the transition graph of
the rcwa mapping~\mbox{\texttt{f}} for modulus~\mbox{\texttt{m}}. 



 See also \texttt{OrbitsModulo} (\ref{OrbitsModulo:G, m}) for rcwa groups. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> OrbitsModulo(Comm(a,b),9);
  [ [ 0 ], [ 1, 4, 5, 6, 7 ], [ 2 ], [ 3 ], [ 8 ] ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{FactorizationOnConnectedComponents (f, m)}}
\logpage{[ 2, 8, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{FactorizationOnConnectedComponents( f, m )\index{FactorizationOnConnectedComponents@\texttt{FactorizationOnConnectedComponents}!f, m}
\label{FactorizationOnConnectedComponents:f, m}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The set of restrictions of the rcwa mapping~\mbox{\texttt{f}} to the weakly-connected components of its transition graph $\Gamma_{f,m}$. 



 The product of the returned mappings is~\mbox{\texttt{f}}. They have pairwise disjoint supports, hence any two of them commute. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> sigma :=   ClassTransposition(1,4,2,4)  * ClassTransposition(1,4,3,4)
  >             * ClassTransposition(3,9,6,18) * ClassTransposition(1,6,3,9);;
  gap> List(FactorizationOnConnectedComponents(sigma,36),Support);
  [ 33(36) U 34(36) U 35(36), 9(36) U 10(36) U 11(36), 
    <union of 23 residue classes (mod 36)> \ [ -6, 3 ] ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{TransitionMatrix (f, m)}}
\logpage{[ 2, 8, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{TransitionMatrix( f, m )\index{TransitionMatrix@\texttt{TransitionMatrix}!f, m}
\label{TransitionMatrix:f, m}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The transition matrix of the rcwa mapping~\mbox{\texttt{f}} for modulus~\mbox{\texttt{m}}. 



 \index{rcwa mapping@!transition matrix} Let $M$ be this matrix. Then for any two residue classes $r_1(m), r_2(m) \in R/mR$, the entry $M_{r_1(m),r_2(m)}$ is defined by  
\[ M_{r_1(m),r_2(m)} \ := \ \displaystyle{\frac{|R/mR|}{|R/\hat{m}R|}} \cdot
\left|\left\{r(\hat{m}) \in R/\hat{m}R | \ r \in r_1(m) \wedge r^f \in
r_2(m)\right\}\right|, \]
 where $\hat{m}$ is the product of \mbox{\texttt{m}} and the square of the modulus of~\mbox{\texttt{f}}.   The assignment of the residue classes (mod~\mbox{\texttt{m}}) to the rows and columns of the matrix is given by the ordering of the
residues in \texttt{AllResidues(Source(\mbox{\texttt{f}}),\mbox{\texttt{m}})}. 

 The transition matrix is a weighted adjacency matrix of the corresponding
transition graph \texttt{TransitionGraph(\mbox{\texttt{f}},\mbox{\texttt{m}})}. The sums of the rows of a transition matrix are always equal to~1. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(TransitionMatrix(a,5));
  [ [  1/2,  1/4,    0,    0,  1/4 ],
    [    0,  1/4,    0,  1/4,  1/2 ],
    [  1/4,    0,    0,  3/4,    0 ],
    [  1/4,    0,  3/4,    0,    0 ],
    [    0,  1/2,  1/4,    0,  1/4 ] ]
  gap> Display(TransitionMatrix(T,19)*One(GF(7)));
   4 . . . . . . . . . 4 . . . . . . . .
   . . 4 . . . . . . . 4 . . . . . . . .
   . 4 . . . . . . . . . . . 4 . . . . .
   . . . . . 4 . . . . . 4 . . . . . . .
   . . 4 . . . . . . . . . . . . . 4 . .
   . . . . . . . . 4 . . . 4 . . . . . .
   4 . . 4 . . . . . . . . . . . . . . .
   . . . . . . . . . . . 4 . 4 . . . . .
   . . . 4 4 . . . . . . . . . . . . . .
   . . . . . . . . . . . . . . 1 . . . .
   . . . . . 4 4 . . . . . . . . . . . .
   . . . . . . . . . . . . . . . 4 . 4 .
   . . . . . . 4 . . 4 . . . . . . . . .
   . 4 . . . . . . . . . . . . . . 4 . .
   . . . . . . . 4 . . . . 4 . . . . . .
   . . . . 4 . . . . . . . . . . . . 4 .
   . . . . . . . . 4 . . . . . . 4 . . .
   . . . . . . . 4 . . . . . . . . . . 4
   . . . . . . . . . 4 . . . . . . . . 4
  
\end{Verbatim}
 }

 \pagebreak[4] 

\subsection{\textcolor{Chapter }{Sources (f)}}
\logpage{[ 2, 8, 5 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Sources( f )\index{Sources@\texttt{Sources}!f}
\label{Sources:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 A list of unions of residue classes modulo the modulus~$m$ of the rcwa mapping~\mbox{\texttt{f}}, as described below. 



 The returned list contains an entry for any strongly connected component of
the transition graph of~\mbox{\texttt{f}} for modulus~$m$ which has only outgoing edges. The list entry corresponding to a given such
strongly connected component is the union of the vertices which belong to the
respective component. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> [ Sources(kappa), Sources(a) ];
  [ [  ], [  ] ]
  gap> Sources(nu*nu^a);
  [ 2(6) ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Sinks (f)}}
\logpage{[ 2, 8, 6 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Sinks( f )\index{Sinks@\texttt{Sinks}!f}
\label{Sinks:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 A list of unions of residue classes modulo the modulus~$m$ of the rcwa mapping~\mbox{\texttt{f}}, as described below. 



 The returned list contains an entry for any strongly connected component of
the transition graph of~\mbox{\texttt{f}} for modulus~$m$ which has only ingoing edges. The list entry corresponding to a given such
strongly connected component is the union of the vertices which belong to the
respective component. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> [ Sinks(kappa), Sinks(a) ];
  [ [  ], [  ] ]
  gap> Sinks(nu*nu^a);
  [ 3(6) ]
  gap> Sinks(Product(List([[1,4,2,4],[1,4,3,4],[3,6,7,12],[2,4,3,6]],
  >                       ClassTransposition)));
  [ 3(6) U 1(12) ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Loops (f)}}
\logpage{[ 2, 8, 7 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Loops( f )\index{Loops@\texttt{Loops}!f}
\label{Loops:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The list of non-isolated vertices of the transition graph of the rcwa mapping~\mbox{\texttt{f}} for modulus \texttt{Modulus(\mbox{\texttt{f}})} which carry a loop. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Loops(kappa);
  [ 10(12) ]
  gap> Loops(a);
  [ 0(4), 1(4), 3(4) ]
  gap> Loops(nu*nu^a);
  [ 2(6), 3(6) ]
  
\end{Verbatim}
 }

 }

 \pagebreak[4]  
\section{\textcolor{Chapter }{Trajectories}}\label{sec:Trajectories}
\logpage{[ 2, 9, 0 ]}
{
  

\subsection{\textcolor{Chapter }{Trajectory (f, n, length)}}
\logpage{[ 2, 9, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Trajectory( f, n, length )\index{Trajectory@\texttt{Trajectory}!f, n, length}
\label{Trajectory:f, n, length}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Trajectory( f, n, length, m )\index{Trajectory@\texttt{Trajectory}!f, n, length, m}
\label{Trajectory:f, n, length, m}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Trajectory( f, n, terminal )\index{Trajectory@\texttt{Trajectory}!f, n, terminal}
\label{Trajectory:f, n, terminal}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Trajectory( f, n, terminal, m )\index{Trajectory@\texttt{Trajectory}!f, n, terminal, m}
\label{Trajectory:f, n, terminal, m}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 The first \mbox{\texttt{length}} iterates in the trajectory of the rcwa mapping~\mbox{\texttt{f}} starting at~\mbox{\texttt{n}}, resp. the initial part of the trajectory of the rcwa mapping~\mbox{\texttt{f}} starting at~\mbox{\texttt{n}} which ends at the first occurence of an iterate in the set \mbox{\texttt{terminal}}. If the argument \mbox{\texttt{m}} is given, the iterates are reduced (mod~\mbox{\texttt{m}}). 



 To save memory when computing long trajectories containing huge iterates, the
reduction (mod~\mbox{\texttt{m}}) is done immediately after any iteration. In place of the ring element~\mbox{\texttt{n}}, the methods also accept a finite set of ring elements or a union of residue
classes. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Trajectory(T,27,16); Trajectory(T,27,25,5);
  [ 27, 41, 62, 31, 47, 71, 107, 161, 242, 121, 182, 91, 137, 206, 103, 155 ]
  [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 0, 3, 0, 0, 3, 0, 3, 0, 0, 3 ]
  gap> Trajectory(T,15,[1]); Trajectory(T,15,[1],2);
  [ 15, 23, 35, 53, 80, 40, 20, 10, 5, 8, 4, 2, 1 ]
  [ 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1 ]
  gap> Trajectory(T,ResidueClass(Integers,3,0),Integers);
  [ 0(3), 0(3) U 5(9), 0(3) U 5(9) U 7(9) U 8(27), 
    <union of 20 residue classes (mod 27)>, <union of 73 residue classes (mod 
      81)>, Z \ 10(81) U 37(81), Integers ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Trajectory (f, n, length, whichcoeffs)}}
\logpage{[ 2, 9, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Trajectory( f, n, length, whichcoeffs )\index{Trajectory@\texttt{Trajectory}!f, n, length, whichcoeffs}
\label{Trajectory:f, n, length, whichcoeffs}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Trajectory( f, n, terminal, whichcoeffs )\index{Trajectory@\texttt{Trajectory}!f, n, terminal, whichcoeffs}
\label{Trajectory:f, n, terminal, whichcoeffs}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 Either the list \texttt{c} of triples of coprime coefficients such that for any~\texttt{k} it holds that \texttt{\mbox{\texttt{n}}{\textasciicircum}(\mbox{\texttt{f}}{\textasciicircum}(k-1)) = (c[k][1]*\mbox{\texttt{n}} + c[k][2])/c[k][3]} or the last entry of that list, depending on whether \mbox{\texttt{whichcoeffs}} is \texttt{"AllCoeffs"} or \texttt{"LastCoeffs"}. 



 The meanings of the arguments \mbox{\texttt{length}} and \mbox{\texttt{terminal}} are the same as in the methods for the operation \texttt{Trajectory} described above. In general, computing only the last coefficient triple (\mbox{\texttt{whichcoeffs}} = \texttt{"LastCoeffs"}) needs considerably less memory than computing the entire list. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Trajectory(T,27,[1],"LastCoeffs");
  [ 36472996377170786403, 195820718533800070543, 1180591620717411303424 ]
  gap> (last[1]*27+last[2])/last[3];
  1
  gap> Trajectory(r,x^3,[x^3+x^2+x],"AllCoeffs");
  [ [ Z(2)^0, 0*Z(2), Z(2)^0 ], [ x^2+x+Z(2)^0, x^2, x^2+Z(2)^0 ],
    [ x^4+x^2+Z(2)^0, x, x^4+Z(2)^0 ],
    [ x^6+x^5+x^3+x+Z(2)^0, x^3+x^2+x, x^6+x^4+x^2+Z(2)^0 ] ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IncreasingOn (f)}}
\logpage{[ 2, 9, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IncreasingOn( f )\index{IncreasingOn@\texttt{IncreasingOn}!f}
\label{IncreasingOn:f}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{DecreasingOn( f )\index{DecreasingOn@\texttt{DecreasingOn}!f}
\label{DecreasingOn:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The union of all residue classes $r(m)$ such that $|R/a_{r(m)}R| > |R/c_{r(m)}R|$ resp. $|R/a_{r(m)}R| < |R/c_{r(m)}R|$, where $R$ denotes the source, $m$ the modulus and $a_{r(m)}$, $b_{r(m)}$ and $c_{r(m)}$ the coefficients of~\mbox{\texttt{f}} as introduced in the definition of an rcwa mapping. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List([1..3],k->IncreasingOn(T^k));
  [ 1(2), 3(4), 3(4) U 1(8) U 6(8) ]
  gap> List([1..3],k->DecreasingOn(T^k));
  [ 0(2), Z \ 3(4), 0(4) U 2(8) U 5(8) ]
  gap> List([1..3],k->IncreasingOn(a^k));
  [ 0(2), Z \ 1(8) U 7(8), 0(4) U 2(16) U 5(16) U 11(16) U 14(16) ]
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{ Localizations of rcwa mappings of the integers }}\label{sec:Localizations}
\logpage{[ 2, 10, 0 ]}
{
  

\subsection{\textcolor{Chapter }{LocalizedRcwaMapping (f, p)}}
\logpage{[ 2, 10, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{LocalizedRcwaMapping( f, p )\index{LocalizedRcwaMapping@\texttt{LocalizedRcwaMapping}!f, p}
\label{LocalizedRcwaMapping:f, p}
}\hfill{\scriptsize (function)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{SemilocalizedRcwaMapping( f, pi )\index{SemilocalizedRcwaMapping@\texttt{SemilocalizedRcwaMapping}!f, pi}
\label{SemilocalizedRcwaMapping:f, pi}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 The rcwa mapping of $\Z_{(p)}$ resp. $\Z_{(\pi)}$ with the same coefficients as the rcwa mapping \mbox{\texttt{f}} of~$\Z$. 



 The argument \mbox{\texttt{p}} resp. \mbox{\texttt{pi}} must be a prime resp. a set of primes, and the argument~\mbox{\texttt{f}} must be an rcwa mapping of~$\Z$ whose modulus is a power of~\mbox{\texttt{p}}, resp. whose modulus has only prime divisors which lie in~\mbox{\texttt{pi}}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Cycle(LocalizedRcwaMapping(T,2),131/13);
  [ 131/13, 203/13, 311/13, 473/13, 716/13, 358/13, 179/13, 275/13, 419/13, 
    635/13, 959/13, 1445/13, 2174/13, 1087/13, 1637/13, 2462/13, 1231/13, 
    1853/13, 2786/13, 1393/13, 2096/13, 1048/13, 524/13, 262/13 ]
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{ Extracting roots of rcwa mappings }}\label{sec:ExtractingRoots}
\logpage{[ 2, 11, 0 ]}
{
  

\subsection{\textcolor{Chapter }{Root (f, k)}}
\logpage{[ 2, 11, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Root( f, k )\index{Root@\texttt{Root}!f, k}
\label{Root:f, k}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 An rcwa mapping \texttt{g} such that \texttt{g{\textasciicircum}\mbox{\texttt{k}}=\mbox{\texttt{f}}}, provided that such a mapping exists and that there is a method available
which can determine it. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Root(Comm(a,b),3)^3 = Comm(a,b);
  true
  
\end{Verbatim}
 }

 }

 \pagebreak[4]  
\section{\textcolor{Chapter }{ Special functions for non-bijective mappings }}\label{sec:NonBijectiveMappings}
\logpage{[ 2, 12, 0 ]}
{
  

\subsection{\textcolor{Chapter }{RightInverse (f)}}
\logpage{[ 2, 12, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RightInverse( f )\index{RightInverse@\texttt{RightInverse}!f}
\label{RightInverse:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 A right inverse of the injective rcwa mapping~\mbox{\texttt{f}}, i.e. a mapping $g$ such that \mbox{\texttt{f}}$g$~=~1. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> RcwaMapping([[2,0,1]]); Display(RightInverse(last));
  Rcwa mapping of Z: n -> 2n
  
  Rcwa mapping of Z with modulus 2
  
                  n mod 2                |                 n^f
  ---------------------------------------+--------------------------------------
    0                                    | n/2
    1                                    | n
  
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{CommonRightInverse (l, r)}}
\logpage{[ 2, 12, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{CommonRightInverse( l, r )\index{CommonRightInverse@\texttt{CommonRightInverse}!l, r}
\label{CommonRightInverse:l, r}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 A mapping $d$ such that \mbox{\texttt{l}}$d$ = \mbox{\texttt{r}}$d$ = 1. 



 The mappings \mbox{\texttt{l}} and \mbox{\texttt{r}} must be injective, and their images must form a partition of their source. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(CommonRightInverse(RcwaMapping([[2,0,1]]),RcwaMapping([[2,1,1]])));
  
  Rcwa mapping of Z with modulus 2
  
                  n mod 2                |                 n^f
  ---------------------------------------+--------------------------------------
    0                                    | n/2
    1                                    | (n - 1)/2
  
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{ImageDensity (f)}}
\logpage{[ 2, 12, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{ImageDensity( f )\index{ImageDensity@\texttt{ImageDensity}!f}
\label{ImageDensity:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The \emph{image density} of the rcwa mapping~\mbox{\texttt{f}}. 



 In the notation introduced in the definition of an rcwa mapping, the \emph{image density} of an rcwa mapping~$f$ is defined by $\frac{1}{m}$ $\sum_{r(m) \in R/mR} |R/c_{r(m)}R|/|R/a_{r(m)}R|$. The image density of an injective rcwa mapping is $\leq 1$, and the image density of a surjective rcwa mapping is $\geq 1$ (this can be seen easily). Thus in particular the image density of a bijective
rcwa mapping is~1. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List( [ T, a, RcwaMapping([[2,0,1]]) ], ImageDensity );
  [ 4/3, 1, 1/2 ]
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{ Probabilistic guesses on the behaviour of trajectories }}\label{sec:BehaviourOfTrajectories}
\logpage{[ 2, 13, 0 ]}
{
  This section describes some functionality for getting ``educated guesses'' concerning the overall behaviour of the trajectories of a given rcwa mapping.
Its contents have deliberately been separated from the documentation of the
non-probabilistic functionality related to trajectories of rcwa mappings. 

\subsection{\textcolor{Chapter }{LikelyContractionCentre (f, maxn, bound)}}
\logpage{[ 2, 13, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{LikelyContractionCentre( f, maxn, bound )\index{LikelyContractionCentre@\texttt{LikelyContractionCentre}!f, maxn, bound}
\label{LikelyContractionCentre:f, maxn, bound}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 A list of ring elements (see below). 



 This operation tries to compute the \emph{contraction centre} of the rcwa mapping \mbox{\texttt{f}}. Assuming its existence this is the uniquely-determined finite subset $S_0$ of the source of~\mbox{\texttt{f}} on which \mbox{\texttt{f}} induces a permutation and which intersects nontrivially with any trajectory of~\mbox{\texttt{f}}. The mapping~\mbox{\texttt{f}} is assumed to be \emph{contracting}, i.e. to have such a contraction centre. As in general contraction centres
are likely not computable, the methods for this operation are probabilistic
and may return wrong results. The argument \mbox{\texttt{maxn}} is a bound on the starting value and \mbox{\texttt{bound}} is a bound on the elements of the trajectories to be searched. If the limit \mbox{\texttt{bound}} is exceeded, an Info message on Info level~3 of \texttt{InfoRCWA} is given. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> S0 := LikelyContractionCentre(T,100,1000);
  #I  Warning: `LikelyContractionCentre' is highly probabilistic.
  The returned result can only be regarded as a rough guess.
  See ?LikelyContractionCentre for information on how to improve this guess.
  [ -136, -91, -82, -68, -61, -55, -41, -37, -34, -25, -17, -10, -7, -5, -1, 0, 
    1, 2 ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{GuessedDivergence (f)}}
\logpage{[ 2, 13, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{GuessedDivergence( f )\index{GuessedDivergence@\texttt{GuessedDivergence}!f}
\label{GuessedDivergence:f}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 A floating point value which is intended to be a rough guess on how fast the
trajectories of the rcwa mapping~\mbox{\texttt{f}} diverge (return value greater than~1) or converge (return value smaller than~1). 



 Nothing particular is guaranteed. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List( [ T, a ], GuessedDivergence );
  #I  Warning: GuessedDivergence: no particular return value is guaranteed.
  #I  Warning: GuessedDivergence: no particular return value is guaranteed.
  [ 0.866025, 1.06066 ]
  
\end{Verbatim}
 }

 }

 \pagebreak[4]  
\section{\textcolor{Chapter }{The categories and families of rcwa mappings}}\label{sec:CategoriesOfRcwaMappings}
\logpage{[ 2, 14, 0 ]}
{
  

\subsection{\textcolor{Chapter }{IsRcwaMapping (f)}}
\logpage{[ 2, 14, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsRcwaMapping( f )\index{IsRcwaMapping@\texttt{IsRcwaMapping}!f}
\label{IsRcwaMapping:f}
}\hfill{\scriptsize (filter)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsRcwaMappingOfZ( f )\index{IsRcwaMappingOfZ@\texttt{IsRcwaMappingOfZ}!f}
\label{IsRcwaMappingOfZ:f}
}\hfill{\scriptsize (filter)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsRcwaMappingOfZ{\_}pi( f )\index{IsRcwaMappingOfZpi@\texttt{IsRcwaMappingOfZ{\_}pi}!f}
\label{IsRcwaMappingOfZpi:f}
}\hfill{\scriptsize (filter)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsRcwaMappingOfGFqx( f )\index{IsRcwaMappingOfGFqx@\texttt{IsRcwaMappingOfGFqx}!f}
\label{IsRcwaMappingOfGFqx:f}
}\hfill{\scriptsize (filter)}}\\
\textbf{\indent Returns:\ }
 \texttt{true} if \mbox{\texttt{f}} is an rcwa mapping resp. an rcwa mapping of the ring of integers resp. an rcwa
mapping of a semilocalization of the ring of integers resp. an rcwa mapping of
a polynomial ring in one variable over a finite field, and \texttt{false} otherwise. 

}

 

\subsection{\textcolor{Chapter }{RcwaMappingsFamily (R)}}
\logpage{[ 2, 14, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RcwaMappingsFamily( R )\index{RcwaMappingsFamily@\texttt{RcwaMappingsFamily}!R}
\label{RcwaMappingsFamily:R}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 The family of rcwa mappings of the ring~\mbox{\texttt{R}}. 

}

 }

  }

         
\chapter{\textcolor{Chapter }{Residue Class-Wise Affine Groups}}\label{ch:RcwaGroups}
\logpage{[ 3, 0, 0 ]}
{
  This chapter describes the functionality provided by this package for
computing with residue class-wise affine groups.  
\section{\textcolor{Chapter }{Constructing residue class-wise affine groups}}\label{sec:ContructingRcwaGroups}
\logpage{[ 3, 1, 0 ]}
{
  Residue class-wise affine groups can be constructed using either \texttt{Group}, \texttt{GroupByGenerators} or \texttt{GroupWithGenerators}, as usual (see the \textsf{GAP} reference manual). 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> g := RcwaMapping([[1,0,1],[1,1,1],[3,6,1],
  >                      [1,0,3],[1,1,1],[3,6,1],
  >                      [1,0,1],[1,1,1],[3,-21,1]]);;
  gap> h := RcwaMapping([[1,0,1],[1,1,1],[3,6,1],
  >                      [1,0,3],[1,1,1],[3,-21,1],
  >                      [1,0,1],[1,1,1],[3,6,1]]);;
  gap> List([g,h],Order);
  [ 9, 9 ]
  gap> G := Group(g,h);
  <rcwa group over Z with 2 generators>
  gap> Size(G);
  infinity
  
\end{Verbatim}
 There are methods for the operations \texttt{Display} and \texttt{Print} which are applicable to rcwa groups. 

 All rcwa groups over the ring $R$ are subgroups of RCWA($R$). The group RCWA($R$) is not finitely generated, thus cannot be constructed in the way described
above. It is handled as a special case: 

\subsection{\textcolor{Chapter }{RCWA (R)}}
\logpage{[ 3, 1, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RCWA( R )\index{RCWA@\texttt{RCWA}!R}
\label{RCWA:R}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 The group RCWA(\mbox{\texttt{R}}) of all residue class-wise affine permutations of the ring~\mbox{\texttt{R}}. 



 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> RCWA_Z := RCWA(Integers);
  RCWA(Z)
  gap> Size(RCWA_Z);
  infinity
  gap> IsFinitelyGeneratedGroup(RCWA_Z);
  false
  gap> One(RCWA_Z);
  IdentityMapping( Integers )
  gap> IsSolvable(RCWA_Z);
  false
  gap> IsPerfect(RCWA_Z);
  false
  gap> Centre(RCWA_Z);
  Trivial rcwa group over Z
  gap> IsSubgroup(RCWA_Z, Group(RcwaMapping((1,2,3),[1..4]),
  >                             RcwaMapping(2,[[0,1],[1,0],[2,3],[3,2]])));
  true
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Random (RCWA( Integers ))}}
\logpage{[ 3, 1, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Random( RCWA{\_}Z )\index{Random@\texttt{Random}!RCWA( Integers )}
\label{Random:RCWA( Integers )}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 A pseudo-random element of the group RCWA($\mathbb{Z}$). 



 This method is designed to be suitable for generating interesting examples. No
particular distribution is guaranteed -- in fact, the author has no idea what
a ``reasonable'' random distribution on RCWA($\mathbb{Z}$) should be. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> elm := Random(RCWA_Z);
  <bijective rcwa mapping of Z with modulus 60>
  gap> Display(elm);
  
  Bijective rcwa mapping of Z with modulus 60
  
                 n mod 60                |                 n^f
  ---------------------------------------+--------------------------------------
     0  4  6  8 10 14 16 18 20 24 26 28  | 
    30 34 36 38 40 44 46 48 50 54 56 58  | -3n - 5
     1                                   | (n - 1)/5
     2 22 42                             | (3n + 24)/5
     3  9 15 21 27 33 39 45 51 57        | (-5n + 12)/3
     5 11 17 29 35 41 47 59              | -n + 2
     7 13 19 25 37 43 49 55              | -n
    12 32 52                             | (3n + 4)/5
    23                                   | (n - 3)/5
    31                                   | (n + 19)/5
    53                                   | (n + 17)/5
  
  
\end{Verbatim}
 The elements returned by this method are obtained by multiplying class shifts
(see \texttt{ClassShift} (\ref{ClassShift:r, m})), class reflections (see \texttt{ClassReflection} (\ref{ClassReflection:r, m})) and class transpositions (see \texttt{ClassTransposition} (\ref{ClassTransposition:r1, m1, r2, m2})). These factors are stored as an attribute value: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Perform(FactorizationIntoCSCRCT(elm),Display);
  
  Rcwa mapping of Z with modulus 6
  
                  n mod 6                |    n^ClassTransposition(0,2,5,6)
  ---------------------------------------+--------------------------------------
    0 2 4                                | 3n + 5
    1 3                                  | n
    5                                    | (n - 5)/3
  
  
  Rcwa mapping of Z with modulus 6
  
                  n mod 6                |    n^ClassTransposition(0,2,3,6)
  ---------------------------------------+--------------------------------------
    0 2 4                                | 3n + 3
    1 5                                  | n
    3                                    | (n - 3)/3
  
  
  Rcwa mapping of Z with modulus 10
  
                 n mod 10                |    n^ClassTransposition(0,2,1,10)
  ---------------------------------------+--------------------------------------
     0  2  4  6  8                       | 5n + 1
     1                                   | (n - 1)/5
     3  5  7  9                          | n
  
  
  Rcwa mapping of Z with modulus 4
  
                  n mod 4                |          n^ClassShift(2,4)
  ---------------------------------------+--------------------------------------
    0 1 3                                | n
    2                                    | n + 4
  
  Rcwa mapping of Z: n -> -n
  
  Rcwa mapping of Z with modulus 2
  
                  n mod 2                |        n^ClassReflection(0,2)
  ---------------------------------------+--------------------------------------
    0                                    | -n
    1                                    | n
  
  
\end{Verbatim}
 }

 \pagebreak[4] \noindent Another way of constructing an rcwa group is taking the image of an rcwa
representation: 

\subsection{\textcolor{Chapter }{IsomorphismRcwaGroupOverZ (G)}}
\logpage{[ 3, 1, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsomorphismRcwaGroupOverZ( G )\index{IsomorphismRcwaGroupOverZ@\texttt{IsomorphismRcwaGroupOverZ}!G}
\label{IsomorphismRcwaGroupOverZ:G}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsomorphismRcwaGroup( G )\index{IsomorphismRcwaGroup@\texttt{IsomorphismRcwaGroup}!G}
\label{IsomorphismRcwaGroup:G}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 A monomorphism from the group \mbox{\texttt{G}} to~RCWA($\Z$). 



 Currently, \texttt{IsomorphismRcwaGroup} works for finite groups, for free products of finite groups and for free
groups. The method for free products of finite groups uses the Table-Tennis
Lemma (cp. e.g. Section~II.B. in~\cite{LaHarpe00}), and the method for free groups uses an adaptation of the construction given
on page~27 in~\cite{LaHarpe00} from PSL(2,$\mathbb{C}$) to RCWA($\mathbb{Z}$). 

 In case \mbox{\texttt{G}} is a finite-degree permutation group, the image under a specific embedding can
be obtained by \texttt{RcwaGroupByPermGroup(\mbox{\texttt{G}})}. The resulting group~\texttt{H} satisfies the relation \texttt{Action(H{\textasciicircum}ClassShift(0,1),[1..LargestMovedPoint(\mbox{\texttt{G}})]) =~G}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> F := FreeProduct(Group((1,2)(3,4),(1,3)(2,4)),Group((1,2,3)),
  >                     SymmetricGroup(3));
  <fp group on the generators [ f1, f2, f3, f4, f5 ]>
  gap> phi := IsomorphismRcwaGroup(F);
  [ f1, f2, f3, f4, f5 ] -> [ <bijective rcwa mapping of Z with modulus 12>,
    <bijective rcwa mapping of Z with modulus 24>,
    <bijective rcwa mapping of Z with modulus 12>,
    <bijective rcwa mapping of Z with modulus 72>,
    <bijective rcwa mapping of Z with modulus 36> ]
  gap> G := Image(phi);
  <wild rcwa group over Z with 5 generators>
  gap> RelatorsOfFpGroup(F); # For illustrational purposes, do some checks:
  [ f1^2, f1^-1*f2*f1*f2^-1, f2^2, f3^3, f4^2, f5^2, f4*f5*f4*f5*f4*f5 ]
  gap> ForAll([ G.1^2, G.1^-1*G.2*G.1*G.2^-1, G.2^2, G.3^3,
  >             G.4^2, G.5^2, (G.4*G.5)^3 ], IsOne);
  true
  gap> S := AllResidueClassesModulo(3);
  [ 0(3), 1(3), 2(3) ]
  gap> nonids := grp->Difference(AsList(grp),[One(grp)]);;
  gap> List(S{[2,3]},Si->List(nonids(Group(G.1,G.2)),g->Si^g));
  [ [ 3(12), 6(24), 0(24) ], [ 9(12), 18(24), 12(24) ] ]
  gap> Union(Flat(last));
  0(3)
  gap> List(S{[1,3]},Si->List(nonids(Group(G.3)),g->Si^g));
  [ [ 1(12), 4(12) ], [ 7(12), 10(12) ] ]
  gap> Union(Flat(last));
  1(3)
  gap> List(S{[1,2]},Si->List(nonids(Group(G.4,G.5)),g->Si^g));
  [ [ 8(24), 5(36), 17(36), 2(24), 11(36) ],
    [ 20(24), 23(36), 35(36), 14(24), 29(36) ] ]
  gap> Union(Flat(last));
  2(3)
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> phi := IsomorphismRcwaGroup(FreeGroup(2)); F2 := Image(phi);;
  [ f1, f2 ] -> [ <wild bijective rcwa mapping of Z with modulus 8>, 
    <wild bijective rcwa mapping of Z with modulus 8> ]
  gap> Difference(Integers,ResidueClass(0,4))^F2.1;
  1(4)
  gap> Difference(Integers,ResidueClass(2,4))^F2.2;
  3(4)
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{Attributes and properties of rcwa groups}}\label{sec:AttributesAndProperties}
\logpage{[ 3, 2, 0 ]}
{
  

\subsection{\textcolor{Chapter }{Modulus (G)}}
\logpage{[ 3, 2, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Modulus( G )\index{Modulus@\texttt{Modulus}!G}
\label{Modulus:G}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Mod( G )\index{Mod@\texttt{Mod}!G}
\label{Mod:G}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 The modulus of the rcwa group~\mbox{\texttt{G}}. 



 The \emph{Modulus} of an rcwa group is the lcm of the moduli of its elements in case such an lcm
exists and 0 otherwise. 

 See also \texttt{IsTame} (\ref{IsTame:G}). 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> g1 := RcwaMapping((1,2),[1..2]);
  <bijective rcwa mapping of Z with modulus 2, of order 2>
  gap> g2 := RcwaMapping((1,2,3),[1..3]);
  <bijective rcwa mapping of Z with modulus 3, of order 3>
  gap> g3 := RcwaMapping((1,2,3,4,5),[1..5]);
  <bijective rcwa mapping of Z with modulus 5, of order 5>
  gap> G := Group(g1,g2,g3);
  <rcwa group over Z with 3 generators>
  gap> Modulus(G);
  30
  gap> a := RcwaMapping([[3,0,2],[3,1,4],[3,0,2],[3,-1,4]]);; SetName(a,"a");
  gap> Modulus(Group(a));
  0
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IsTame (G)}}
\logpage{[ 3, 2, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsTame( G )\index{IsTame@\texttt{IsTame}!G}
\label{IsTame:G}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
 \texttt{true} if the rcwa group~\mbox{\texttt{G}} is tame and \texttt{false} otherwise. 



 An rcwa group is called \emph{tame} if its modulus is not equal to~0. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> b := ClassShift(1,4) * a;; SetName(b,"b");
  gap> c := ClassShift(3,4) * a;; SetName(c,"c");
  gap> List( [ G, Group(a,b), Group(Comm(a,b),Comm(a,c)) ], IsTame );
  [ true, false, true ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{PrimeSet (G)}}
\logpage{[ 3, 2, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{PrimeSet( G )\index{PrimeSet@\texttt{PrimeSet}!G}
\label{PrimeSet:G}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The prime set of the rcwa group~\mbox{\texttt{G}}. 



 The \emph{prime set} of an rcwa group is the union of the prime sets of its elements. 

 See also~\texttt{PrimeSet} (\ref{PrimeSet:f}) for rcwa mappings. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> PrimeSet(G);
  [ 2, 3, 5 ]
  
\end{Verbatim}
 }

 An rcwa group is called \emph{integral} resp. \emph{class-wise order-preserving} if all of its elements are so. There are corresponding methods available for \texttt{IsIntegral} and \texttt{IsClassWiseOrderPreserving}. }

  
\section{\textcolor{Chapter }{ Membership testing, order computation, permutation- / matrix representations }}\label{sec:MembershipOrderMatPerm}
\logpage{[ 3, 3, 0 ]}
{
  \index{rcwa group@!element testing} 

\subsection{\textcolor{Chapter }{\texttt{\symbol{92}}in (g, G)}}
\logpage{[ 3, 3, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{\texttt{\symbol{92}}in( g, G )\index{in@\texttt{\texttt{\symbol{92}}in}!g, G}
\label{in:g, G}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 \texttt{true} if the rcwa mapping~\mbox{\texttt{g}} is an element of the rcwa group~\mbox{\texttt{G}} and \texttt{false} if not. 



 This method tries to decide whether \mbox{\texttt{g}} is an element of~\mbox{\texttt{G}} or not. It can always decide this question if \mbox{\texttt{G}} is tame and class-wise order-preserving. For wild groups only a number of easy
cases are covered. On Info level~3 of \texttt{InfoRCWA} the method gives information on reasons why \mbox{\texttt{g}} is an element of~\mbox{\texttt{G}} or~not. 

 The direct product of two free groups of rank~2 can faithfully be represented as an rcwa group. In~\cite{Mihailova58} it is shown that this implies that in general the membership problem for rcwa
groups is algorithmically undecidable. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
  gap> u in G;
  false
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Size (G)}}
\logpage{[ 3, 3, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Size( G )\index{Size@\texttt{Size}!G}
\label{Size:G}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 The order of the rcwa group~\mbox{\texttt{G}}. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Size(G);
  265252859812191058636308480000000
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IsomorphismPermGroup (G)}}
\logpage{[ 3, 3, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsomorphismPermGroup( G )\index{IsomorphismPermGroup@\texttt{IsomorphismPermGroup}!G}
\label{IsomorphismPermGroup:G}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 An isomorphism from the finite rcwa group~\mbox{\texttt{G}} to a finite-degree permutation group. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsomorphismPermGroup(Group(g1,g2));
  [ <bijective rcwa mapping of Z with modulus 2, of order 2>, 
    <bijective rcwa mapping of Z with modulus 3, of order 3> ] -> 
  [ (1,2)(3,4)(5,6), (1,2,3)(4,5,6) ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IsomorphismMatrixGroup (G)}}
\logpage{[ 3, 3, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsomorphismMatrixGroup( G )\index{IsomorphismMatrixGroup@\texttt{IsomorphismMatrixGroup}!G}
\label{IsomorphismMatrixGroup:G}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 An isomorphism from the rcwa group~\mbox{\texttt{G}} to a matrix group, provided that \mbox{\texttt{G}} embeds into a matrix group and that there is a suitable method available. Both
conditions are fulfilled if \mbox{\texttt{G}} is tame. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> g := RcwaMapping([[2,2,1],[1, 4,1],[1,0,2],[2,2,1],[1,-4,1],[1,-2,1]]);;
  gap> h := RcwaMapping([[2,2,1],[1,-2,1],[1,0,2],[2,2,1],[1,-1,1],[1, 1,1]]);;
  gap> SetName(g,"g"); SetName(h,"h");
  gap> phi := IsomorphismMatrixGroup(Group(g,h));;
  gap> FieldOfMatrixGroup(Image(phi));
  Rationals
  gap> DegreeOfMatrixGroup(Image(phi));
  14
  gap> Display(GeneratorsOfGroup(Image(phi))[1]*One(GF(5)));
   . . . . . . 1 1 . . . . . .
   . . . . . . . 1 . . . . . .
   . . . . . . . . . . 3 . . .
   . . . . . . . . . . . 1 . .
   . . . . . . . . 1 3 . . . .
   . . . . . . . . . 1 . . . .
   . . . . . . . . . . . . 3 .
   . . . . . . . . . . . . . 1
   . . 1 4 . . . . . . . . . .
   . . . 1 . . . . . . . . . .
   2 2 . . . . . . . . . . . .
   . 1 . . . . . . . . . . . .
   . . . . 2 2 . . . . . . . .
   . . . . . 1 . . . . . . . .
  
\end{Verbatim}
 }

 }

 \pagebreak[4]  
\section{\textcolor{Chapter }{ Factoring elements into generators }}\label{sec:ElementFactorization}
\logpage{[ 3, 4, 0 ]}
{
  

\subsection{\textcolor{Chapter }{PreImagesRepresentative (phi, g)}}
\logpage{[ 3, 4, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{PreImagesRepresentative( phi, g )\index{PreImagesRepresentative@\texttt{PreImagesRepresentative}!phi, g}
\label{PreImagesRepresentative:phi, g}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 A representative of the set of preimages of~\mbox{\texttt{g}} under the homomorphism~\mbox{\texttt{phi}} from a free group to an rcwa group over~$\mathbb{Z}$. 



 This method can be used for factoring elements of rcwa groups over $\mathbb{Z}$ into generators. It can also be used for finding nontrivial relations among
the generators if the respective group is not free and the method returns a
factorization which does not happen to be equal to one which is known a
priori. The homomorphism \mbox{\texttt{phi}} must map the generators of the free group to the generators of the rcwa group
one-by-one. This method is also suitable for wild groups. The implementation
is based on \texttt{RepresentativeActionPreImage} (\ref{RepresentativeActionPreImage:G, src, dest, act, F}). 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(g,h);
  <rcwa group over Z with 2 generators>
  gap> phi := EpimorphismFromFreeGroup(G);
  [ g, h ] -> [ g, h ]
  gap> PreImagesRepresentative(phi,h*g^3*h^2*g^-1*h*g*h^-3);
  h*g^3*h^2*g^-1*h*g*h^-3
  gap> nu := RcwaMapping([[1,1,1]]);
  Rcwa mapping of Z: n -> n + 1
  gap> SetName(nu,"nu");
  gap> G := Group(a,nu);
  <rcwa group over Z with 2 generators>
  gap> IsTame(G);
  false
  gap> phi := EpimorphismFromFreeGroup(G);
  [ a, nu ] -> [ a, nu ]
  gap> F := Source(phi);
  <free group on the generators [ a, nu ]>
  gap> w := Comm(F.1,Comm(F.1,F.2^2));
  a^-1*nu^-2*a^-1*nu^2*a*nu^-2*a*nu^2
  gap> f := w^phi;
  <bijective rcwa mapping of Z with modulus 18>
  gap> IsTame(f);
  false
  gap> pre := PreImagesRepresentative(phi,f);
  a^-2*nu^-2*a^2*nu^2
  gap> one := w*pre^-1; # pre <> w --> We have a non-trivial relation!
  a^-1*nu^-2*a^-1*nu^2*a*nu^-2*a^-1*nu^2*a^2
  gap> one^phi;
  IdentityMapping( Integers )
  
\end{Verbatim}
 }

 \pagebreak[4] 

\subsection{\textcolor{Chapter }{PreImagesRepresentatives (phi, g)}}
\logpage{[ 3, 4, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{PreImagesRepresentatives( phi, g )\index{PreImagesRepresentatives@\texttt{PreImagesRepresentatives}!phi, g}
\label{PreImagesRepresentatives:phi, g}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 A list of representatives of the set of preimages of~\mbox{\texttt{g}} under the homomorphism~\mbox{\texttt{phi}} from a free group to an rcwa group over~$\mathbb{Z}$. 



 Quite frequently, computing several preimages is not harder than computing
just one, i.e. often several preimages are found simultaneously. This
operation is called by \texttt{PreImagesRepresentative} (\ref{PreImagesRepresentative:phi, g}), which simply chooses the shortest representative. For a slightly more
concise description see there. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(g,h);
  <rcwa group over Z with 2 generators>
  gap> phi := EpimorphismFromFreeGroup(G);
  [ g, h ] -> [ g, h ]
  gap> f := g^3*h*g^-4*h^5*g;
  <bijective rcwa mapping of Z with modulus 12>
  gap> RCWAInfo(2);
  gap> pre := PreImagesRepresentatives(phi,f);
  #I  Orbit lengths after extension step 1: [ 4, 5 ]
  #I  |Candidates| = 1
  #I  Orbit lengths after extension step 1: [ 5, 5 ]
  #I  Orbit lengths after extension step 2: [ 17, 15 ]
  #I  Orbit lengths after extension step 3: [ 52, 39 ]
  #I  |Candidates| = 1
  #I  Orbit lengths after extension step 1: [ 5, 5 ]
  #I  Orbit lengths after extension step 2: [ 17, 15 ]
  #I  Orbit lengths after extension step 3: [ 53, 43 ]
  #I  Orbit lengths after extension step 4: [ 158, 119 ]
  #I  |Candidates| = 1
  #I  Orbit lengths after extension step 1: [ 5, 5 ]
  #I  Orbit lengths after extension step 2: [ 17, 17 ]
  #I  Orbit lengths after extension step 3: [ 53, 53 ]
  #I  Orbit lengths after extension step 4: [ 159, 158 ]
  #I  Orbit lengths after extension step 5: [ 472, 462 ]
  #I  Orbit lengths after extension step 6: [ 1356, 1309 ]
  #I  Orbit lengths after extension step 7: [ 3822, 3643 ]
  #I  |Candidates| = 11
  [ g^3*h*g^3*h^5*g, g^-3*h^-4*g^-3*h^-1*g*h*g, g^3*h*g^-4*h^5*g ]
  gap> RCWAInfo(0);
  gap> List(pre,Length);
  [ 13, 14, 14 ]
  gap> Set(List(pre,w->w^phi)) = [f];
  true
  gap> w := pre[1]*pre[2]^-1;
  g^3*h*g^3*h^4*g^-1*h*g^3*h^4*g^3
  gap> Length(w);
  23
  gap> w^phi; # A relation of length 23.
  IdentityMapping( Integers )
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{The action of an rcwa group on the underlying ring R}}\label{sec:ActionOnR}
\logpage{[ 3, 5, 0 ]}
{
  The support (set of moved points) of an rcwa group can be determined by \texttt{Support} or \texttt{MovedPoints} (these are synonyms). Sometimes testing for transitivity on the underlying
ring is feasible. This is e.g. the case for tame groups over~$\Z$. Further it is often possible to determine group elements which map a given
tuple of elements of the underlying ring to a given other tuple, if such
elements exist. 

\subsection{\textcolor{Chapter }{IsTransitive (G, Integers)}}
\logpage{[ 3, 5, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsTransitive( G, Integers )\index{IsTransitive@\texttt{IsTransitive}!G, Integers}
\label{IsTransitive:G, Integers}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 \texttt{true} if the rcwa group~\mbox{\texttt{G}} acts transitively on~$\Z$ and \texttt{false} otherwise. 



 If \mbox{\texttt{G}} is wild, this may fail or run into an infinite loop. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(g,h);;
  gap> RCWAInfo(3);
  gap> IsTransitive(G,Integers);
  #I  IsTransitive: testing for finiteness and searching short orbits ...
  #I  IsTame: balancedness criterion.
  #I  IsTame:`dead end' criterion.
  #I  IsTame: loop criterion.
  #I  IsTame:`finite order or integral power' criterion.
  
    [ ... ]
  
  #I  Order: the 4th power of the argument is RcwaMapping(
  [ [ 1, 12, 1 ], [ 1, 12, 1 ], [ 1, -6, 2 ], [ 2, -10, 1 ], [ 1, -7, 1 ],
    [ 2, -8, 1 ], [ 1, 12, 1 ], [ 1, 12, 1 ], [ 1, -10, 2 ], [ 2, -10, 1 ],
    [ 1, -7, 1 ], [ 2, -8, 1 ] ] );
  There is a `class shift' on the residue class 0(12).
  #I  Trying probabilistic random walk, initial m = 12
  #I  checking modulus ...
  #I  Size: use action on respected partition.
  #I  KernelOfActionOnRespectedPartition: gen. #1, lng = 1
  #I  KernelOfActionOnRespectedPartition: gen. #2, lng = 2
  
                         [ ... ]
  
  #I  KernelOfActionOnRespectedPartition: gen. #14, lng = 10
  #I  Searching for class shifts ...
  #I  ... in generators
  #I  ... in commutators of the generators
  #I  The cyclic group generated by RcwaMapping(
  [ [ 1, -9, 1 ], [ 1, 0, 1 ], [ 1, 6, 1 ], [ 1, -3, 1 ], [ 1, 0, 1 ],
    [ 1, 6, 1 ] ] ) acts transitively on the residue class 2(6).
  #I  OrbitUnion: initial set = ResidueClassUnion( Integers, 6, [ 2 ] )
  #I  Image = Integers
  true
  gap> RCWAInfo(0);
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{RepresentativeAction (G, src, dest, act)}}
\logpage{[ 3, 5, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RepresentativeAction( G, src, dest, act )\index{RepresentativeAction@\texttt{RepresentativeAction}!G, src, dest, act}
\label{RepresentativeAction:G, src, dest, act}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 An element of \mbox{\texttt{G}} which maps \mbox{\texttt{src}} to~\mbox{\texttt{dest}} under the action given by~\mbox{\texttt{act}}. 



 If an element satisfying this condition does not exist, this method either
returns \texttt{fail} or runs into an infinite loop. The problem of whether \mbox{\texttt{src}} and \mbox{\texttt{dest}} lie in the same orbit under the action of \mbox{\texttt{G}} in general seems to be hard. The method is based on \texttt{RepresentativeActionPreImage} (\ref{RepresentativeActionPreImage:G, src, dest, act, F}), and it basically just computes an image under an epimorphism. As this
involves multiplications of rcwa mappings, this can be quite expensive if the
group \mbox{\texttt{G}} is wild, the preimage is a rather long word and coefficient explosion happens
to occur. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(a,b);
  <rcwa group over Z with 2 generators>
  gap> elm := RepresentativeAction(G,[7,4,9],[4,5,13],OnTuples);
  <bijective rcwa mapping of Z with modulus 12>
  gap> Display(elm);
  
  Bijective rcwa mapping of Z with modulus 12
  
                 n mod 12                |                 n^f
  ---------------------------------------+--------------------------------------
     0  2  3  6  8 11                    | n
     1  7 10                             | n - 3
     4                                   | n + 1
     5  9                                | n + 4
  
  gap> List([7,4,9],n->n^elm);
  [ 4, 5, 13 ]
  gap> elm := RepresentativeAction(G,[5,4,9],[13,5,4],OnTuples);
  <bijective rcwa mapping of Z with modulus 9>
  gap> Display(elm);
  
  Bijective rcwa mapping of Z with modulus 9
  
                  n mod 9                |                 n^f
  ---------------------------------------+--------------------------------------
    0                                    | 4n/9
    1                                    | (8n - 26)/9
    2                                    | (8n + 2)/9
    3                                    | (8n + 3)/9
    4                                    | (16n - 19)/9
    5                                    | (16n + 37)/9
    6                                    | (8n + 33)/9
    7                                    | (16n - 49)/9
    8                                    | (16n + 7)/9
  
  gap> List([5,4,9],n->n^elm);
  [ 13, 5, 4 ]
  gap> RepresentativeAction(G,[7,4,9],[4,5,8],OnTuples);
  <bijective rcwa mapping of Z with modulus 256>
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{RepresentativeActionPreImage (G, src, dest, act, F)}}
\logpage{[ 3, 5, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RepresentativeActionPreImage( G, src, dest, act, F )\index{RepresentativeActionPreImage@\texttt{RepresentativeActionPreImage}!G, src, dest, act, F}
\label{RepresentativeActionPreImage:G, src, dest, act, F}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The result of \texttt{RepresentativeAction(\mbox{\texttt{G}},\mbox{\texttt{src}},\mbox{\texttt{dest}},\mbox{\texttt{act}})} as word in generators. 



 The argument \mbox{\texttt{F}} is a free group whose generators are used as letters of the returned word.
Note that the dependency is just in the opposite direction than suggested
above (\texttt{RepresentativeAction} calls \texttt{RepresentativeActionPreImage}) and that the evaluation of the word sometimes takes much more time than its
determination. This causes \texttt{RepresentativeActionPreImage} sometimes to be much faster than \texttt{RepresentativeAction}. The used algorithm is not inefficient, as the last two of the examples below
suggest. It is based on computing balls of increasing radius around \mbox{\texttt{src}} and \mbox{\texttt{dest}} until they intersect nontrivially. It avoids multiplying rcwa mappings. Of
course the other warnings given in the description of \texttt{RepresentativeAction} (\ref{RepresentativeAction:G, src, dest, act}) apply to this operation as well. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> F := FreeGroup("a","b");;
  gap> w := RepresentativeActionPreImage(G,[7,4,9],[4,5,13],OnPoints,F);
  b^-1*a*b*a^-1
  gap> w := RepresentativeActionPreImage(G,[5,4,9],[13,5,4],OnTuples,F);
  b^-1*a^-1*b*a^-1
  gap> w := RepresentativeActionPreImage(G,[7,4,9],[4,5,8],OnPoints,F);
  b^2*a^2
  gap> phi := GroupHomomorphismByImages(F,G,[F.1,F.2],[a,b]);
  [ a, b ] -> [ a, b ]
  gap> w^phi;
  <bijective rcwa mapping of Z with modulus 256>
  gap> w^phi = RepresentativeAction(G,[7,4,9],[4,5,8],OnPoints);
  true
  gap> List([7,4,9],n->n^(w^phi));
  [ 4, 5, 8 ]
  gap> w := RepresentativeActionPreImage(G,[37,4,9],[4,51,8],OnPoints,F);
  a^-1*b^-1*a*b^4*a
  gap> w^phi;
  <bijective rcwa mapping of Z with modulus 4608>
  gap> w := RepresentativeActionPreImage(G,[37,4,9],[4,51,8],OnTuples,F);
  b*a^6*b*a^-3*b^-3*a^-1*b*a^2
  gap> w := RepresentativeActionPreImage(G,[17,14,9],[4,51,8],OnPoints,F);
  a^-1*b^-1*a^3*b^2*a*b*a*b^-1*a^2
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{RepresentativeAction (RCWA( Integers ), P1, P2)}}
\logpage{[ 3, 5, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RepresentativeAction( RCWA{\_}Z, P1, P2 )\index{RepresentativeAction@\texttt{RepresentativeAction}!RCWA( Integers ), P1, P2}
\label{RepresentativeAction:RCWA( Integers ), P1, P2}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 An element of RCWA($\Z$) which maps the partition~\mbox{\texttt{P1}} to~\mbox{\texttt{P2}}. 



 The arguments \mbox{\texttt{P1}} and \mbox{\texttt{P2}} must be partitions of the underlying ring $R$ into the same number of disjoint unions of residue classes. The method
recognizes the option \texttt{IsTame}. If this option is set, the returned mapping is tame provided that there is a
tame mapping which satisfies the given condition. If the option \texttt{IsTame} is not set and the partitions \mbox{\texttt{P1}} and \mbox{\texttt{P2}} both consist entirely of single residue classes, then the returned mapping is
affine on any residue class in~\mbox{\texttt{P1}}. \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> rc := function(r,m) return ResidueClass(DefaultRing(m),m,r); end;;
  gap> P1 := [rc(0,3),rc(1,3),rc(2,9),rc(5,9),rc(8,9)];
  [ 0(3), 1(3), 2(9), 5(9), 8(9) ]
  gap> P2 := [rc(0,2),rc(1,8),rc(5,16),rc(3,4),rc(13,16)];
  [ 0(2), 1(8), 5(16), 3(4), 13(16) ]
  gap> elm := RepresentativeAction(RCWA(Integers),P1,P2);
  <rcwa mapping of Z with modulus 9>
  gap> Display(elm);
  
  Rcwa mapping of Z with modulus 9
  
                  n mod 9                |                 n^f
  ---------------------------------------+--------------------------------------
    0 3 6                                | 2n/3
    1 4 7                                | (8n - 5)/3
    2                                    | (16n + 13)/9
    5                                    | (4n + 7)/9
    8                                    | (16n - 11)/9
  
  gap> P1^elm = P2;
  true
  gap> elm := RepresentativeAction(RCWA(Integers),P1,P2:IsTame);
  <tame rcwa mapping of Z with modulus 1152>
  gap> P := RespectedPartition(elm);;
  gap> Length(P);
  313
  gap> elm := RepresentativeAction(RCWA(Integers),
  >                                [rc(1,3),Union(rc(0,3),rc(2,3))],
  >                                [Union(rc(2,5),rc(4,5)),
  >                                 Union(rc(0,5),rc(1,5),rc(3,5))]);
  <rcwa mapping of Z with modulus 6>
  gap> [rc(1,3),Union(rc(0,3),rc(2,3))]^elm;
  [ 2(5) U 4(5), Z \ 2(5) U 4(5) ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{ShortOrbits (G, S, maxlng)}}
\logpage{[ 3, 5, 5 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{ShortOrbits( G, S, maxlng )\index{ShortOrbits@\texttt{ShortOrbits}!G, S, maxlng}
\label{ShortOrbits:G, S, maxlng}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 A list of all finite orbits of the rcwa group~\mbox{\texttt{G}} of maximal length~\mbox{\texttt{maxlng}}, which intersect nontrivially with the set~\mbox{\texttt{S}}. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> A5 := RcwaGroupByPermGroup(AlternatingGroup(5));;
  gap> ShortOrbits(A5,[-10..10],100);
  [ [ -10, -9, -8, -7, -6 ], [ -5, -4, -3, -2, -1 ], [ 0, 1, 2, 3, 4 ], 
    [ 5, 6, 7, 8, 9 ], [ 10, 11, 12, 13, 14 ] ]
  gap> Action(A5,last[2]);
  Group([ (1,2,3,4,5), (3,4,5) ])
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(Comm(a,b),Comm(a,c));;
  gap> orb := ShortOrbits(G,[-15..15],100);
  [ [ -15, -12, -7, -6, -5, -4, -3, -2, -1, 1 ], 
    [ -33, -30, -24, -21, -16, -14, -13, -11, -10, -8 ], [ -9 ], [ 0 ], 
    [ 2, 3, 4, 5, 6, 7, 8, 10, 12, 15 ], [ 9 ], 
    [ 11, 13, 14, 16, 17, 19, 21, 24, 30, 33 ] ]
  gap> Action(G,orb[1]);
  Group([ (2,5,8,10,7,6), (1,3,6,9,4,5) ])
  gap> ShortOrbits(Group(u),[-30..30],100);
  [ [ -13, -8, -7, -5, -4, -3, -2 ], [ -10, -6 ], [ -1 ], [ 0 ], [ 1, 2 ], 
    [ 3, 5 ], [ 24, 36, 39, 40, 44, 48, 60, 65, 67, 71, 80, 86, 93, 100, 112, 
        128, 138, 155, 167, 187, 230, 248, 312, 446, 520, 803, 867, 1445 ] ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{OrbitsModulo (G, m)}}
\logpage{[ 3, 5, 6 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{OrbitsModulo( G, m )\index{OrbitsModulo@\texttt{OrbitsModulo}!G, m}
\label{OrbitsModulo:G, m}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 A partition of \texttt{[0..\mbox{\texttt{m}}-1]} such that $i$ and $j$ lie in the same subset if and only if there is an element $g$ of \mbox{\texttt{G}} which moves an element from the residue class $i(m)$ to the residue class $j(m)$. 



 The argument \mbox{\texttt{G}} must be an rcwa group over $\mathbb{Z}$. See also \texttt{OrbitsModulo} (\ref{OrbitsModulo:f, m}) for rcwa mappings. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> OrbitsModulo(G,36);
  [ [ 0 ], [ 1, 11, 13, 14, 16, 17, 19, 21, 24, 29, 30, 31, 32, 33, 34, 35 ], 
    [ 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 20, 22, 23, 25, 26, 28 ], [ 9 ], [ 18 ], 
    [ 27 ] ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Ball (G, p, d, act)}}
\logpage{[ 3, 5, 7 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Ball( G, p, d, act )\index{Ball@\texttt{Ball}!G, p, d, act}
\label{Ball:G, p, d, act}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Ball( G, g, d )\index{Ball@\texttt{Ball}!G, g, d}
\label{Ball:G, g, d}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 The ball of radius~\mbox{\texttt{d}} around the point~\mbox{\texttt{p}} under the action~\mbox{\texttt{act}} of the group~\mbox{\texttt{G}}, resp. the ball of radius~\mbox{\texttt{d}} around the element~\mbox{\texttt{g}} in the group~\mbox{\texttt{G}}. 



 All balls are understood w.r.t. \texttt{GeneratorsOfGroup(\mbox{\texttt{G}})}. As element tests can be expensive, the latter method does not check whether \mbox{\texttt{g}} is indeed an element of~\mbox{\texttt{G}}. The methods require that point comparisons resp. element comparisons are
cheap. They are not only applicable to rcwa groups. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> for d in [1..4] do Print(Ball(G,1,d,OnPoints),"\n"); od;
  [ -3, -2, 1 ]
  [ -5, -4, -3, -2, 1 ]
  [ -15, -12, -7, -6, -5, -4, -3, -2, -1, 1 ]
  [ -15, -12, -7, -6, -5, -4, -3, -2, -1, 1 ]
  gap> List([1..11],d->Length(Ball(G,[1,2,3],d,OnSets)));
  [ 5, 17, 51, 127, 245, 324, 343, 357, 360, 360, 360 ]
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List([1..11],d->Length(Ball(G,[1,2,3],d,OnTuples)));
  [ 5, 17, 51, 143, 325, 557, 662, 695, 713, 720, 720 ]
  gap> Ball(Group((1,2),(2,3),(3,4),(4,5),(5,6)),(),2);
  [ (), (5,6), (4,5), (4,5,6), (4,6,5), (3,4), (3,4)(5,6), (3,4,5), (3,5,4), 
    (2,3), (2,3)(5,6), (2,3)(4,5), (2,3,4), (2,4,3), (1,2), (1,2)(5,6), 
    (1,2)(4,5), (1,2)(3,4), (1,2,3), (1,3,2) ]
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{Conjugacy in RCWA(R)}}\label{sec:ConjugacyInRCWA_R}
\logpage{[ 3, 6, 0 ]}
{
  

\subsection{\textcolor{Chapter }{IsConjugate (RCWA( Integers ), f, g)}}
\logpage{[ 3, 6, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsConjugate( RCWA{\_}Z, f, g )\index{IsConjugate@\texttt{IsConjugate}!RCWA( Integers ), f, g}
\label{IsConjugate:RCWA( Integers ), f, g}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 \texttt{true} if the bijective rcwa mappings \mbox{\texttt{f}} and~\mbox{\texttt{g}} are conjugate in RCWA($\mathbb{Z}$), and \texttt{false} otherwise. 



 This may fail or run into an infinite loop. In particular the support for wild
rcwa mappings is currently very poor, since the author does not know a way to
solve the conjugacy problem for these. Some easy cases are handled anyway. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsConjugate(RCWA(Integers),g,h);
  false
  gap> IsConjugate(RCWA(Integers),g,g^a);
  true
  gap> IsConjugate(RCWA(Integers),a,b);
  false
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{RepresentativeAction (RCWA( Integers ), f, g)}}
\logpage{[ 3, 6, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RepresentativeAction( RCWA{\_}Z, f, g )\index{RepresentativeAction@\texttt{RepresentativeAction}!RCWA( Integers ), f, g}
\label{RepresentativeAction:RCWA( Integers ), f, g}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 An rcwa mapping~\texttt{x} such that \texttt{\mbox{\texttt{f}}{\textasciicircum}x = \mbox{\texttt{g}}}, if such an \texttt{x} exists and \texttt{fail} otherwise. 



 This method currently works only for tame rcwa mappings of~$\Z$, since the author does not know a way to solve the conjugacy problem for wild
rcwa mappings. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> elm := RepresentativeAction(RCWA(Integers),h,h^g);
  <bijective rcwa mapping of Z with modulus 12>
  gap> h^elm = h^g; # check ...
  true
  gap> Order(elm);
  infinity
  gap> cent := g*elm^-1;
  <bijective rcwa mapping of Z with modulus 12>
  gap> Comm(cent,h); # cent must lie in the centralizer of h in RCWA(Z).
  IdentityMapping( Integers )
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Order(cent);; Display(cent);
  
  Bijective rcwa mapping of Z with modulus 12, of order 12
  
                 n mod 12                |                 n^f
  ---------------------------------------+--------------------------------------
     0  4  6 10                          | n - 1
     1  7                                | 2n
     2                                   | (n - 2)/2
     3  9                                | 2n + 2
     5 11                                | n + 2
     8                                   | n/2
  
  gap> cent in Group(h); # This particular element is even a power of h.
  true
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{ShortCycles (f, maxlng)}}
\logpage{[ 3, 6, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{ShortCycles( f, maxlng )\index{ShortCycles@\texttt{ShortCycles}!f, maxlng}
\label{ShortCycles:f, maxlng}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 All ``single'' finite cycles of the rcwa mapping~\mbox{\texttt{f}} of length at most \mbox{\texttt{maxlng}}. 



 In this context, ``single'' finite cycles are finite cycles which do not belong to an infinite series.
This means that there is no constant~$m$ such that adding any multiple of~$m$ to the elements of the cycle always yields another cycle. Since \textsf{GAP}-permutations cannot move negative integers, rationals or polynomials, the
cycles are returned as lists. For example, the list \texttt{[-3,1,2,-2]} denotes the cycle \texttt{(-3,1,2,-2)}. Permutations with different sets of finite cycle lengths are obviously not
conjugate. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> ShortCycles(a,5);
  [ [ 0 ], [ 1 ], [ -1 ], [ 2, 3 ], [ -3, -2 ], [ 4, 6, 9, 7, 5 ], 
    [ -9, -7, -5, -4, -6 ] ]
  gap> ShortCycles(u,2);
  [ [ 0 ], [ -1 ], [ 1, 2 ], [ 3, 5 ], [ -10, -6 ] ]
  gap> ShortCycles(Comm(a,b),10);
  [  ]
  gap> ShortCycles(a*b,2);
  [ [ 0 ], [ 2 ], [ 3 ], [ -26 ], [ 7 ], [ -3 ], [ -1 ] ]
  gap> v := RcwaMapping([[-1,2,1],[1,-1,1],[1,-1,1]]);;
  gap> w := RcwaMapping([[-1,3,1],[1,-1,1],[1,-1,1],[1,-1,1]]);;
  gap> List( [ v, w ], Order );
  [ 6, 8 ]
  gap> [ ShortCycles(v,10), ShortCycles(w,10) ];
  [ [ [ 0, 2, 1 ] ], [ [ 0, 3, 2, 1 ] ] ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{NrConjugacyClassesOfRCWAZOfOrder (ord)}}
\logpage{[ 3, 6, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{NrConjugacyClassesOfRCWAZOfOrder( ord )\index{NrConjugacyClassesOfRCWAZOfOrder@\texttt{NrConjugacyClassesOfRCWAZOfOrder}!ord}
\label{NrConjugacyClassesOfRCWAZOfOrder:ord}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 The number of conjugacy classes of RCWA($\Z$) of elements of order~\mbox{\texttt{ord}}. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> NrConjugacyClassesOfRCWAZOfOrder(2);
  infinity
  gap> NrConjugacyClassesOfRCWAZOfOrder(105);
  218
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{Restriction and induction}}\label{sec:RestrictionAndInduction}
\logpage{[ 3, 7, 0 ]}
{
  

\subsection{\textcolor{Chapter }{Restriction (g, f)}}
\logpage{[ 3, 7, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Restriction( g, f )\index{Restriction@\texttt{Restriction}!g, f}
\label{Restriction:g, f}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The \emph{restriction} of the rcwa mapping \mbox{\texttt{g}} by the injective rcwa mapping~\mbox{\texttt{f}}. 



 By definition, the restriction $g_f$ of the rcwa mapping \mbox{\texttt{g}} by the injective rcwa mapping~\mbox{\texttt{f}} is the uniquely determined rcwa mapping which satisfies $f \cdot g_f = g \cdot f$ and fixes the complement of the image of \mbox{\texttt{f}} pointwisely. If \mbox{\texttt{f}} is bijective, the restriction of \mbox{\texttt{g}} by \mbox{\texttt{f}} is just the conjugate of \mbox{\texttt{g}} under~\mbox{\texttt{f}}. See also \texttt{Restriction} (\ref{Restriction:G, f}) for rcwa groups. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Comm(Restriction(a,RcwaMapping([[2,0,1]])),
  >         Restriction(u,RcwaMapping([[2,1,1]])));
  IdentityMapping( Integers )
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Restriction (G, f)}}
\logpage{[ 3, 7, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Restriction( G, f )\index{Restriction@\texttt{Restriction}!G, f}
\label{Restriction:G, f}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The \emph{restriction} of the rcwa group \mbox{\texttt{G}} by the injective rcwa mapping~\mbox{\texttt{f}}. 



 By definition, the restriction of the rcwa group~\mbox{\texttt{G}} by the injective rcwa mapping~\mbox{\texttt{f}} consists of the restrictions of the elements of~\mbox{\texttt{G}} by~\mbox{\texttt{f}}. The restriction of~\mbox{\texttt{G}} by~\mbox{\texttt{f}} acts on the image of~\mbox{\texttt{f}} and fixes its complement pointwisely. See also \texttt{Restriction} (\ref{Restriction:g, f}) for rcwa mappings. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Restriction(Group(a,b),RcwaMapping([[5,3,1]]));
  <rcwa group over Z with 2 generators>
  gap> MovedPoints(G);
  3(5) \ [ -2, 3 ]
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{Induction (g, f)}}
\logpage{[ 3, 7, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Induction( g, f )\index{Induction@\texttt{Induction}!g, f}
\label{Induction:g, f}
}\hfill{\scriptsize (operation)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{Induction( G, f )\index{Induction@\texttt{Induction}!G, f}
\label{Induction:G, f}
}\hfill{\scriptsize (operation)}}\\
\textbf{\indent Returns:\ }
 The \emph{induction} of the rcwa mapping \mbox{\texttt{g}} resp. the rcwa group \mbox{\texttt{G}} by the injective rcwa mapping~\mbox{\texttt{f}}. 



 By definition, induction is the right inverse of restriction. This means that
it is \texttt{Induction(Restriction(\mbox{\texttt{g}},\mbox{\texttt{f}}),\mbox{\texttt{f}}) = \mbox{\texttt{g}}} resp. \texttt{Induction(Restriction(\mbox{\texttt{G}},\mbox{\texttt{f}}),\mbox{\texttt{f}}) = \mbox{\texttt{G}}}. The mapping~\mbox{\texttt{g}} resp. the group~\mbox{\texttt{G}} must not move points outside the image of~\mbox{\texttt{f}}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Induction(G,RcwaMapping([[5,3,1]])) = Group(a,b);
  true
  
\end{Verbatim}
 }

 Restriction monomorphisms permit forming direct products and wreath products
of rcwa groups, regardless of whether they are tame or not: 

\subsection{\textcolor{Chapter }{DirectProduct (G1, G2, ...)}}
\logpage{[ 3, 7, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{DirectProduct( G1, G2, ... )\index{DirectProduct@\texttt{DirectProduct}!G1, G2, ...}
\label{DirectProduct:G1, G2, ...}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 An rcwa group isomorphic to the direct product of the rcwa groups over~$\mathbb{Z}$ given as arguments. 



 There is certainly no unique or canonical way to embed a direct product of
rcwa groups into RCWA($\mathbb{Z}$). This method chooses to embed the groups \mbox{\texttt{G1}}, \mbox{\texttt{G2}}, \mbox{\texttt{G3}}~... via restrictions by $n \mapsto mn$, $n \mapsto mn+1$, $n \mapsto mn+2$~..., where $m$ denotes the number of groups given as arguments. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := DirectProduct(Group(g,h),Group(a,b),Group(u));;
  gap> Embedding(G,1);
  [ g, h ] -> [ <bijective rcwa mapping of Z with modulus 18, of order 7>, 
    <bijective rcwa mapping of Z with modulus 18, of order 12> ]
  gap> List([1..3],i->MovedPoints(Image(Embedding(G,i))));
  [ 0(3), 1(3) \ [ -2, 1 ], 2(3) \ [ -1, 2 ] ]
  gap> Image(Projection(G,2)) = Group(a,b);
  true
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{WreathProduct (G, P)}}
\logpage{[ 3, 7, 5 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{WreathProduct( G, P )\index{WreathProduct@\texttt{WreathProduct}!G, P}
\label{WreathProduct:G, P}
}\hfill{\scriptsize (method)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{WreathProduct( G, Z )\index{WreathProduct@\texttt{WreathProduct}!G, Z}
\label{WreathProduct:G, Z}
}\hfill{\scriptsize (method)}}\\
\textbf{\indent Returns:\ }
 An rcwa group isomorphic to the wreath product of the rcwa group~\mbox{\texttt{G}} over~$\mathbb{Z}$ with the finite permutation group~\mbox{\texttt{P}}, resp. with the infinite cyclic group~\mbox{\texttt{Z}}. 



 There is certainly no unique or canonical way to embed a wreath product of
rcwa groups into RCWA($\mathbb{Z}$). The first-mentioned method embeds the \texttt{DegreeAction(\mbox{\texttt{P}})}th direct power of~\mbox{\texttt{G}} using the method for \texttt{DirectProduct}, and lets the permutation group~\mbox{\texttt{P}} act naturally on the set of residue classes modulo \texttt{DegreeAction(\mbox{\texttt{P}})}. The second-mentioned method restricts the group~\mbox{\texttt{G}} to the residue class~3(4), and maps the generator of the infinite cyclic group \mbox{\texttt{Z}} to \texttt{ClassTransposition(0,2,1,2) * ClassTransposition(0,2,1,4)}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(g,h);;
  gap> IsTame(G); Size(G);    
  true
  infinity
  gap> H := WreathProduct(G,AlternatingGroup(5));
  <tame rcwa group over Z with 12 generators, of size infinity>
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Embedding(H,1);
  [ g, h ] -> [ <bijective rcwa mapping of Z with modulus 30, of order 7>, 
    <bijective rcwa mapping of Z with modulus 30, of order 12> ]
  gap> Embedding(H,2);
  [ (1,2,3,4,5), (3,4,5) ] -> 
  [ <bijective rcwa mapping of Z with modulus 5, of order 5>, 
    <bijective rcwa mapping of Z with modulus 5, of order 3> ]
  gap> H := WreathProduct(G,Group(ClassShift(0,1)));
  <wild rcwa group over Z with 3 generators>
  gap> Support(Image(Embedding(H,1)));
  3(4)
  gap> Embedding(H,2);
  [ ClassShift(0,1) ] -> [ <wild bijective rcwa mapping of Z with modulus 4> ]
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{Special attributes for tame rcwa groups}}\label{sec:MethodsForTameGroups}
\logpage{[ 3, 8, 0 ]}
{
  There is a couple of attributes which a priori make only sense for tame rcwa
groups. In the sequel, these attributes are described in detail. 

 With their help, various structural information about a given tame rcwa group
can be obtained. For example there are methods for \texttt{IsSolvable} and \texttt{IsPerfect} available for tame rcwa groups (the latter works in some cases by other means
also for wild groups). Often it is also feasible to compute the derived
subgroup of a tame rcwa group. 

\subsection{\textcolor{Chapter }{RespectedPartition (G)}}
\logpage{[ 3, 8, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RespectedPartition( G )\index{RespectedPartition@\texttt{RespectedPartition}!G}
\label{RespectedPartition:G}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RespectedPartition( sigma )\index{RespectedPartition@\texttt{RespectedPartition}!sigma}
\label{RespectedPartition:sigma}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 A \emph{respected partition} of \mbox{\texttt{G}} resp. \mbox{\texttt{sigma}}. 



 A \emph{respected partition} of~\mbox{\texttt{G}} resp.~\mbox{\texttt{sigma}} is a partition of the underlying ring~$R$ into a finite number of residue classes on which~\mbox{\texttt{G}} resp. the cyclic group generated by~\mbox{\texttt{sigma}} acts in a natural way as a permutation group, and on whose elements all
elements of~\mbox{\texttt{G}} resp. all powers of~\mbox{\texttt{sigma}} are affine. In the author's thesis it is shown that such a partition exists if
and only if \mbox{\texttt{G}} resp. \mbox{\texttt{sigma}} is tame (see Theorem~2.5.8). 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(g,h);; Size(G);
  infinity
  gap> P := RespectedPartition(G);
  [ 0(6), 1(6), 3(6), 4(6), 5(6), 2(12), 8(12) ]
  gap> Permutation(g,P);
  (1,6,2,5,3,7,4)
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{ActionOnRespectedPartition (G)}}
\logpage{[ 3, 8, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{ActionOnRespectedPartition( G )\index{ActionOnRespectedPartition@\texttt{ActionOnRespectedPartition}!G}
\label{ActionOnRespectedPartition:G}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 The action of the tame rcwa group~\mbox{\texttt{G}} on \texttt{RespectedPartition(\mbox{\texttt{G}})}. 



 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> H := ActionOnRespectedPartition(G);
  Group([ (1,6,2,5,3,7,4), (1,6,2,5)(3,7,4) ])
  gap> H = Action(G,P);
  true
  gap> StructureDescription(H);
  "S7"
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IntegralConjugate (G)}}
\logpage{[ 3, 8, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IntegralConjugate( G )\index{IntegralConjugate@\texttt{IntegralConjugate}!G}
\label{IntegralConjugate:G}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IntegralConjugate( f )\index{IntegralConjugate@\texttt{IntegralConjugate}!f}
\label{IntegralConjugate:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 Some integral conjugate of the tame rcwa group~\mbox{\texttt{G}} resp. of the tame bijective rcwa mapping~\mbox{\texttt{f}} in the group RCWA($\mathbb{Z}$). 



 In the author's thesis it is shown that such a conjugate exists (see Theorem~2.5.14). There are usually infinitely many such conjugates, and methods for
this operation may choose any of them. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(IntegralConjugate(g));
  
  Bijective rcwa mapping of Z with modulus 7, of order 7
  
                  n mod 7                |                 n^f
  ---------------------------------------+--------------------------------------
    0                                    | n + 5
    1                                    | n + 3
    2                                    | n + 4
    3 6                                  | n - 3
    4                                    | n - 2
    5                                    | n - 4
  
  gap> RespectedPartition(IntegralConjugate(G));
  [ 0(7), 1(7), 2(7), 3(7), 4(7), 5(7), 6(7) ]
  gap> Action(IntegralConjugate(G),last);
  Group([ (1,6,2,5,3,7,4), (1,6,2,5)(3,7,4) ])
  gap> last = ActionOnRespectedPartition(G);
  true
  
\end{Verbatim}
 }

 \pagebreak[4] 

\subsection{\textcolor{Chapter }{IntegralizingConjugator (G)}}
\logpage{[ 3, 8, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IntegralizingConjugator( G )\index{IntegralizingConjugator@\texttt{IntegralizingConjugator}!G}
\label{IntegralizingConjugator:G}
}\hfill{\scriptsize (attribute)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IntegralizingConjugator( f )\index{IntegralizingConjugator@\texttt{IntegralizingConjugator}!f}
\label{IntegralizingConjugator:f}
}\hfill{\scriptsize (attribute)}}\\
\textbf{\indent Returns:\ }
 An rcwa mapping mapping~\mbox{\texttt{x}} such that \mbox{\texttt{G}}{\textasciicircum}\mbox{\texttt{x}} resp. \mbox{\texttt{f}}{\textasciicircum}\mbox{\texttt{x}} is integral. 



 While there are usually infinitely many such rcwa mappings, it is taken care
that the returned ones always satisfy the relations \texttt{\mbox{\texttt{G}}{\textasciicircum}IntegralizingConjugator(\mbox{\texttt{G}}) = IntegralConjugate(\mbox{\texttt{G}})} resp. \texttt{\mbox{\texttt{f}}{\textasciicircum}IntegralizingConjugator(\mbox{\texttt{f}}) = IntegralConjugate(\mbox{\texttt{f}})}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(IntegralizingConjugator(g));
  
  Bijective rcwa mapping of Z with modulus 12
  
                 n mod 12                |                 n^f
  ---------------------------------------+--------------------------------------
     0  6                                | 7n/6
     1  7                                | (7n - 1)/6
     2                                   | (7n + 46)/12
     3  9                                | (7n - 9)/6
     4 10                                | (7n - 10)/6
     5 11                                | (7n - 11)/6
     8                                   | (7n + 16)/12
  
  
\end{Verbatim}
 }

 }

  
\section{\textcolor{Chapter }{The categories of rcwa groups}}\label{sec:CategoriesOfRcwaGroups}
\logpage{[ 3, 9, 0 ]}
{
  

\subsection{\textcolor{Chapter }{IsRcwaGroup (G)}}
\logpage{[ 3, 9, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsRcwaGroup( G )\index{IsRcwaGroup@\texttt{IsRcwaGroup}!G}
\label{IsRcwaGroup:G}
}\hfill{\scriptsize (filter)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsRcwaGroupOverZ( G )\index{IsRcwaGroupOverZ@\texttt{IsRcwaGroupOverZ}!G}
\label{IsRcwaGroupOverZ:G}
}\hfill{\scriptsize (filter)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsRcwaGroupOverZ{\_}pi( G )\index{IsRcwaGroupOverZpi@\texttt{IsRcwaGroupOverZ{\_}pi}!G}
\label{IsRcwaGroupOverZpi:G}
}\hfill{\scriptsize (filter)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsRcwaGroupOverGFqx( G )\index{IsRcwaGroupOverGFqx@\texttt{IsRcwaGroupOverGFqx}!G}
\label{IsRcwaGroupOverGFqx:G}
}\hfill{\scriptsize (filter)}}\\
\textbf{\indent Returns:\ }
 \texttt{true} if \mbox{\texttt{G}} is an rcwa group resp. an rcwa group over the ring of integers resp. an rcwa
group over a semilocalization of the ring of integers resp. an rcwa group over
a polynomial ring in one variable over a finite field, and \texttt{false} otherwise. 

}

 }

  }

         
\chapter{\textcolor{Chapter }{Examples}}\label{ch:Examples}
\logpage{[ 4, 0, 0 ]}
{
 This chapter discusses a number of ``nice'' examples of rcwa mappings and -groups in detail. All of them show different
aspects of the package, and the order in which they appear is entirely
arbitrary. In particular they are not ordered by degree of interestingness or
difficulty. The rcwa mappings defined in this chapter can be found in the file \texttt{pkg/rcwa/examples/examples.g}, so there is no need to extract them from the manual files. This file can be
read into the current \textsf{GAP} session by issueing \texttt{RCWAReadExamples( );}.  
\section{\textcolor{Chapter }{ Factoring Collatz' permutation of the integers }}\label{sec:FactoringTheCollatzPermutation}
\logpage{[ 4, 1, 0 ]}
{
  In 1932, Lothar Collatz mentioned in his notebook the following permutation of
the integers: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Collatz := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);;
  gap> SetName(Collatz,"Collatz"); Display(Collatz);
  
  Rcwa mapping of Z with modulus 3
  
                  n mod 3                |              n^Collatz
  ---------------------------------------+--------------------------------------
    0                                    | 2n/3
    1                                    | (4n - 1)/3
    2                                    | (4n + 1)/3
  
  
\end{Verbatim}
 The cycle structure of this permutation has not been completely determined
yet. In particular it is not known whether the cycle containing~8 is finite or infinite. There are a few finite cycles: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List(ShortOrbits(Group(Collatz),[-100..100],100),
  >         orb->Cycle(Collatz,Minimum(orb)));
  [ [ -111, -74, -99, -66, -44, -59, -79, -105, -70, -93, -62, -83 ], 
    [ -9, -6, -4, -5, -7 ], [ -3, -2 ], [ -1 ], [ 0 ], [ 1 ], [ 2, 3 ], 
    [ 4, 5, 7, 9, 6 ], [ 44, 59, 79, 105, 70, 93, 62, 83, 111, 74, 99, 66 ] ]
  
\end{Verbatim}
 Nevertheless, the factorization routine included in this package can determine
a factorization of this permutation into involutions interchanging two
disjoint residue classes, each (for reasons of saving a bit space in this
manual, we factor the inverse mapping instead and revert the list afterwards): 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> RCWAInfo(2); # Switch Info output on.
  gap> Reversed(Factorization(Collatz^-1:ExpandPrimeSwitches));
  #I  Modulus(<g>) = 4, Multiplier(<g>) = 3, Divisor(<g>) = 4
  #I  Dividing by PrimeSwitch(3) from the right.
  #I  Modulus(<g>) = 16, Multiplier(<g>) = 3, Divisor(<g>) = 4
  #I  Dividing by PrimeSwitch(3) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 6, Divisor(<g>) = 4
  #I  Dividing by PrimeSwitch(3) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 12
  #I  p = 3, kmult = 1, kdiv = 1
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 3(6), 5(12), 7(12), 8(12), 0(48) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 6(8) ]
  #I  Found 5 pairs.
  #I  After filtering and splitting: 5 pairs.
  #I  Dividing by ClassTransposition(3,6,6,8) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(6,8,5,12) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(6,8,7,12) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(6,8,8,12) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(6,8,0,48) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 6, Divisor(<g>) = 4
  #I  Dividing by PrimeSwitch(3) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 12
  #I  p = 3, kmult = 1, kdiv = 1
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 7(12), 8(12), 0(96) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 6(8) ]
  #I  Found 3 pairs.
  #I  After filtering and splitting: 3 pairs.
  #I  Dividing by ClassTransposition(6,8,7,12) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(6,8,8,12) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(6,8,0,96) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 12, Divisor(<g>) = 4
  #I  Dividing by PrimeSwitch(3) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 24, Divisor(<g>) = 12
  #I  p = 3, kmult = 1, kdiv = 1
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 7(12), 0(192) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 2(4) ]
  #I  Found 2 pairs.
  #I  After filtering and splitting: 2 pairs.
  #I  Dividing by ClassTransposition(2,4,7,12) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 24, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(2,4,0,192) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 24, Divisor(<g>) = 4
  #I  Dividing by PrimeSwitch(3) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 48, Divisor(<g>) = 12
  #I  p = 3, kmult = 1, kdiv = 1
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 0(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 2(4) ]
  #I  Found 1 pairs.
  #I  After filtering and splitting: 1 pairs.
  #I  Dividing by ClassTransposition(2,4,0,384) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 128, Divisor(<g>) = 4
  #I  p = 2, kmult = 7, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 384(1536) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 2(4), 3(6), 1(12), 11(12) ]
  #I  Found 3 pairs.
  #I  After filtering and splitting: 5 pairs.
  #I  Dividing by ClassTransposition(2,12,384,1536) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 32, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(1,12,384,1536) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 32, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(6,12,384,1536) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 32, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(10,12,384,1536) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 32, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(11,12,384,1536) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 32, Divisor(<g>) = 4
  #I  p = 2, kmult = 5, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 0(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 3(6), 1(12), 6(12), 10(12), 11(12) ]
  #I  Found 4 pairs.
  #I  After filtering and splitting: 4 pairs.
  #I  Dividing by ClassTransposition(1,12,0,384) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 16, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(6,12,0,384) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 16, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(10,12,0,384) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 16, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(11,12,0,384) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 16, Divisor(<g>) = 4
  #I  p = 2, kmult = 4, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 48(192), 192(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 3(6), 6(12), 10(12), 11(12) ]
  #I  Found 3 pairs.
  #I  After filtering and splitting: 3 pairs.
  #I  Dividing by ClassTransposition(6,12,48,192) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 16, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(10,12,48,192) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 16, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(11,12,48,192) from the right.
  #I  Modulus(<g>) = 48, Multiplier(<g>) = 16, Divisor(<g>) = 4
  #I  p = 2, kmult = 4, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 192(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 3(6), 10(12), 11(12) ]
  #I  Splitting classes being divided by q*p^kdiv.
  #I  Found 4 pairs.
  #I  After filtering and splitting: 4 pairs.
  #I  Dividing by ClassTransposition(10,24,192,384) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 8, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(11,24,192,384) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 8, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(22,24,192,384) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 8, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(23,24,192,384) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 8, Divisor(<g>) = 4
  #I  p = 2, kmult = 3, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 72(96), 96(192), 0(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 3(6), 11(12), 22(24) ]
  #I  Found 2 pairs.
  #I  After filtering and splitting: 2 pairs.
  #I  Dividing by ClassTransposition(11,12,72,96) from the right.
  #I  Dividing by ClassTransposition(22,24,96,192) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 8, Divisor(<g>) = 4
  #I  p = 2, kmult = 3, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 0(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 3(6) ]
  #I  Splitting classes being divided by q*p^kdiv.
  #I  Splitting classes being divided by q*p^kdiv.
  #I  Splitting classes being divided by q*p^kdiv.
  #I  Found 8 pairs.
  #I  After filtering and splitting: 8 pairs.
  #I  Dividing by ClassTransposition(3,48,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(9,48,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(15,48,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(21,48,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(27,48,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(33,48,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(39,48,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(45,48,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  p = 2, kmult = 2, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 44(48), 48(96), 192(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 9(12), 15(24), 27(48) ]
  #I  Found 2 pairs.
  #I  After filtering and splitting: 2 pairs.
  #I  Dividing by ClassTransposition(9,12,44,48) from the right.
  #I  Dividing by ClassTransposition(15,24,48,96) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 4, Divisor(<g>) = 4
  #I  p = 2, kmult = 2, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 192(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 27(48) ]
  #I  Splitting classes being divided by q*p^kdiv.
  #I  Found 2 pairs.
  #I  After filtering and splitting: 2 pairs.
  #I  Dividing by ClassTransposition(27,96,192,384) from the right.
  #I  Modulus(<g>) = 384, Multiplier(<g>) = 2, Divisor(<g>) = 4
  #I  Dividing by ClassTransposition(75,96,192,384) from the right.
  #I  Modulus(<g>) = 384, Multiplier(<g>) = 2, Divisor(<g>) = 4
  #I  p = 2, kmult = 1, kdiv = 2
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 4(24), 17(24), 24(48), 96(192), 0(384) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 75(96) ]
  #I  Found 1 pairs.
  #I  After filtering and splitting: 1 pairs.
  #I  Dividing by ClassTransposition(75,96,0,384) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 2, Divisor(<g>) = 2
  #I  p = 2, kmult = 1, kdiv = 1
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 4(24), 17(24), 24(48), 96(192) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 7(12), 5(24), 12(24), 16(24), 75(96) ]
  #I  Found 6 pairs.
  #I  After filtering and splitting: 6 pairs.
  #I  Dividing by ClassTransposition(7,12,4,24) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 2, Divisor(<g>) = 2
  #I  Dividing by ClassTransposition(7,12,17,24) from the right.
  #I  Dividing by ClassTransposition(5,24,24,48) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 2, Divisor(<g>) = 2
  #I  Dividing by ClassTransposition(12,24,24,48) from the right.
  #I  Modulus(<g>) = 192, Multiplier(<g>) = 2, Divisor(<g>) = 2
  #I  Dividing by ClassTransposition(16,24,24,48) from the right.
  #I  Dividing by ClassTransposition(75,96,96,192) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 2, Divisor(<g>) = 2
  #I  p = 2, kmult = 1, kdiv = 1
  #I  Images of classes being multiplied by q*p^kmult:
  #I  [ 17(24) ]
  #I  Images of classes being divided by q*p^kdiv:
  #I  [ 12(24), 16(24) ]
  #I  Splitting classes being multiplied by q*p^kmult.
  #I  Found 4 pairs.
  #I  After filtering and splitting: 4 pairs.
  #I  Dividing by ClassTransposition(12,24,17,48) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 2, Divisor(<g>) = 2
  #I  Dividing by ClassTransposition(16,24,17,48) from the right.
  #I  Dividing by ClassTransposition(12,24,41,48) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 2, Divisor(<g>) = 2
  #I  Dividing by ClassTransposition(16,24,41,48) from the right.
  #I  Modulus(<g>) = 96, Multiplier(<g>) = 1, Divisor(<g>) = 1
  #I  Determining largest sources of affine mappings.
  #I  Computing respected partition.
  #I  Computing induced permutation on respected partition 
  [ 12(24), 14(24), 18(24), 13(48), 22(48), 23(48), 28(48), 31(48), 32(48), 
    33(48), 34(48), 35(48), 39(48), 44(48), 45(48), 0(96), 1(96), 2(96), 3(96), 
    4(96), 5(96), 6(96), 7(96), 8(96), 9(96), 10(96), 11(96), 15(96), 16(96), 
    17(96), 19(96), 20(96), 21(96), 24(96), 25(96), 26(96), 27(96), 29(96), 
    30(96), 37(96), 40(96), 41(96), 43(96), 46(96), 47(96), 48(96), 49(96), 
    50(96), 51(96), 52(96), 53(96), 54(96), 55(96), 56(96), 57(96), 58(96), 
    59(96), 63(96), 64(96), 65(96), 67(96), 68(96), 69(96), 72(96), 73(96), 
    74(96), 75(96), 77(96), 78(96), 85(96), 88(96), 89(96), 91(96), 94(96), 
    95(96) ].
  #I  Factoring the rest into class shifts.
  #I  Checking the result.
  [ ClassTransposition(4,6,7,12), ClassTransposition(2,6,1,12), 
    ClassTransposition(2,4,5,6), ClassTransposition(0,4,1,6), 
    ClassTransposition(5,6,4,8), ClassTransposition(1,6,0,8), 
    ClassTransposition(4,6,7,12), ClassTransposition(2,6,1,12), 
    ClassTransposition(2,4,5,6), ClassTransposition(0,4,1,6), 
    ClassTransposition(5,6,4,8), ClassTransposition(1,6,0,8), 
    ClassTransposition(4,6,7,12), ClassTransposition(2,6,1,12), 
    ClassTransposition(2,4,5,6), ClassTransposition(0,4,1,6), 
    ClassTransposition(5,6,4,8), ClassTransposition(1,6,0,8), 
    ClassTransposition(3,6,6,8), ClassTransposition(6,8,5,12), 
    ClassTransposition(6,8,7,12), ClassTransposition(6,8,8,12), 
    ClassTransposition(6,8,0,48), ClassTransposition(4,6,7,12), 
    ClassTransposition(2,6,1,12), ClassTransposition(2,4,5,6), 
    ClassTransposition(0,4,1,6), ClassTransposition(5,6,4,8), 
    ClassTransposition(1,6,0,8), ClassTransposition(6,8,7,12), 
    ClassTransposition(6,8,8,12), ClassTransposition(6,8,0,96), 
    ClassTransposition(4,6,7,12), ClassTransposition(2,6,1,12), 
    ClassTransposition(2,4,5,6), ClassTransposition(0,4,1,6), 
    ClassTransposition(5,6,4,8), ClassTransposition(1,6,0,8), 
    ClassTransposition(2,4,7,12), ClassTransposition(2,4,0,192), 
    ClassTransposition(4,6,7,12), ClassTransposition(2,6,1,12), 
    ClassTransposition(2,4,5,6), ClassTransposition(0,4,1,6), 
    ClassTransposition(5,6,4,8), ClassTransposition(1,6,0,8), 
    ClassTransposition(2,4,0,384), ClassTransposition(2,12,384,1536), 
    ClassTransposition(1,12,384,1536), ClassTransposition(6,12,384,1536), 
    ClassTransposition(10,12,384,1536), ClassTransposition(11,12,384,1536), 
    ClassTransposition(1,12,0,384), ClassTransposition(6,12,0,384), 
    ClassTransposition(10,12,0,384), ClassTransposition(11,12,0,384), 
    ClassTransposition(6,12,48,192), ClassTransposition(10,12,48,192), 
    ClassTransposition(11,12,48,192), ClassTransposition(10,24,192,384), 
    ClassTransposition(11,24,192,384), ClassTransposition(22,24,192,384), 
    ClassTransposition(23,24,192,384), ClassTransposition(11,12,72,96), 
    ClassTransposition(22,24,96,192), ClassTransposition(3,48,0,384), 
    ClassTransposition(9,48,0,384), ClassTransposition(15,48,0,384), 
    ClassTransposition(21,48,0,384), ClassTransposition(27,48,0,384), 
    ClassTransposition(33,48,0,384), ClassTransposition(39,48,0,384), 
    ClassTransposition(45,48,0,384), ClassTransposition(9,12,44,48), 
    ClassTransposition(15,24,48,96), ClassTransposition(27,96,192,384), 
    ClassTransposition(75,96,192,384), ClassTransposition(75,96,0,384), 
    ClassTransposition(7,12,4,24), ClassTransposition(7,12,17,24), 
    ClassTransposition(5,24,24,48), ClassTransposition(12,24,24,48), 
    ClassTransposition(16,24,24,48), ClassTransposition(75,96,96,192), 
    ClassTransposition(12,24,17,48), ClassTransposition(16,24,17,48), 
    ClassTransposition(12,24,41,48), ClassTransposition(16,24,41,48), 
    ClassTransposition(3,96,43,96), ClassTransposition(3,96,40,96), 
    ClassTransposition(3,96,26,96), ClassTransposition(3,96,30,96), 
    ClassTransposition(3,96,24,96), ClassTransposition(3,96,25,96), 
    ClassTransposition(3,96,29,96), ClassTransposition(3,96,11,96), 
    ClassTransposition(3,96,10,96), ClassTransposition(3,96,15,96), 
    ClassTransposition(3,96,27,96), ClassTransposition(3,96,51,96), 
    ClassTransposition(3,96,91,96), ClassTransposition(3,96,88,96), 
    ClassTransposition(3,96,74,96), ClassTransposition(3,96,78,96), 
    ClassTransposition(3,96,72,96), ClassTransposition(3,96,73,96), 
    ClassTransposition(3,96,77,96), ClassTransposition(3,96,59,96), 
    ClassTransposition(3,96,58,96), ClassTransposition(3,96,63,96), 
    ClassTransposition(3,96,75,96), ClassTransposition(0,96,95,96), 
    ClassTransposition(0,96,94,96), ClassTransposition(0,96,85,96), 
    ClassTransposition(0,96,89,96), ClassTransposition(0,96,49,96), 
    ClassTransposition(0,96,53,96), ClassTransposition(0,96,57,96), 
    ClassTransposition(0,96,56,96), ClassTransposition(0,96,55,96), 
    ClassTransposition(0,96,52,96), ClassTransposition(0,96,69,96), 
    ClassTransposition(0,96,68,96), ClassTransposition(0,96,65,96), 
    ClassTransposition(0,96,67,96), ClassTransposition(0,96,64,96), 
    ClassTransposition(0,96,50,96), ClassTransposition(0,96,54,96), 
    ClassTransposition(0,96,48,96), ClassTransposition(0,96,47,96), 
    ClassTransposition(0,96,46,96), ClassTransposition(0,96,37,96), 
    ClassTransposition(0,96,41,96), ClassTransposition(0,96,1,96), 
    ClassTransposition(0,96,5,96), ClassTransposition(0,96,9,96), 
    ClassTransposition(0,96,8,96), ClassTransposition(0,96,7,96), 
    ClassTransposition(0,96,4,96), ClassTransposition(0,96,21,96), 
    ClassTransposition(0,96,20,96), ClassTransposition(0,96,17,96), 
    ClassTransposition(0,96,19,96), ClassTransposition(0,96,16,96), 
    ClassTransposition(0,96,2,96), ClassTransposition(0,96,6,96), 
    ClassTransposition(13,48,35,48), ClassTransposition(13,48,34,48), 
    ClassTransposition(13,48,39,48), ClassTransposition(13,48,33,48), 
    ClassTransposition(13,48,32,48), ClassTransposition(13,48,31,48), 
    ClassTransposition(13,48,28,48), ClassTransposition(13,48,45,48), 
    ClassTransposition(13,48,44,48), ClassTransposition(13,48,23,48), 
    ClassTransposition(13,48,22,48), ClassTransposition(12,24,14,24), 
    ClassTransposition(12,24,18,24) ]
  gap> Product(last) = Collatz; # Check the result.
  true
  gap> Length(last2);
  159
  gap> RCWAInfo(0); # Switch Info output off again.
  
\end{Verbatim}
 See the end of Section \ref{sec:WildButFiniteCycles} for a much smaller factorization task which is performed ``manually'' for purposes of illustration. }

  
\section{\textcolor{Chapter }{ An rcwa mapping which seems to be contracting, but very slow }}\label{sec:SlowlyContractingMapping}
\logpage{[ 4, 2, 0 ]}
{
  The iterates of an integer under the Collatz mapping $T$ seem to approach its contraction centre -- this is the finite set where all
trajectories end up after a finite number of steps -- rather quickly and do
not get very large before doing so (of course this is a purely heuristic
statement as the $3n+1$ Conjecture has not been proven so far!): 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> T := RcwaMapping([[1,0,2],[3,1,2]]);;
  gap> S0 := LikelyContractionCentre(T,100,1000);
  #I  Warning: `LikelyContractionCentre' is highly probabilistic.
  The returned result can only be regarded as a rough guess.
  See ?LikelyContractionCentre for information on how to improve this guess.
  [ -136, -91, -82, -68, -61, -55, -41, -37, -34, -25, -17, -10, -7, -5, -1, 0, 
    1, 2 ]
  gap> S0^T = S0; # This holds by definition of the contraction centre.
  true
  gap> Trajectory(T,27,S0);
  [ 27, 41, 62, 31, 47, 71, 107, 161, 242, 121, 182, 91, 137, 206, 103, 155, 
    233, 350, 175, 263, 395, 593, 890, 445, 668, 334, 167, 251, 377, 566, 283, 
    425, 638, 319, 479, 719, 1079, 1619, 2429, 3644, 1822, 911, 1367, 2051, 
    3077, 4616, 2308, 1154, 577, 866, 433, 650, 325, 488, 244, 122, 61, 92, 46, 
    23, 35, 53, 80, 40, 20, 10, 5, 8, 4, 2 ]
  gap> List([1..40],n->Length(Trajectory(T,n,S0)));
  [ 1, 1, 5, 2, 4, 6, 11, 3, 13, 5, 10, 7, 7, 12, 12, 4, 9, 14, 14, 6, 6, 11, 
    11, 8, 16, 8, 70, 13, 13, 13, 67, 5, 18, 10, 10, 15, 15, 15, 23, 7 ]
  gap> Maximum(List([1..1000],n->Length(Trajectory(T,n,S0))));
  113
  gap> Maximum(List([1..1000],n->Maximum(Trajectory(T,n,S0))));
  125252
  
\end{Verbatim}
 \pagebreak[4] The following mapping also seems to be contracting, but its trajectories are
much longer: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> f6 := RcwaMapping([[ 1,0,6],[ 5, 1,6],[ 7,-2,6],
  >                       [11,3,6],[11,-2,6],[11,-1,6]]);;
  gap> SetName(f6,"f6");
  gap> Display(f6);
  
  Rcwa mapping of Z with modulus 6
  
                  n mod 6                |                 n^f6
  ---------------------------------------+--------------------------------------
    0                                    | n/6
    1                                    | (5n + 1)/6
    2                                    | (7n - 2)/6
    3                                    | (11n + 3)/6
    4                                    | (11n - 2)/6
    5                                    | (11n - 1)/6
  
  gap> S0 := LikelyContractionCentre(f6,1000,100000);;
  #I  Warning: `LikelyContractionCentre' is highly probabilistic.
  The returned result can only be regarded as a rough guess.
  gap> Trajectory(f6,25,S0);
  [ 25, 21, 39, 72, 12, 2 ]
  gap> List([1..100],n->Length(Trajectory(f6,n,S0)));
  [ 2, 2, 3, 4, 2, 2, 3, 2, 2, 5, 7, 2, 8, 17, 3, 16, 2, 4, 17, 6, 5, 2, 5, 5, 
    6, 2, 4, 2, 15, 2, 2, 3, 2, 5, 13, 3, 2, 3, 4, 2, 8, 4, 4, 2, 7, 19, 23517, 
    3, 9, 3, 2, 18, 14, 2, 20, 23512, 14, 2, 6, 6, 2, 4, 19, 12, 23511, 8, 
    23513, 10, 2, 13, 13, 3, 2, 23517, 7, 20, 7, 9, 9, 6, 12, 8, 6, 18, 14, 
    23516, 31, 12, 23545, 4, 21, 19, 5, 2, 17, 17, 13, 19, 6, 23515 ]
  gap> Maximum(Trajectory(f6,47,S0));;
  736339177776247330443187705477107581873369010805146980871580925673774229545698\
  886054
  
\end{Verbatim}
 Computing the trajectory of 3224 takes quite a while -- this trajectory
ascends to about $3 \cdot 10^{2197}$, before it approaches the fixed point~2 after 19949562 steps. 

 When constructing the mapping \texttt{f6}, the denominators of the partial mappings have been chosen to be equal and
the numerators have been chosen to be numbers coprime to the common
denominator, whose product is just a little bit smaller than the \texttt{Modulus(f6)}th power of the denominator. In the example we have $5 \cdot 7 \cdot 11^3 = 46585$ and $6^6 = 46656$. 

 Although the trajectories of \texttt{T} are much shorter than those of \texttt{f6}, it seems likely that this does not make the problem of deciding whether the
mapping~\texttt{T} is contracting essentially easier -- even for mappings with much shorter
trajectories than~\texttt{T} the problem seems to be equally hard. A solution can usually only be found in
trivial cases, i.e. for example when there is some $k$ such that applying the $k$th power of the respective mapping to any integer decreases its absolute
value. }

 \pagebreak[4]  
\section{\textcolor{Chapter }{Checking a result by P. Andaloro}}\label{sec:AndaloroResult}
\logpage{[ 4, 3, 0 ]}
{
  In \cite{Andaloro00}, P.~Andaloro has shown that proving that trajectories of integers $n \in 1(16)$ under the Collatz mapping always contain~1 would be sufficient to prove the $3n+1$ Conjecture. In the sequel, this result is verified by \textsf{RCWA}. Checking that the union of the images of the residue class 1(16) under
powers of the Collatz mapping~$T$ contains $\mathbb{Z} \setminus 0(3)$ is obviously enough. Thus we proceed by setting $S := 1(16)$ and successively uniting the set $S$ with its image under~$T$: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> S := ResidueClass(Integers,16,1);
  1(16)
  gap> S := Union(S,S^T);
  1(16) U 2(24)
  gap> S := Union(S,S^T);
  1(12) U 2(24) U 17(48) U 33(48)
  gap> S := Union(S,S^T);
  <union of 30 residue classes (mod 144)>
  gap> S := Union(S,S^T);
  <union of 42 residue classes (mod 144)>
  gap> S := Union(S,S^T);
  <union of 172 residue classes (mod 432)>
  gap> S := Union(S,S^T);
  <union of 676 residue classes (mod 1296)>
  gap> S := Union(S,S^T);
  <union of 810 residue classes (mod 1296)>
  gap> S := Union(S,S^T);
  <union of 2638 residue classes (mod 3888)>
  gap> S := Union(S,S^T);
  <union of 33 residue classes (mod 48)>
  gap> S := Union(S,S^T);
  <union of 33 residue classes (mod 48)>
  gap> Union(S,ResidueClass(Integers,3,0)); # Et voila ...
  Integers
  
\end{Verbatim}
 Further similar computations are shown in Section~\ref{sec:CollatzImagesAndPreImages}. }

 \pagebreak[4]  
\section{\textcolor{Chapter }{Two examples by Matthews and Leigh}}\label{sec:MatthewsLeighExamples}
\logpage{[ 4, 4, 0 ]}
{
  In \cite{MatthewsLeigh87}, K.~R.~Matthews and G.~M.~Leigh have shown that two trajectories of the following (surjective, but not
injective) mappings are acyclic (mod~$x$) and divergent: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> x := Indeterminate(GF(4),1);; SetName(x,"x");
  gap> R := PolynomialRing(GF(2),1);
  GF(2)[x]
  gap> ML1 := RcwaMapping(R,x,[[1,0,x],[(x+1)^3,1,x]]*One(R));;
  gap> ML2 := RcwaMapping(R,x,[[1,0,x],[(x+1)^2,1,x]]*One(R));;
  gap> SetName(ML1,"ML1"); SetName(ML2,"ML2");
  gap> Display(ML1);
  
  Rcwa mapping of GF(2)[x] with modulus x
  
           P mod x          |                       P^ML1
  --------------------------+---------------------------------------------------
   0*Z(2)                   | P/x
   Z(2)^0                   | ((x^3+x^2+x+Z(2)^0)*P + Z(2)^0)/x
  
  gap> Display(ML2);
  
  Rcwa mapping of GF(2)[x] with modulus x
  
           P mod x          |                       P^ML2
  --------------------------+---------------------------------------------------
   0*Z(2)                   | P/x
   Z(2)^0                   | ((x^2+Z(2)^0)*P + Z(2)^0)/x
  
  gap> List([ML1,ML2],IsSurjective);
  [ true, true ]
  gap> List([ML1,ML2],IsInjective);
  [ false, false ]
  gap> traj1 := Trajectory(ML1,One(R),16);
  [ Z(2)^0, x^2+x+Z(2)^0, x^4+x^2+x, x^3+x+Z(2)^0, x^5+x^4+x^2, x^4+x^3+x, 
    x^3+x^2+Z(2)^0, x^5+x^2+Z(2)^0, x^7+x^6+x^5+x^3+Z(2)^0, 
    x^9+x^7+x^6+x^5+x^3+x+Z(2)^0, x^11+x^10+x^8+x^7+x^6+x^5+x^2, 
    x^10+x^9+x^7+x^6+x^5+x^4+x, x^9+x^8+x^6+x^5+x^4+x^3+Z(2)^0, 
    x^11+x^8+x^7+x^6+x^4+x+Z(2)^0, x^13+x^12+x^11+x^8+x^7+x^6+x^4, 
    x^12+x^11+x^10+x^7+x^6+x^5+x^3 ]
  gap> traj2 := Trajectory(ML2,(x^3+x+1)*One(R),16);
  [ x^3+x+Z(2)^0, x^4+x+Z(2)^0, x^5+x^3+x^2+x+Z(2)^0, x^6+x^3+Z(2)^0, 
    x^7+x^5+x^4+x^2+x, x^6+x^4+x^3+x+Z(2)^0, x^7+x^4+x^3+x+Z(2)^0, 
    x^8+x^6+x^5+x^4+x^3+x+Z(2)^0, x^9+x^6+x^3+x+Z(2)^0, 
    x^10+x^8+x^7+x^5+x^4+x+Z(2)^0, x^11+x^8+x^7+x^5+x^4+x^3+x^2+x+Z(2)^0, 
    x^12+x^10+x^9+x^8+x^7+x^5+Z(2)^0, x^13+x^10+x^7+x^4+x, 
    x^12+x^9+x^6+x^3+Z(2)^0, x^13+x^11+x^10+x^8+x^7+x^5+x^4+x^2+x, 
    x^12+x^10+x^9+x^7+x^6+x^4+x^3+x+Z(2)^0 ]
  
\end{Verbatim}
 \pagebreak[4] The pattern which Matthews and Leigh used to show the divergence of the above
trajectories can be recognized easily by looking at the corresponding Markov
chains with the two states 0~mod~$x$ and 1~mod~$x$: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> traj1modx := Trajectory(ML1,One(R),400,x);;
  gap> traj2modx := Trajectory(ML2,(x^3+x+1)*One(R),600,x);;
  gap> List(traj1modx{[1..200]},val->Position([Zero(R),One(R)],val)-1);
  [ 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 
    1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
  gap> List(traj2modx{[1..200]},val->Position([Zero(R),One(R)],val)-1);
  [ 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 
    1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 
    0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
  
\end{Verbatim}
 What is important here are the lengths of the intervals between two changes
from one state to the other: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> ChangePoints := l -> Filtered([1..Length(l)-1],pos->l[pos]<>l[pos+1]);;
  gap> Diffs := l -> List([1..Length(l)-1],pos->l[pos+1]-l[pos]);;
  gap> Diffs(ChangePoints(traj1modx)); # The pattern in the first ...
  [ 1, 1, 2, 4, 2, 2, 4, 8, 4, 4, 8, 16, 8, 8, 16, 32, 16, 16, 32, 64, 32, 32, 
    64 ]
  gap> Diffs(ChangePoints(traj2modx)); # ... and in the second example.
  [ 1, 7, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 193, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
  gap> Diffs(ChangePoints(last)); # Make this a bit more obvious.
  [ 1, 3, 1, 7, 1, 15, 1, 31, 1, 63, 1 ]
  
\end{Verbatim}
 This looks clearly acyclic, thus the trajectories diverge. Needless to say
however that this computational evidence does not replace the proof along
these lines given in the article cited above, but just sheds a light on the
idea behind it. }

  
\section{\textcolor{Chapter }{Exploring the structure of a wild rcwa group}}\label{sec:StructureOfWildGroup}
\logpage{[ 4, 5, 0 ]}
{
  In this example, a simple attempt to should be made to investigate the
structure of a given wild group by finding orders of torsion elements. In
general, determining the structure of a given wild group computationally seems
to be a very hard task. First of all, the group in question has to be defined: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
  gap> SetName(u,"u");
  gap> Display(u);
  
  Rcwa mapping of Z with modulus 5
  
                  n mod 5                |                 n^u
  ---------------------------------------+--------------------------------------
    0                                    | 3n/5
    1                                    | (9n + 1)/5
    2                                    | (3n - 1)/5
    3                                    | (9n - 2)/5
    4                                    | (9n + 4)/5
  
  gap> nu := RcwaMapping([[1,1,1]]);
  Rcwa mapping of Z: n -> n + 1
  gap> SetName(nu,"nu");
  gap> G := Group(u,nu);
  <rcwa group over Z with 2 generators>
  gap> IsTame(G);
  false
  
\end{Verbatim}
 Now we would like to know which orders torsion elements of~\texttt{G} can have -- taking a look at the above generators it seems to make sense to
try commutators: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> l := Filtered([0..100],k->IsTame(Comm(u,nu^k)));
  [ 0, 2, 3, 5, 6, 9, 10, 12, 13, 15, 17, 18, 20, 21, 24, 25, 27, 28, 30, 32, 
    33, 35, 36, 39, 40, 42, 43, 45, 47, 48, 50, 51, 54, 55, 57, 58, 60, 62, 63, 
    65, 66, 69, 70, 72, 73, 75, 77, 78, 80, 81, 84, 85, 87, 88, 90, 92, 93, 95, 
    96, 99, 100 ]
  gap> List(l,k->Order(Comm(u,nu^k)));
  [ 1, 6, 5, 3, 5, 5, 3, infinity, 7, infinity, 7, 5, 3, infinity, infinity, 3, 
    5, 7, infinity, 7, infinity, 3, 5, 5, 3, 5, infinity, infinity, infinity, 
    5, 3, 5, 5, 3, infinity, 7, infinity, 7, 5, 3, infinity, infinity, 3, 5, 7, 
    infinity, 7, infinity, 3, 5, 5, 3, 5, infinity, infinity, infinity, 5, 3, 
    5, 5, 3 ]
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(Comm(u,nu^13));
  
  Bijective rcwa mapping of Z with modulus 9
  
                  n mod 9                |                 n^f
  ---------------------------------------+--------------------------------------
    0 3 6                                | n + 5
    1 4 7                                | 3n - 9
    2 8                                  | n - 11
    5                                    | (n + 16)/3
  
  gap> Order(Comm(u,nu^13));
  7
  gap> u2 := u^2;
  <wild bijective rcwa mapping of Z with modulus 25>
  gap> Filtered([1..16],k->IsTame(Comm(u2,nu^k))); # k < 15 -> commutator wild!
  [ 15 ]
  gap> Order(Comm(u2,nu^15));
  infinity
  gap> u2nu17 := Comm(u2,nu^17);
  <bijective rcwa mapping of Z with modulus 81>
  gap> orbs := ShortOrbits(Group(u2nu17),[-100..100],100);;
  gap> List(orbs,Length);
  [ 72, 72, 73, 72, 73, 72, 72, 73, 72, 72, 72, 73, 72, 72, 73, 72, 72, 73, 72, 
    72, 73, 72, 72 ]
  gap> Lcm(last);
  5256
  gap> u2nu17^5256; # This element has indeed order 2^3*3^2*73 = 5256.
  IdentityMapping( Integers )
  gap> u2nu18 := Comm(u2,nu^18);
  <bijective rcwa mapping of Z with modulus 81>
  gap> orbs := ShortOrbits(Group(u2nu18),[-100..100],100);;
  gap> List(orbs,Length);
  [ 22, 22, 22, 21, 22, 22, 22, 21, 21, 22, 22, 21, 22, 21, 22, 22, 21, 22, 22, 
    21, 22, 22, 21 ]
  gap> Lcm(last);
  462
  gap> u2nu18^462; # This is an element of order 2*3*7*11 = 462.
  IdentityMapping( Integers )
  gap> Order(Comm(u2,nu^20));
  29
  gap> Order(Comm(u2,nu^25));
  9
  gap> Order(Comm(u2,nu^30));
  15
  
\end{Verbatim}
 Thus even this rather simple-minded approach reveals various different orders
of torsion elements, and the involved primes are also not all quite ``small''. }

  
\section{\textcolor{Chapter }{A wild rcwa mapping which has only finite cycles}}\label{sec:WildButFiniteCycles}
\logpage{[ 4, 6, 0 ]}
{
  Some wild rcwa mappings of $\mathbb{Z}$ have only finite cycles. In this section, a permutation is examined which can
be shown to be such a mapping and which is likely to be something like a ``minimal'' example. 

 Over $R$ = GF($q$)[$x$], constructing such mappings is easy since the degree function gives rise to
a partition of $R$ into finite sets which is left invariant by suitable wild rcwa mappings. Over $R = \mathbb{Z}$ however the situation looks different -- there is no such ``natural'' partition into finite sets which can be fixed by a wild rcwa mapping. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> kappa := RcwaMapping([[1,0,1],[1,0,1],[3,2,2],[1,-1,1],
  >                          [2,0,1],[1,0,1],[3,2,2],[1,-1,1],
  >                          [1,1,3],[1,0,1],[3,2,2],[2,-2,1]]);;
  gap> SetName(kappa,"kappa");
  gap> List([-5..5],k->Modulus(kappa^k));
  [ 7776, 1296, 432, 72, 24, 1, 12, 72, 144, 864, 1728 ]
  gap> Display(kappa);
  
  Bijective rcwa mapping of Z with modulus 12
  
                 n mod 12                |               n^kappa
  ---------------------------------------+--------------------------------------
     0  1  5  9                          | n
     2  6 10                             | (3n + 2)/2
     3  7                                | n - 1
     4                                   | 2n
     8                                   | (n + 1)/3
    11                                   | 2n - 2
  
  gap> List([-32..32],n->Length(Cycle(kappa,n)));
  [ 4, 1, 4, 4, 7, 1, 10, 10, 1, 1, 4, 4, 7, 1, 10, 10, 4, 1, 7, 7, 1, 1, 7, 7, 
    4, 1, 4, 4, 2, 1, 1, 2, 1, 1, 4, 4, 4, 1, 7, 7, 4, 1, 7, 7, 1, 1, 10, 10, 
    7, 1, 4, 4, 7, 1, 10, 10, 1, 1, 4, 4, 4, 1, 13, 13, 7 ]
  gap> List([2..14],k->Maximum(List([1..2^k],n->Length(Cycle(kappa,n)))));
  [ 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 ]
  gap> List([2..14],k->Length(Cycle(kappa,2^k-2)));
  [ 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 ]
  gap> Cycle(kappa,2^12-2);
  [ 4094, 6142, 9214, 13822, 20734, 31102, 46654, 69982, 104974, 157462, 
    236194, 354292, 708584, 236195, 472388, 157463, 314924, 104975, 209948, 
    69983, 139964, 46655, 93308, 31103, 62204, 20735, 41468, 13823, 27644, 
    9215, 18428, 6143, 12284, 4095 ]
  gap> last mod 12;
  [ 2, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 4, 8, 11, 8, 11, 8, 11, 8, 11, 
    8, 11, 8, 11, 8, 11, 8, 11, 8, 11, 8, 11, 8, 3 ]
  gap> lengthstatistics := Collected(List(ShortOrbits(Group(kappa),
  >                                                   [1..12^4],100),Length));
  [ [ 1, 6912 ], [ 4, 1728 ], [ 7, 864 ], [ 10, 432 ], [ 13, 216 ], 
    [ 16, 108 ], [ 19, 54 ], [ 22, 27 ], [ 25, 13 ], [ 28, 7 ], [ 31, 3 ], 
    [ 34, 2 ], [ 37, 1 ], [ 40, 1 ] ]
  
\end{Verbatim}
 \pagebreak[4] We would like to determine a partition of~$\Z$ into unions of cycles of equal length: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> C := [Difference(Integers,MovedPoints(kappa))];; pow := [kappa^0];;
  gap> rc := function(r,m) return ResidueClass(r,m); end;;
  gap> for i in [1..3] do
  >      Add(pow,kappa^i);
  >      C[i+1] := Difference(rc(2,4),
  >                           Union(Union(C{[1..i]}),
  >                                 Union(List([0..i],
  >                                       j->Intersection(rc(2,4)^pow[j+1],
  >                                                       rc(2,4)^(pow[i-j+1]^-1))))));
  >    od;
  gap> C;
  [ 1(4) U 0(12) U [ -2 ], 2(24) U 18(24), 6(48) U 38(48) U 10(72) U 58(72), 
    <union of 38 residue classes (mod 864)> ]
  gap> List(C,S->Length(Cycle(kappa,S)));
  [ 1, 4, 7, 10 ]
  gap> Cycle(kappa,C[1]);
  [ 1(4) U 0(12) U [ -2 ] ]
  gap> Cycle(kappa,C[2]);
  [ 2(24) U 18(24), 4(36) U 28(36), 8(72) U 56(72), 3(24) U 19(24) ]
  gap> cycle7 := Cycle(kappa,C[3]);;
  gap> for S in cycle7 do View(S); Print("\n"); od;
  6(48) U 38(48) U 10(72) U 58(72)
  10(72) U 58(72) U 16(108) U 88(108)
  16(108) U 88(108) U 32(216) U 176(216)
  11(72) U 59(72) U 32(216) U 176(216)
  11(72) U 59(72) U 20(144) U 116(144)
  7(48) U 39(48) U 20(144) U 116(144)
  6(48) U 7(48) U 38(48) U 39(48)
  gap> cycle10 := Cycle(kappa,C[4]);;
  gap> for S in cycle10 do View(S); Print("\n"); od;
  <union of 38 residue classes (mod 864)>
  <union of 38 residue classes (mod 1296)>
  <union of 12 residue classes (mod 648)>
  <union of 12 residue classes (mod 648)>
  <union of 22 residue classes (mod 1296)>
  <union of 12 residue classes (mod 432)>
  <union of 22 residue classes (mod 864)>
  <union of 12 residue classes (mod 288)>
  <union of 14 residue classes (mod 288)>
  <union of 16 residue classes (mod 288)>
  gap> List(cycle10,Density);
  [ 19/432, 19/648, 1/54, 1/54, 11/648, 1/36, 11/432, 1/24, 7/144, 1/18 ]
  gap> List(last,Float);
  [ 0.0439815, 0.029321, 0.0185185, 0.0185185, 0.0169753, 0.0277778, 0.025463, 
    0.0416667, 0.0486111, 0.0555556 ]
  gap> Sum(last2);
  47/144
  gap> Density(Union(cycle10));
  47/432
  
\end{Verbatim}
 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> P := List(C,S->Union(Cycle(kappa,S)));;
  gap> for S in P do View(S); Print("\n"); od;
  1(4) U 0(12) U [ -2 ]
  <union of 18 residue classes (mod 72)>
  <union of 78 residue classes (mod 432)>
  <union of 282 residue classes (mod 2592)>
  gap> P2 := AsUnionOfFewClasses(P[2]);
  [ 2(24), 3(24), 18(24), 19(24), 4(36), 28(36), 8(72), 56(72) ]
  gap> Permutation(kappa,P2);
  (1,5,7,2)(3,6,8,4)
  gap> P3 := AsUnionOfFewClasses(P[3]);
  [ 6(48), 7(48), 38(48), 39(48), 10(72), 11(72), 58(72), 59(72), 16(108), 
    88(108), 20(144), 116(144), 32(216), 176(216) ]
  gap> Permutation(kappa,P3);
  (1,5,9,13,6,11,2)(3,7,10,14,8,12,4)
  gap> P4 := AsUnionOfFewClasses(P[4]);
  [ 14(96), 15(96), 78(96), 79(96), 22(144), 23(144), 118(144), 119(144), 
    34(216), 35(216), 178(216), 179(216), 44(288), 236(288), 52(324), 268(324), 
    68(432), 356(432), 104(648), 536(648) ]
  gap> Permutation(kappa,P4);
  (1,5,9,15,19,10,17,6,13,2)(3,7,11,16,20,12,18,8,14,4)
  gap> List(P,S->Set(List(Intersection([1..12^4],S),n->Length(Cycle(kappa,n)))));
  [ [ 1 ], [ 4 ], [ 7 ], [ 10 ] ]
  gap> Set(List(Intersection([1..12^4],Difference(Integers,Union(P))),
  >             n->Length(Cycle(kappa,n))));
  [ 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 ]
  
\end{Verbatim}
 Finally, the permutation \texttt{kappa} should be factored into involutions (this time ``by hand'', for purposes of illustration): 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> elm1 := kappa;
  kappa
  gap> Multpk(elm1,2,1)^elm1;
  8(12)
  gap> Multpk(elm1,2,-1)^elm1;
  4(6)
  gap> Multpk(elm1,3,1)^elm1;
  4(6)
  gap> Multpk(elm1,3,-1)^elm1;
  3(4)
  gap> fact1 := RcwaMapping([[rc(4,6),rc(8,12)]]);
  <rcwa mapping of Z with modulus 12>
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> elm2 := elm1/fact1;
  <bijective rcwa mapping of Z with modulus 12>
  gap> Display(elm2);
  
  Bijective rcwa mapping of Z with modulus 12
  
                 n mod 12                |                 n^f
  ---------------------------------------+--------------------------------------
     0  1  4  5  9                       | n
     2  6 10                             | 3n + 2
     3  7 11                             | n - 1
     8                                   | (n + 1)/3
  
  gap> Multpk(elm2,3,1)^elm2;
  8(12)
  gap> Multpk(elm2,3,-1)^elm2;
  3(4)
  gap> fact2 := RcwaMapping([[rc(3,4),rc(8,12)]]);
  <rcwa mapping of Z with modulus 12>
  gap> elm3 := elm2/fact2;
  <bijective rcwa mapping of Z with modulus 4>
  gap> Display(elm3);
  
  Bijective rcwa mapping of Z with modulus 4
  
                  n mod 4                |                 n^f
  ---------------------------------------+--------------------------------------
    0 1                                  | n
    2                                    | n + 1
    3                                    | n - 1
  
  gap> fact3 := RcwaMapping([[rc(2,4),rc(3,4)]]);
  <rcwa mapping of Z with modulus 4>
  gap> elm4 := elm3/fact3;
  IdentityMapping( Integers )
  gap> kappafacts := [ fact3, fact2, fact1 ];
  [ <bijective rcwa mapping of Z with modulus 4>, 
    <bijective rcwa mapping of Z with modulus 12>, 
    <bijective rcwa mapping of Z with modulus 12> ]
  gap> List(kappafacts,Order);
  [ 2, 2, 2 ]
  gap> kappa = Product(kappafacts);
  true
  
\end{Verbatim}
 }

 \pagebreak[4]  
\section{\textcolor{Chapter }{An abelian rcwa group over a polynomial ring}}\label{sec:AbelianModularGroup}
\logpage{[ 4, 7, 0 ]}
{
  In this section, a wild rcwa group over GF(4)[$x$] should be invstigated, which happens to be abelian. Of course in general,
rcwa groups also over this ring are usually far from being abelian (see
below). We start by defining this group: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> x := Indeterminate(GF(4),1);; SetName(x,"x");
  gap> R := PolynomialRing(GF(4),1);
  GF(2^2)[x]
  gap> e := One(GF(4));;
  gap> p := x^2 + x + e;;    q := x^2 + e;;
  gap> r := x^2 + x + Z(4);; s := x^2 + x + Z(4)^2;;
  gap> cg := List( AllResidues(R,x^2), pol -> [ p, p * pol mod q, q ] );;
  gap> ch := List( AllResidues(R,x^2), pol -> [ r, r * pol mod s, s ] );;
  gap> g := RcwaMapping( R, q, cg );
  <rcwa mapping of GF(2^2)[x] with modulus x^2+Z(2)^0>
  gap> h := RcwaMapping( R, s, ch );
  <rcwa mapping of GF(2^2)[x] with modulus x^2+x+Z(2^2)^2>
  gap> List([g,h],Order);
  [ infinity, infinity ]
  gap> List([g,h],IsTame);
  [ false, false ]
  gap> G := Group(g,h);
  <rcwa group over GF(2^2)[x] with 2 generators>
  gap> IsAbelian(G);
  true
  
\end{Verbatim}
 Now we compute the action of the group \texttt{G} on one of its orbits, and make some statistics of the orbits of \texttt{G} containing polynomials of degree less than~4: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> orb := Orbit(G,x^5);
  [ x^5, x^5+x^4+x^2+Z(2)^0, x^5+x^3+x^2+Z(2^2)*x+Z(2)^0, x^5+x^3, 
    x^5+x^4+x^3+x^2+Z(2^2)^2*x+Z(2^2)^2, x^5+x, x^5+x^4+x^3, x^5+x^2+Z(2^2)^2*x,
    x^5+x^4+x^2+x, x^5+x^3+x^2+Z(2^2)^2*x+Z(2)^0, x^5+x^4+Z(2^2)*x+Z(2^2), 
    x^5+x^3+x, x^5+x^4+x^3+x^2+Z(2^2)*x+Z(2^2), x^5+x^4+x^3+x+Z(2)^0, 
    x^5+x^2+Z(2^2)*x, x^5+x^4+Z(2^2)^2*x+Z(2^2)^2 ]
  gap> H := Action(G,orb);
  Group([ (1,2,4,7,6,9,12,14)(3,5,8,11,10,13,15,16), 
    (1,3,6,10)(2,5,9,13)(4,8,12,15)(7,11,14,16) ])
  gap> IsAbelian(H); # check ...
  true
  gap> Exponent(H);
  8
  gap> Collected(List(ShortOrbits(G,AllResidues(R,x^4),100),Length));
  [ [ 1, 4 ], [ 2, 6 ], [ 4, 12 ], [ 8, 24 ] ]
  
\end{Verbatim}
 \pagebreak[4] Changing the generators a little causes the group structure to change a lot: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> cg[1][2] := cg[1][2] + (x^2 + e) * p * q;;
  gap> ch[7][2] := ch[7][2] + x * r * s;;
  gap> g := RcwaMapping( R, q, cg );; h := RcwaMapping( R, s, ch );;
  gap> G := Group(g,h);
  <rcwa group over GF(2^2)[x] with 2 generators>
  gap> orb := Orbit(G,Zero(R));;
  gap> Length(orb);
  87
  gap> Collected(List(orb,DegreeOfLaurentPolynomial));
  [ [ 1, 2 ], [ 2, 4 ], [ 3, 16 ], [ 4, 64 ], [ infinity, 1 ] ]
  gap> H := Action(G,orb);
  <permutation group with 2 generators>
  gap> IsNaturalAlternatingGroup(H);
  true
  gap> orb := Orbit(G,x^6);;
  gap> Length(orb);
  512
  gap> H := Action(G,orb);
  <permutation group with 2 generators>
  gap> IsNaturalSymmetricGroup(H) or IsNaturalAlternatingGroup(H);
  false
  gap> blk := Blocks(H,[1..512]);;
  gap> List(blk,Length);
  [ 128, 128, 128, 128 ]
  gap> Action(H,blk,OnSets);
  Group([ (1,2)(3,4), (1,3)(2,4) ])
  
\end{Verbatim}
 Thus the modified group has a quotient isomorphic to the alternating group of
degree~87, and a quotient isomorphic to some wreath product or a subgroup thereof
acting transitively, but not primitively on 512 points. }

  
\section{\textcolor{Chapter }{An rcwa representation of a small group}}\label{sec:SubgroupOfS9}
\logpage{[ 4, 8, 0 ]}
{
  In the sequel, an rcwa representation of the 3-Sylow-subgroup of the symmetric
group on 9~points is given. Of course this group has a very nice permutation
representation, hence for computational purposes one does not gain anything
here. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> r := RcwaMapping([[1,0,1],[1,1,1],[3,-3,1],
  >                      [1,0,3],[1,1,1],[3,-3,1],
  >                      [1,0,1],[1,1,1],[3,-3,1]]);;
  gap> s := RcwaMapping([[1,0,1],[1,1,1],[3,6,1],
  >                      [1,0,3],[1,1,1],[3,6,1],
  >                      [1,0,1],[1,1,1],[3,-21,1]]);;
  gap> SetName(r,"r"); SetName(s,"s");
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(r);
  
  Rcwa mapping of Z with modulus 9
  
                  n mod 9                |                 n^r
  ---------------------------------------+--------------------------------------
    0 6                                  | n
    1 4 7                                | n + 1
    2 5 8                                | 3n - 3
    3                                    | n/3
  
  gap> Display(s);
  
  Rcwa mapping of Z with modulus 9
  
                  n mod 9                |                 n^s
  ---------------------------------------+--------------------------------------
    0 6                                  | n
    1 4 7                                | n + 1
    2 5                                  | 3n + 6
    3                                    | n/3
    8                                    | 3n - 21
  
  gap> G := Group(r,s);
  <rcwa group over Z with 2 generators>
  gap> H := SylowSubgroup(SymmetricGroup(9),3);
  Group([ (1,2,3), (4,5,6), (7,8,9), (1,4,7)(2,5,8)(3,6,9) ])
  gap> phi := InverseGeneralMapping(IsomorphismGroups(G,H));;
  gap> (1,2,3)^phi;
  <bijective rcwa mapping of Z with modulus 27>
  
\end{Verbatim}
 }

  
\section{\textcolor{Chapter }{ An rcwa representation of the symmetric group on 10 points }}\label{sec:RcwaRepresentationOfS10}
\logpage{[ 4, 9, 0 ]}
{
  In this section, an rcwa representation of the symmetric group on 10~points should be investigated. We start by defining some bijections of
infinite order and computing commutators: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> a := RcwaMapping([[3,0,2],[3, 1,4],[3,0,2],[3,-1,4]]);;
  gap> b := RcwaMapping([[3,0,2],[3,13,4],[3,0,2],[3,-1,4]]);;
  gap> c := RcwaMapping([[3,0,2],[3, 1,4],[3,0,2],[3,11,4]]);;
  gap> SetName(a,"a"); SetName(b,"b"); SetName(c,"c");
  gap> List([a,b,c],Order);
  [ infinity, infinity, infinity ]
  gap> ab := Comm(a,b);; ac := Comm(a,c);; bc := Comm(b,c);;
  gap> SetName(ab,"[a,b]"); SetName(ac,"[a,c]"); SetName(bc,"[b,c]");
  gap> List([ab,ac,bc],Order);
  [ 6, 6, 12 ]
  
\end{Verbatim}
 Now we would like to have a look at [\mbox{\texttt{a}},\mbox{\texttt{b}}] ... 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(ab);
  
  Bijective rcwa mapping of Z with modulus 18, of order 6
  
                 n mod 18                |               n^[a,b]
  ---------------------------------------+--------------------------------------
     0  2  3  8  9 11 12 17              | n
     1 10                                | 2n - 5
     4  7 13 16                          | n + 3
     5 14                                | 2n - 4
     6                                   | (n + 2)/2
    15                                   | (n - 5)/2
  
  
\end{Verbatim}
 ... form the group generated by [\mbox{\texttt{a}},\mbox{\texttt{b}}] and [\mbox{\texttt{a}},\mbox{\texttt{c}}] and compute its action on one of its orbits: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(ab,ac);
  <rcwa group over Z with 2 generators>
  gap> orb := Orbit(G,1);
  [ -15, -12, -7, -6, -5, -4, -3, -2, -1, 1 ]
  gap> H := Action(G,orb);
  Group([ (2,5,8,10,7,6), (1,3,6,9,4,5) ])
  gap> Size(H);
  3628800
  gap> Size(G); # G acts faithfully on orb.
  3628800
  
\end{Verbatim}
 Hence the group \mbox{\texttt{G}} is isomorphic to the symmetric group on 10~points and acts faithfully on the orbit containing~1. Another question is which groups arise if we take as generators either \mbox{\texttt{ab}}, \mbox{\texttt{ac}} or \mbox{\texttt{bc}} and the mapping~\mbox{\texttt{t}}, which maps each integer to its additive inverse: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> t := ClassReflection(0,1);;
  gap> Display(t);
  Bijective rcwa mapping of Z: n -> -n
  gap> G := Group(ab,t);
  <rcwa group over Z with 2 generators>
  gap> Size(G);
  7257600
  gap> phi := IsomorphismPermGroup(G);
  [ [a,b], ClassReflection(0,1) ] -> [ (2,3,4,6,9,13)(5,8,11,16,10,15),
    (1,2)(3,5)(4,7)(6,10)(8,12)(9,14)(11,17)(13,18)(15,19)(16,20) ]
  gap> H := Group((1,2),(1,2,3,4,5,6,7,8,9,10),(11,12));;
  gap> IsomorphismGroups(Image(phi),H) <> fail; # G = C2 x S10
  true
  
\end{Verbatim}
 Thus the group generated by \mbox{\texttt{ab}} and \mbox{\texttt{t}} is isomorphic to ${\rm C}_2 \times {\rm S}_{10}$ . The next group is an extension of a perfect group of order~960: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(ac,t);;
  gap> Size(G);
  3840
  gap> H := Image(IsomorphismPermGroup(G));;
  gap> P := DerivedSubgroup(H);;
  gap> Size(P);
  960
  gap> IsPerfect(P);
  true
  gap> PerfectGroup(PerfectIdentification(P));
  A5 2^4'
  
\end{Verbatim}
 The last group is infinite: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(bc,t);;
  gap> Size(G);
  infinity
  gap> Order(bc*t);
  infinity
  gap> Modulus(G);
  18
  gap> RespectedPartition(G);
  [ 1(9), 2(9), 4(9), 5(9), 7(9), 8(9), 0(18), 3(18), 6(18), 9(18), 12(18),
    15(18) ]
  gap> ActionOnRespectedPartition(G);
  Group([ (1,5,8,2,4,12)(3,9,6,11), (1,6)(2,5)(3,4)(8,12)(9,11) ])
  gap> IsNaturalSymmetricGroup(last);
  true
  gap> RankOfKernelOfActionOnRespectedPartition(G:ProperSubgroupAllowed);
  9
  
\end{Verbatim}
 }

 \pagebreak[4]  
\section{\textcolor{Chapter }{Checking for solvability}}\label{sec:CheckingForSolvability}
\logpage{[ 4, 10, 0 ]}
{
  Is the group generated by the mappings \mbox{\texttt{a}} and \mbox{\texttt{b}} from the last paragraph solvable? 

 This group is wild. Presently there is no general method available for testing
wild rcwa groups for solvability. But nevertheless, for the given group this
question can be decided to the negative. The idea is to find a subgroup \mbox{\texttt{U}} which acts on a finite set \mbox{\texttt{S}} of integers, and induces on \mbox{\texttt{S}} a non-solvable finite permutation group: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(a,b);;
  gap> ShortOrbits(Group(Comm(a,b)),[-10..10],100);
  [ [ -10 ], [ -9 ], [ -30, -21, -14, -13, -11, -8 ], [ -7 ], [ -6 ], 
    [ -12, -5, -4, -3, -2, 1 ], [ -1 ], [ 0 ], [ 2 ], [ 3 ], 
    [ 4, 5, 6, 7, 10, 15 ], [ 8 ], [ 9 ] ]
  gap> S := [ 4, 5, 6, 7, 10, 15 ];;
  gap> Cycle(Comm(a,b),4);
  [ 4, 7, 10, 15, 5, 6 ]
  gap> elm := RepresentativeAction(G,S,Permuted(S,(1,4)),OnTuples);
  <bijective rcwa mapping of Z with modulus 81>
  gap> List(S,n->n^elm);
  [ 7, 5, 6, 4, 10, 15 ]
  gap> U := Group(Comm(a,b),elm);
  <rcwa group over Z with 2 generators>
  gap> Action(U,S);
  Group([ (1,4,5,6,2,3), (1,4) ])
  gap> IsNaturalSymmetricGroup(last);
  true
  
\end{Verbatim}
 Thus, the subgroup \mbox{\texttt{U}} induces on \mbox{\texttt{S}} a natural symmetric group of degree~6. Hence the group~\mbox{\texttt{G}} is not solvable, as claimed. We finish this example by factoring the group
element \mbox{\texttt{elm}} into generators: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> F := FreeGroup("a","b");
  <free group on the generators [ a, b ]>
  gap> RepresentativeActionPreImage(G,S,Permuted(S,(1,4)),OnTuples,F);
  a^-2*b^-2*a*b*a^-1*b*a*b^-2*a
  gap> a^-2*b^-2*a*b*a^-1*b*a*b^-2*a = elm;
  true
  
\end{Verbatim}
 }

 \pagebreak[4]  
\section{\textcolor{Chapter }{Some examples over (semi)localizations of the integers}}\label{sec:LocalExample}
\logpage{[ 4, 11, 0 ]}
{
  We start with something one can observe when trying to ``transfer'' an rcwa mapping from the ring of integers to one of its localizations (we take
the mapping \texttt{a} from the previous examples): 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> a2 := LocalizedRcwaMapping(a,2);
  <rcwa mapping of Z_( 2 ) with modulus 4>
  gap> IsSurjective(a2); # As expected
  true
  gap> IsInjective(a2); # Why not??
  false
  gap> 0^a2;
  0
  gap> (1/3)^a2; # That's the reason!
  0
  
\end{Verbatim}
 The above can also be explained easily by pointing out that the modulus of the
inverse of \texttt{a} is 3, and that 3 is a unit of $\mathbb{Z}_{(2)}$. Moving to $\mathbb{Z}_{(2,3)}$ solves this problem: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> a23 := SemilocalizedRcwaMapping(a,[2,3]);
  <rcwa mapping of Z_( 2, 3 ) with modulus 4>
  gap> IsBijective(a23);
  true
  
\end{Verbatim}
 We get additional finite cycles, e.g.: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List(ShortOrbits(Group(a23),[0..50]/5,50),orb->Cycle(a23,orb[1]));
  [ [ 0 ], [ 1/5, 2/5, 3/5 ], 
    [ 4/5, 6/5, 9/5, 8/5, 12/5, 18/5, 27/5, 19/5, 13/5, 11/5, 7/5 ], [ 1 ], 
    [ 2, 3 ], [ 14/5, 21/5, 17/5 ], 
    [ 16/5, 24/5, 36/5, 54/5, 81/5, 62/5, 93/5, 71/5, 52/5, 78/5, 117/5, 89/5, 
        68/5, 102/5, 153/5, 116/5, 174/5, 261/5, 197/5, 149/5, 113/5, 86/5, 
        129/5, 98/5, 147/5, 109/5, 83/5, 61/5, 47/5, 34/5, 51/5, 37/5, 29/5, 
        23/5 ], [ 4, 6, 9, 7, 5 ] ]
  gap> List(last,Length);
  [ 1, 3, 11, 1, 2, 3, 34, 5 ]
  gap> List(ShortOrbits(Group(a23),[0..50]/7,50),orb->Cycle(a23,orb[1]));
  [ [ 0 ], [ -1/7, 1/7 ], [ 2/7, 3/7, 4/7, 6/7, 9/7, 5/7 ], [ 1 ], [ 2, 3 ], 
    [ 4, 6, 9, 7, 5 ] ]
  gap> List(last,Length);
  [ 1, 2, 6, 1, 2, 5 ]
  
\end{Verbatim}
 \pagebreak[4] But the group structure remains invariant under the ``transfer'' of a group with prime set $\{2,3\}$ from $\Z$ to $\Z_{(2,3)}$: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> b23 := SemilocalizedRcwaMapping(b,[2,3]);;
  gap> c23 := SemilocalizedRcwaMapping(c,[2,3]);;
  gap> ab23 := Comm(a23,b23);
  <rcwa mapping of Z_( 2, 3 ) with modulus 18>
  gap> ac23 := Comm(a23,c23);
  <rcwa mapping of Z_( 2, 3 ) with modulus 18>
  gap> G := Group(ab23,ac23);
  <rcwa group over Z_( 2, 3 ) with 2 generators>
  gap> S := Intersection(Enumerator(Rationals){[1..200]},Z_pi([2,3]));
  [ -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -12/5, -11/5, -2, -9/5, -12/7, 
    -8/5, -11/7, -10/7, -7/5, -9/7, -6/5, -8/7, -12/11, -1, -10/11, -6/7, 
    -9/11, -4/5, -8/11, -5/7, -7/11, -3/5, -4/7, -6/11, -5/11, -3/7, -2/5, 
    -4/11, -2/7, -3/11, -1/5, -2/11, -1/7, -1/11, 0, 1/13, 1/11, 1/7, 2/13, 
    2/11, 1/5, 3/13, 3/11, 2/7, 4/13, 4/11, 5/13, 2/5, 3/7, 5/11, 6/13, 7/13, 
    6/11, 4/7, 3/5, 8/13, 7/11, 9/13, 5/7, 8/11, 10/13, 4/5, 9/11, 11/13, 6/7, 
    10/11, 12/13, 1, 12/11, 8/7, 13/11, 6/5, 9/7, 7/5, 10/7, 11/7, 8/5, 12/7, 
    9/5, 2, 11/5, 12/5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ]
  gap> orbs := ShortOrbits(G,S,50);;
  gap> List(orbs,Length);
  [ 10, 10, 1, 10, 1, 10, 10, 10, 10, 10, 1, 10, 10, 10, 1, 10, 10, 10, 10, 10, 
    10, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 
    10, 1, 10, 10, 10, 1, 1, 10, 1, 10 ]
  gap> ForAll(orbs,orb->IsNaturalSymmetricGroup(Action(G,orb)));
  true
  
\end{Verbatim}
 ``Transferring'' a non-invertible rcwa mapping from the ring of integers to some of its
(semi)localizations can also turn it into an invertible one: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> v := RcwaMapping([[6,0,1],[1,-7,2],[6,0,1],[1,-1,1],
  >                      [6,0,1],[1, 1,2],[6,0,1],[1,-1,1]]);;
  gap> SetName(v,"v");
  gap> Display(v);
  
  Rcwa mapping of Z with modulus 8
  
                  n mod 8                |                 n^v
  ---------------------------------------+--------------------------------------
    0 2 4 6                              | 6n
    1                                    | (n - 7)/2
    3 7                                  | n - 1
    5                                    | (n + 1)/2
  
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsInjective(v);
  true
  gap> IsSurjective(v);
  false
  gap> Image(v);
  Z \ 4(12) U 8(12)
  gap> Difference(Integers,last);
  4(12) U 8(12)
  gap> v2 := LocalizedRcwaMapping(v,2);
  <rcwa mapping of Z_( 2 ) with modulus 8>
  gap> IsBijective(v2);
  true
  gap> Display(v2^-1);
  
  Bijective rcwa mapping of Z_( 2 ) with modulus 4
  
                  n mod 4                |                 n^f
  ---------------------------------------+--------------------------------------
    0                                    | 1/3 n / 2
    1                                    | 2 n + 7
    2                                    | n + 1
    3                                    | 2 n - 1
  
  gap> S := ResidueClass(Z_pi(2),2,0);; l := [S];;
  gap> for i in [1..10] do Add(l,l[Length(l)]^v2); od;
  gap> l; # Visibly v2 is wild ...
  [ 0(2), 0(4), 0(8), 0(16), 0(32), 0(64), 0(128), 0(256), 0(512), 0(1024), 
    0(2048) ]
  gap> w2 := RcwaMapping(Z_pi(2),[[1,0,2],[2,-1,1],[1,1,1],[2,-1,1]]);;
  gap> v2w2 := Comm(v2,w2);; SetName(v2w2,"[v2,w2]"); v2w2^-1;;
  gap> Display(v2w2);
  
  Bijective rcwa mapping of Z_( 2 ) with modulus 8
  
                  n mod 8                |              n^[v2,w2]
  ---------------------------------------+--------------------------------------
    0 3 4 7                              | n
    1                                    | n + 4
    2 6                                  | 3 n
    5                                    | n - 4
  
  
\end{Verbatim}
 Again, viewed as an rcwa mapping of the integers the commutator given at the
end of the example would not be surjective. }

 \pagebreak[4]  
\section{\textcolor{Chapter }{ Twisting 257-cycles into an rcwa mapping with modulus 32 }}\label{sec:Twisting257CyclesToModulus32}
\logpage{[ 4, 12, 0 ]}
{
  We define an rcwa mapping \mbox{\texttt{x}} of order~257 with modulus~32. The easiest way to construct such a mapping is to prescribe a transition
graph and then to assign suitable affine mappings to its vertices. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> x := RcwaMapping(
  >           [[ 16,  2,  1], [ 16, 18,  1], [  1, 16,  1], [ 16, 18,  1],
  >            [  1, 16,  1], [ 16, 18,  1], [  1, 16,  1], [ 16, 18,  1],
  >            [  1, 16,  1], [ 16, 18,  1], [  1, 16,  1], [ 16, 18,  1],
  >            [  1, 16,  1], [ 16, 18,  1], [  1, 16,  1], [ 16, 18,  1],
  >            [  1,  0, 16], [ 16, 18,  1], [  1,-14,  1], [ 16, 18,  1],
  >            [  1,-14,  1], [ 16, 18,  1], [  1,-14,  1], [ 16, 18,  1],
  >            [  1,-14,  1], [ 16, 18,  1], [  1,-14,  1], [ 16, 18,  1],
  >            [  1,-14,  1], [ 16, 18,  1], [  1,-14,  1], [  1,-31,  1]]);;
  gap> SetName(x,"x"); Display(x);
  
  Rcwa mapping of Z with modulus 32
  
                 n mod 32                |                 n^x
  ---------------------------------------+--------------------------------------
     0                                   | 16n + 2
     1  3  5  7  9 11 13 15 17 19 21 23  | 
    25 27 29                             | 16n + 18
     2  4  6  8 10 12 14                 | n + 16
    16                                   | n/16
    18 20 22 24 26 28 30                 | n - 14
    31                                   | n - 31
  
  gap> Order(x);
  257
  gap> Cycle(x,[1],0);
  [ 0, 2, 18, 4, 20, 6, 22, 8, 24, 10, 26, 12, 28, 14, 30, 16, 1, 34, 50, 36, 
    52, 38, 54, 40, 56, 42, 58, 44, 60, 46, 62, 48, 3, 66, 82, 68, 84, 70, 86, 
    72, 88, 74, 90, 76, 92, 78, 94, 80, 5, 98, 114, 100, 116, 102, 118, 104, 
    120, 106, 122, 108, 124, 110, 126, 112, 7, 130, 146, 132, 148, 134, 150, 
    136, 152, 138, 154, 140, 156, 142, 158, 144, 9, 162, 178, 164, 180, 166, 
    182, 168, 184, 170, 186, 172, 188, 174, 190, 176, 11, 194, 210, 196, 212, 
    198, 214, 200, 216, 202, 218, 204, 220, 206, 222, 208, 13, 226, 242, 228, 
    244, 230, 246, 232, 248, 234, 250, 236, 252, 238, 254, 240, 15, 258, 274, 
    260, 276, 262, 278, 264, 280, 266, 282, 268, 284, 270, 286, 272, 17, 290, 
    306, 292, 308, 294, 310, 296, 312, 298, 314, 300, 316, 302, 318, 304, 19, 
    322, 338, 324, 340, 326, 342, 328, 344, 330, 346, 332, 348, 334, 350, 336, 
    21, 354, 370, 356, 372, 358, 374, 360, 376, 362, 378, 364, 380, 366, 382, 
    368, 23, 386, 402, 388, 404, 390, 406, 392, 408, 394, 410, 396, 412, 398, 
    414, 400, 25, 418, 434, 420, 436, 422, 438, 424, 440, 426, 442, 428, 444, 
    430, 446, 432, 27, 450, 466, 452, 468, 454, 470, 456, 472, 458, 474, 460, 
    476, 462, 478, 464, 29, 482, 498, 484, 500, 486, 502, 488, 504, 490, 506, 
    492, 508, 494, 510, 496, 31 ]
  gap> Length(last);
  257
  
\end{Verbatim}
 }

 \pagebreak[4]  
\section{\textcolor{Chapter }{ The behaviour of the moduli of powers }}\label{sec:ModuliOfPowers}
\logpage{[ 4, 13, 0 ]}
{
  In this section some examples are given, which illustrate how different the
series of the moduli of powers of a given rcwa mapping of the integers can
look like. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List([0..4],i->Modulus(a^i));
  [ 1, 4, 16, 64, 256 ]
  gap> List([0..6],i->Modulus(ab^i));
  [ 1, 18, 18, 18, 18, 18, 1 ]
  gap> List([0..3],i->Modulus(r^i));
  [ 1, 9, 9, 1 ]
  gap> List([0..9],i->Modulus(s^i));
  [ 1, 9, 9, 27, 27, 27, 27, 27, 27, 1 ]
  gap> g := RcwaMapping([[2,2,1],[1,4,1],[1,0,2],[2,2,1],[1,-4,1],[1,-2,1]]);;
  gap> List([0..7],i->Modulus(g^i));
  [ 1, 6, 12, 12, 12, 12, 6, 1 ]
  gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
  gap> List([0..3],i->Modulus(u^i));
  [ 1, 5, 25, 125 ]
  gap> v6 := RcwaMapping([[-1,2,1],[1,-1,1],[1,-1,1]]);;
  gap> List([0..6],i->Modulus(v6^i));
  [ 1, 3, 3, 3, 3, 3, 1 ]
  gap> w8 := RcwaMapping([[-1,3,1],[1,-1,1],[1,-1,1],[1,-1,1]]);;
  gap> List([0..8],i->Modulus(w8^i));
  [ 1, 4, 4, 4, 4, 4, 4, 4, 1 ]
  gap> z := RcwaMapping([[2,  1, 1],[1,  1,1],[2, -1,1],[2, -2,1],
  >                      [1,  6, 2],[1,  1,1],[1, -6,2],[2,  5,1],
  >                      [1,  6, 2],[1,  1,1],[1,  1,1],[2, -5,1],
  >                      [1,  0, 1],[1, -4,1],[1,  0,1],[2,-10,1]]);;
  gap> SetName(z,"z");
  gap> IsBijective(z);
  true
  gap> Display(z);
  
  Bijective rcwa mapping of Z with modulus 16
  
                 n mod 16                |                 n^z
  ---------------------------------------+--------------------------------------
     0                                   | 2n + 1
     1  5  9 10                          | n + 1
     2                                   | 2n - 1
     3                                   | 2n - 2
     4  8                                | (n + 6)/2
     6                                   | (n - 6)/2
     7                                   | 2n + 5
    11                                   | 2n - 5
    12 14                                | n
    13                                   | n - 4
    15                                   | 2n - 10
  
  
\end{Verbatim}
 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List([0..25],i->Modulus(z^i));
  [ 1, 16, 32, 64, 64, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 
    256, 512, 512, 512, 512, 512, 512, 1024, 1024, 1024 ]
  gap> e1 := RcwaMapping([[1,4,1],[2,0,1],[1,0,2],[2,0,1]]);;
  gap> e2 := RcwaMapping([[1,4,1],[2,0,1],[1,0,2],[1,0,1],
  >                       [1,4,1],[2,0,1],[1,0,1],[1,0,1]]);;
  gap> List([e1,e2],Order);
  [ infinity, infinity ]
  gap> List([1..20],i->Modulus(e1^i));
  [ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 ]
  gap> List([1..20],i->Modulus(e2^i));
  [ 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4 ]
  gap> SetName(e1,"e1"); SetName(e2,"e2");
  gap> Display(e2);
  
  Bijective rcwa mapping of Z with modulus 8, of order infinity
  
                  n mod 8                |                 n^e2
  ---------------------------------------+--------------------------------------
    0 4                                  | n + 4
    1 5                                  | 2n
    2                                    | n/2
    3 6 7                                | n
  
  gap> e2^2 = Restriction(RcwaMapping([[1,2,1]]),RcwaMapping([[4,0,1]]));
  true
  
\end{Verbatim}
 }

  
\section{\textcolor{Chapter }{ Images and preimages under the Collatz mapping }}\label{sec:CollatzImagesAndPreImages}
\logpage{[ 4, 14, 0 ]}
{
  We have a look at the images of the residue class 1(2) under powers of the
Collatz mapping. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; S0 := ResidueClass(Integers,2,1);;
  gap> S1 := S0^T;
  2(3)
  gap> S2 := S1^T;
  1(3) U 8(9)
  gap> S3 := S2^T;
  2(3) U 4(9)
  gap> S4 := S3^T;
  Z \ 0(3) U 5(9)
  gap> S5 := S4^T;
  Z \ 0(3) U 7(9)
  gap> S6 := S5^T;
  Z \ 0(3)
  gap> S7 := S6^T;
  Z \ 0(3)
  
\end{Verbatim}
 Thus the image gets stable after applying the mapping $T$ for the 6th time. Hence $T^6$ maps the residue class 1(2) surjectively onto the union of the residue classes
1(3) and 2(3), which is setwisely stabilized by $T$. Now we would like to determine the preimages of 1(3) resp. 2(3) in 1(2)
under $T^6$. The residue class 1(2) has to be the disjoint union of these sets. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> U := Intersection(PreImage(T^6,ResidueClass(Integers,3,1)),S0);
  <union of 11 residue classes (mod 64)>
  gap> V := Intersection(PreImage(T^6,ResidueClass(Integers,3,2)),S0);
  <union of 21 residue classes (mod 64)>
  gap> AsUnionOfFewClasses(U);
  [ 1(64), 5(64), 7(64), 9(64), 21(64), 23(64), 29(64), 31(64), 49(64), 51(64), 
    59(64) ]
  gap> AsUnionOfFewClasses(V);
  [ 3(32), 11(32), 13(32), 15(32), 25(32), 17(64), 19(64), 27(64), 33(64), 
    37(64), 39(64), 41(64), 53(64), 55(64), 61(64), 63(64) ]
  gap> Union(U,V) = S0 and Intersection(U,V) = [];  # consistency check
  true
  
\end{Verbatim}
 The images of the residue class 0(3) under powers of~$T$ look as follows: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> S0 := ResidueClass(Integers,3,0);
  0(3)
  gap> S1 := S0^T;
  0(3) U 5(9)
  gap> S2 := S1^T;
  0(3) U 5(9) U 7(9) U 8(27)
  gap> S3 := S2^T;
  <union of 20 residue classes (mod 27)>
  gap> S4 := S3^T;
  <union of 73 residue classes (mod 81)>
  gap> S5 := S4^T;
  Z \ 10(81) U 37(81)
  gap> S6 := S5^T;
  Integers
  gap> S7 := S6^T;
  Integers
  
\end{Verbatim}
 Thus every integer is the image of a multiple of~3 under~$T^6$. This means that it would be sufficient to prove the $3n+1$ Conjecture for multiples of~3. We can obtain the corresponding result for multiples of~5 as follows: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> S := [ResidueClass(Integers,5,0)];
  [ 0(5) ]
  gap> for i in [1..12] do Add(S,S[i]^T); od;
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> for s in S do View(s); Print("\n"); od;
  0(5)
  0(5) U 8(15)
  0(5) U 4(15) U 8(15)
  0(5) U 2(15) U 4(15) U 8(15) U 29(45)
  <union of 73 residue classes (mod 135)>
  <union of 244 residue classes (mod 405)>
  <union of 784 residue classes (mod 1215)>
  <union of 824 residue classes (mod 1215)>
  <union of 2593 residue classes (mod 3645)>
  <union of 2647 residue classes (mod 3645)>
  <union of 2665 residue classes (mod 3645)>
  <union of 2671 residue classes (mod 3645)>
  1(3) U 2(3) U 0(15)
  gap> Union(S[13],ResidueClass(Integers,3,0));
  Integers
  gap> List(S,Si->Float(Density(Si)));
  [ 0.2, 0.266667, 0.333333, 0.422222, 0.540741, 0.602469, 0.645267, 0.678189, 
    0.711385, 0.7262, 0.731139, 0.732785, 0.733333 ]
  
\end{Verbatim}
 }

  
\section{\textcolor{Chapter }{ A group which acts 4-transitively on the positive integers }}\label{sec:FourTransitiveGroup}
\logpage{[ 4, 15, 0 ]}
{
  In this section, we would like to show that the group $G$ generated by the two wild mappings 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> a := RcwaMapping([[3,0,2],[3,1,4],[3,0,2],[3,-1,4]]);;
  gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
  gap> SetName(a,"a"); SetName(u,"u"); G := Group(a,u);;
  
\end{Verbatim}
 which we have already investigated in earlier examples acts 4-transitively on
the set of positive integers. Obviously, it acts on the set of positive
integers. First we show that this action is transitive. We start by checking
in which residue classes sufficiently large positive integers are mapped to
smaller ones by a suitable group element: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List([a,a^-1,u,u^-1],DecreasingOn);
  [ 1(2), 0(3), 0(5) U 2(5), 2(3) ]
  gap> Union(last);
  Z \ 4(30) U 16(30) U 28(30)
  
\end{Verbatim}
 We see that we cannot always choose such a group element from the set of
generators and their inverses -- otherwise the union would be \texttt{Integers}. \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2],DecreasingOn);
  [ 1(2), 0(3), 0(5) U 2(5), 2(3), 1(8) U 7(8), 0(3) U 2(9) U 7(9), 
    0(25) U 12(25) U 17(25) U 20(25), 2(3) U 1(9) U 3(9) ]
  gap> Union(last); # Still not enough ...
  Z \ 4(90) U 58(90) U 76(90)
  gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2,a*u,u*a,(a*u)^-1,(u*a)^-1],
  >         DecreasingOn);
  [ 1(2), 0(3), 0(5) U 2(5), 2(3), 1(8) U 7(8), 0(3) U 2(9) U 7(9), 
    0(25) U 12(25) U 17(25) U 20(25), 2(3) U 1(9) U 3(9), 
    3(5) U 0(10) U 7(20) U 9(20), 0(5) U 2(5), 2(3), 3(9) U 4(9) U 8(9) ]
  gap> Union(last); # ... but that's it!
  Integers
  
\end{Verbatim}
 Finally, we have to deal with ``small'' integers. We use the notation for the coefficients of rcwa mappings introduced
at the beginning of this manual. Let $c_{r(m)} > a_{r(m)}$. Then we easily see that $(a_{r(m)}n+b_{r(m)})/c_{r(m)} > n$ implies $n < b_{r(m)}/(c_{r(m)}-a_{r(m)})$. Thus we can restrict our considerations to integers $n < b_{\rm max}$, where $b_{\rm max}$ is the largest second entry of a coefficient triple of one of the group
elements in our list: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2,a*u,u*a,(a*u)^-1,(u*a)^-1],
  >         f->Maximum(List(Coefficients(f),c->c[2])));
  [ 1, 1, 4, 2, 7, 7, 56, 28, 25, 17, 17, 11 ]
  gap> Maximum(last);
  56
  
\end{Verbatim}
 Thus this upper bound is 56. The rest is easy -- all we have to do is to check
that the orbit containing~1 contains also all other positive integers less than or equal to~56: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> S := [1];;
  gap> while not IsSubset(S,[1..56]) do
  >      S := Union(S,S^a,S^u,S^(a^-1),S^(u^-1));
  >    od;
  gap> IsSubset(S,[1..56]);
  true
  
\end{Verbatim}
 Checking 2-transitivity is computationally harder, and in the sequel we will
omit some steps which are in practice needed to find out ``what~to~do''. The approach taken here is to show that the stabilizer of~1 in~$G$ acts transitively on the set of positive integers greater than~1. We do this by similar means as used above for showing the transitivity of
the action of $G$ on the positive integers. We start by determining all products of at most 5
generators and their inverses, which stabilize~1 (taking at most 4-generator products would not suffice!): \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> gens := [a,u,a^-1,u^-1];;
  gap> tups := Concatenation(List([1..5],k->Tuples([1..4],k)));;
  gap> Length(tups);
  1364
  gap> tups := Filtered(tups,tup->ForAll([[1,3],[3,1],[2,4],[4,2]],
  >                                      l->PositionSublist(tup,l)=fail));;
  gap> Length(tups);
  484
  gap> stab := [];;
  gap> for tup in tups do
  >      n := 1;
  >      for i in tup do n := n^gens[i]; od;
  >      if n = 1 then Add(stab,tup); fi;
  >    od;
  gap> Length(stab);
  118
  gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i])));;
  gap> ForAll(stabelm,elm->1^elm=1); # Check.
  true
  
\end{Verbatim}
 The resulting products have various different not quite small moduli: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> List(stabelm,Modulus);
  [ 4, 3, 16, 25, 9, 81, 64, 100, 108, 100, 25, 75, 27, 243, 324, 243, 256, 
    400, 144, 400, 100, 432, 324, 400, 80, 400, 625, 25, 75, 135, 150, 75, 225, 
    81, 729, 486, 729, 144, 144, 81, 729, 1296, 729, 6561, 1024, 1600, 192, 
    1600, 400, 576, 432, 1600, 320, 1600, 2500, 100, 100, 180, 192, 192, 108, 
    972, 1728, 972, 8748, 1600, 400, 320, 80, 1600, 2500, 300, 2500, 625, 625, 
    75, 675, 75, 75, 135, 405, 600, 120, 600, 1875, 75, 225, 405, 225, 225, 
    675, 243, 2187, 729, 2187, 216, 216, 243, 2187, 1944, 2187, 19683, 576, 
    144, 576, 432, 81, 81, 729, 2187, 5184, 324, 8748, 243, 2187, 19683, 26244, 
    19683 ]
  gap> Lcm(last);
  12597120000
  gap> Collected(Factors(last));
  [ [ 2, 10 ], [ 3, 9 ], [ 5, 4 ] ]
  
\end{Verbatim}
 Similar as before, we determine for any of the above mappings the residue
classes whose elements larger than the largest $b_{r(m)}$ - coefficient of the respective mapping are mapped to smaller integers: \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> decs := List(stabelm,DecreasingOn);;
  gap> List(decs,Modulus);
  [ 2, 3, 8, 25, 9, 9, 16, 100, 12, 50, 25, 75, 27, 81, 54, 81, 64, 400, 48, 
    200, 100, 72, 108, 400, 80, 200, 625, 25, 75, 45, 75, 75, 225, 81, 243, 81, 
    243, 144, 144, 81, 243, 216, 243, 243, 128, 1600, 64, 400, 400, 48, 144, 
    1600, 320, 400, 2500, 100, 100, 60, 96, 192, 108, 324, 144, 324, 972, 400, 
    400, 80, 80, 400, 2500, 100, 1250, 625, 625, 25, 75, 75, 75, 45, 135, 600, 
    120, 150, 1875, 75, 225, 135, 225, 225, 675, 243, 729, 243, 729, 108, 216, 
    243, 729, 162, 729, 2187, 144, 144, 144, 144, 81, 81, 243, 729, 1296, 324, 
    972, 243, 729, 2187, 1458, 2187 ]
  gap> Lcm(last);
  174960000
  
\end{Verbatim}
 Since the least common multiple of the moduli of these unions of residue
classes is as large as 174960000, directly forming their union and checking
whether it is equal to the set of integers would take relatively much time and
memory. However, starting with the set of integers and subtracting the above
sets one-by-one in a suitably chosen order is cheap: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> SortParallel(decs,stabelm,
  >                 function(S1,S2)
  >                   return First([1..100],k->Factorial(k) mod Modulus(S1) = 0)
  >                        < First([1..100],k->Factorial(k) mod Modulus(S2) = 0);
  >                 end);
  gap> S := Integers;;
  gap> for i in [1..Length(decs)] do
  >      S_old := S; S := Difference(S,decs[i]);
  >      if S <> S_old then ViewObj(S); Print("\n"); fi;
  >      if S = [] then maxind := i; break; fi;
  >    od;
  0(2)
  2(6) U 4(6)
  <union of 8 residue classes (mod 30)>
  <union of 19 residue classes (mod 90)>
  <union of 114 residue classes (mod 720)>
  <union of 99 residue classes (mod 720)>
  <union of 57 residue classes (mod 720)>
  <union of 54 residue classes (mod 720)>
  <union of 41 residue classes (mod 720)>
  <union of 35 residue classes (mod 720)>
  <union of 8 residue classes (mod 720)>
  4(720) U 94(720) U 148(720) U 238(720)
  <union of 24 residue classes (mod 5760)>
  <union of 72 residue classes (mod 51840)>
  <union of 48 residue classes (mod 51840)>
  <union of 192 residue classes (mod 259200)>
  <union of 168 residue classes (mod 259200)>
  <union of 120 residue classes (mod 259200)>
  <union of 96 residue classes (mod 259200)>
  <union of 72 residue classes (mod 259200)>
  <union of 60 residue classes (mod 259200)>
  <union of 48 residue classes (mod 259200)>
  <union of 24 residue classes (mod 259200)>
  <union of 12 residue classes (mod 259200)>
  <union of 24 residue classes (mod 777600)>
  <union of 12 residue classes (mod 777600)>
  111604(194400) U 14404(777600) U 208804(777600)
  [  ]
  
\end{Verbatim}
 Similar as above, it remains to check that the ``small'' integers all lie in the orbit containing~2. Obviously, it is sufficient to check that any integer greater than~2 is mapped to a smaller one by some suitably chosen element of the stabilizer
under consideration: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Maximum(List(stabelm{[1..maxind]},
  >                 f->Maximum(List(Coefficients(f),c->c[2]))));
  6581
  gap> Filtered([3..6581],n->Minimum(List(stabelm,elm->n^elm))>=n);
  [ 4 ]
  
\end{Verbatim}
 We have to treat 4 separately: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> 1^(u*a*u^2*a^-1*u);
  1
  gap> 4^(u*a*u^2*a^-1*u);
  3
  
\end{Verbatim}
 Now we know that any positive integer greater than~1 lies in the same orbit under the action of the stabilizer of~1 in~$G$ as~2, thus that this stabilizer acts transitively on $\mathbb{N} \setminus \{1\}$. But this means that we have established the 2-transitivity of the action of $G$ on~$\mathbb{N}$. 

 In the following, we essentially repeat the above steps to show that this
action is indeed 3-transitive: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> tups := Concatenation(List([1..6],k->Tuples([1..4],k)));;
  gap> tups := Filtered(tups,tup->ForAll([[1,3],[3,1],[2,4],[4,2]],
  >                                      l->PositionSublist(tup,l)=fail));;
  gap> stab := [];;
  gap> for tup in tups do
  >      l := [1,2];
  >      for i in tup do l := List(l,n->n^gens[i]); od;
  >      if l = [1,2] then Add(stab,tup); fi;
  >    od;
  gap> Length(stab);
  212
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i])));;
  gap> decs := List(stabelm,DecreasingOn);;
  gap> SortParallel(decs,stabelm,
  >      function(S1,S2) return First([1..100],k->Factorial(k) mod Modulus(S1) = 0)
  >                           < First([1..100],k->Factorial(k) mod Modulus(S2) = 0);
  >      end);
  gap> S := Integers;;
  gap> for i in [1..Length(decs)] do
  >      S_old := S; S := Difference(S,decs[i]);
  >      if S <> S_old then ViewObj(S); Print("\n"); fi;
  >      if S = [] then break; fi;
  >    od;
  Z \ 1(8) U 7(8)
  <union of 151 residue classes (mod 240)>
  <union of 208 residue classes (mod 720)>
  <union of 51 residue classes (mod 720)>
  <union of 45 residue classes (mod 720)>
  <union of 39 residue classes (mod 720)>
  <union of 33 residue classes (mod 720)>
  <union of 23 residue classes (mod 720)>
  <union of 19 residue classes (mod 720)>
  <union of 17 residue classes (mod 720)>
  <union of 16 residue classes (mod 720)>
  <union of 14 residue classes (mod 720)>
  <union of 8 residue classes (mod 720)>
  <union of 7 residue classes (mod 720)>
  238(360) U 4(720) U 148(720) U 454(720)
  <union of 38 residue classes (mod 5760)>
  <union of 37 residue classes (mod 5760)>
  <union of 25 residue classes (mod 5760)>
  <union of 21 residue classes (mod 5760)>
  <union of 17 residue classes (mod 5760)>
  <union of 16 residue classes (mod 5760)>
  <union of 138 residue classes (mod 51840)>
  <union of 48 residue classes (mod 51840)>
  <union of 32 residue classes (mod 51840)>
  <union of 20 residue classes (mod 51840)>
  <union of 16 residue classes (mod 51840)>
  <union of 68 residue classes (mod 259200)>
  <union of 42 residue classes (mod 259200)>
  <union of 32 residue classes (mod 259200)>
  <union of 26 residue classes (mod 259200)>
  <union of 25 residue classes (mod 259200)>
  <union of 11 residue classes (mod 259200)>
  <union of 10 residue classes (mod 259200)>
  <union of 7 residue classes (mod 259200)>
  13414(129600) U 2164(259200) U 66964(259200) U 228964(259200)
  2164(259200) U 66964(259200) U 228964(259200)
  [  ]
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Maximum(List(stabelm,f->Maximum(List(Coefficients(f),c->c[2]))));
  515816
  gap> smallnum := [4..515816];;
  gap> for i in [1..Length(stabelm)] do
  >      smallnum := Filtered(smallnum,n->n^stabelm[i]>=n);
  >    od;
  gap> smallnum;
  [  ]
  
\end{Verbatim}
 The same for 4-transitivity: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> tups := Concatenation(List([1..8],k->Tuples([1..4],k)));;
  gap> tups := Filtered(tups,tup->ForAll([[1,3],[3,1],[2,4],[4,2]],
  >                                      l->PositionSublist(tup,l)=fail));;
  gap> stab := [];;
  gap> for tup in tups do
  >      l := [1,2,3];
  >      for i in tup do l := List(l,n->n^gens[i]); od;
  >      if l = [1,2,3] then Add(stab,tup); fi;
  >    od;
  gap> Length(stab);
  528
  gap> stabelm := [];;
  gap> for i in [1..Length(stab)] do
  >      elm := One(G);
  >      for j in stab[i] do
  >        if Modulus(elm) > 10000 then elm := fail; break; fi;
  >        elm := elm * gens[j];
  >      od;
  >      if elm <> fail then Add(stabelm,elm); fi;
  >    od;
  gap> Length(stabelm);
  334
  gap> decs := List(stabelm,DecreasingOn);;
  gap> SortParallel(decs,stabelm,
  >                 function(S1,S2)
  >                   return First([1..100],k->Factorial(k) mod Modulus(S1) = 0)
  >                        < First([1..100],k->Factorial(k) mod Modulus(S2) = 0);
  >                 end);
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> S := Integers;;
  gap> for i in [1..Length(decs)] do
  >      S_old := S; S := Difference(S,decs[i]);
  >      if S <> S_old then ViewObj(S); Print("\n"); fi;
  >      if S = [] then maxind := i; break; fi;
  >    od;
  Z \ 1(8) U 7(8)
  <union of 46 residue classes (mod 72)>
  <union of 20 residue classes (mod 72)>
  4(18)
  <union of 28 residue classes (mod 576)>
  <union of 22 residue classes (mod 576)>
  <union of 21 residue classes (mod 576)>
  40(72) U 4(144) U 94(144) U 346(576) U 418(576)
  <union of 16 residue classes (mod 576)>
  <union of 15 residue classes (mod 576)>
  4(144) U 94(144) U 346(576) U 418(576)
  <union of 30 residue classes (mod 5184)>
  <union of 26 residue classes (mod 5184)>
  <union of 6 residue classes (mod 1296)>
  <union of 504 residue classes (mod 129600)>
  <union of 324 residue classes (mod 129600)>
  <union of 282 residue classes (mod 129600)>
  <union of 239 residue classes (mod 129600)>
  <union of 218 residue classes (mod 129600)>
  <union of 194 residue classes (mod 129600)>
  <union of 154 residue classes (mod 129600)>
  <union of 97 residue classes (mod 129600)>
  <union of 85 residue classes (mod 129600)>
  <union of 77 residue classes (mod 129600)>
  <union of 67 residue classes (mod 129600)>
  <union of 125 residue classes (mod 259200)>
  <union of 108 residue classes (mod 259200)>
  <union of 107 residue classes (mod 259200)>
  <union of 101 residue classes (mod 259200)>
  <union of 100 residue classes (mod 259200)>
  <union of 84 residue classes (mod 259200)>
  <union of 80 residue classes (mod 259200)>
  <union of 76 residue classes (mod 259200)>
  <union of 70 residue classes (mod 259200)>
  <union of 66 residue classes (mod 259200)>
  <union of 54 residue classes (mod 259200)>
  <union of 53 residue classes (mod 259200)>
  <union of 47 residue classes (mod 259200)>
  <union of 43 residue classes (mod 259200)>
  <union of 31 residue classes (mod 259200)>
  <union of 24 residue classes (mod 259200)>
  <union of 23 residue classes (mod 259200)>
  <union of 13 residue classes (mod 259200)>
  57406(129600) U 115006(129600) U 192676(259200) U 250276(259200)
  57406(129600) U 192676(259200) U 250276(259200) U 374206(388800)
  57406(129600) U 192676(259200) U 250276(259200)
  250276(259200) U 57406(388800) U 316606(388800) U 451876(777600)
  316606(388800) U 451876(777600) U 509476(777600) U 768676(777600)
  <union of 18 residue classes (mod 3110400)>
  451876(777600) U 509476(777600) U 705406(777600) U 768676(777600) U 2649406(
  3110400)
  451876(777600) U 705406(777600) U 768676(777600) U 2649406(3110400)
  451876(777600) U 705406(777600) U 2649406(3110400)
  705406(777600) U 2007076(3110400) U 2649406(3110400) U 2784676(3110400)
  <union of 14 residue classes (mod 9331200)>
  2260606(2332800) U 5759806(9331200) U 5895076(9331200) U 8227876(9331200)
  4593406(6998400) U 15091006(27993600) U 17559076(27993600) U 24557476(
  27993600)
  <union of 14 residue classes (mod 83980800)>
  18590206(20995200) U 24557476(83980800) U 45552676(83980800) U 71078206(
  83980800)
  [  ]
  gap> Maximum(List(stabelm{[1..maxind]},
  >                 f->Maximum(List(Coefficients(f),c->c[2]))));
  58975
  gap> smallnum := [5..58975];;
  gap> for i in [1..maxind] do
  >      smallnum := Filtered(smallnum,n->n^stabelm[i]>=n);
  >    od;
  gap> smallnum;
  [  ]
  
\end{Verbatim}
 There is even some evidence that the degree of transitivity of the action of $G$ on the positive integers is higher than~4: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> phi := EpimorphismFromFreeGroup(G);
  [ a, u ] -> [ a, u ]
  gap> F := Source(phi);
  <free group on the generators [ a, u ]>
  gap> words := List([5..20],
  >                  n->RepresentativeActionPreImage(G,[1,2,3,4,5],
  >                                                    [1,2,3,4,n],OnTuples,F));
  [ <identity ...>, a^-3*u^4*a*u^-2*a^2, a^-2*u*a^-1*u*a^-1*u*a^-1*u*a^-1*u^-1*a
      , a^4*u^-2*a^-4, a^-1*u^-4*a, u^2*a^-1*u^2*a^-1*u^-2, u^-2*a^-2*u^4, 
    a^-1*u^2*a, a^-1*u^-6*a, a^2*u^4*a^2*u^2, u^-4*a*u^-2*a^-3, 
    a^-1*u^-2*a^-3*u^4*a^2, a^3*u^2*a*u^2, a*u^-4*a*u^-4*a^-2, 
    u^-2*a*u^2*a*u^-2, u^-4*a^2*u^2 ]
  
\end{Verbatim}
 \pagebreak[4] }

  
\section{\textcolor{Chapter }{ A group which acts 3-transitively, but not 4-transitively on Z }}\label{sec:ThreeButNotFourTransitiveGroup}
\logpage{[ 4, 16, 0 ]}
{
  In this section, we would like to show that the wild group $G$ generated by the two tame mappings $n \mapsto n + 1$ and $\tau_{1(2),0(4)}$ acts 3-transitively, but not 4-transitively on the set of integers. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := Group(ClassShift(0,1),ClassTransposition(1,2,0,4));
  <rcwa group over Z with 2 generators>
  gap> IsTame(G);
  false
  gap> (G.1^-2*G.2)^3*(G.1^2*G.2)^3; # G is not the free product C_infty * C_2. 
  IdentityMapping( Integers )
  gap> Display(G);
  
  Wild rcwa group over Z, generated by
  
  [
  Tame bijective rcwa mapping of Z: n -> n + 1
  
  Bijective rcwa mapping of Z with modulus 4, of order 2
  
                  n mod 4                |    n^ClassTransposition(1,2,0,4)
  ---------------------------------------+--------------------------------------
    0                                    | (n + 2)/2
    1 3                                  | 2n - 2
    2                                    | n
  
  ]
  
  
\end{Verbatim}
 This group acts transitively on $\mathbb{Z}$, since already the cyclic group generated by the first of the two generators
does so. Next we have to show that it acts 2-transitively. We essentially
proceed as in the example in the previous section, by checking that the
stabilizer of~0 acts transitively on $\mathbb{Z} \setminus \{0\}$. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> gens := [ClassShift(0,1)^-1,ClassTransposition(1,2,0,4),ClassShift(0,1)];;
  gap> tups := Concatenation(List([1..6],k->Tuples([-1,0,1],k)));;
  gap> tups := Filtered(tups,tup->ForAll([[0,0],[-1,1],[1,-1]],
  >                                      l->PositionSublist(tup,l)=fail));;
  gap> Length(tups);
  189
  gap> stab := [];;
  gap> for tup in tups do
  >      n := 0;
  >      for i in tup do n := n^gens[i+2]; od;
  >      if n = 0 then Add(stab,tup); fi;
  >    od;
  gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i+2])));;
  gap> Collected(List(stabelm,Modulus));
  [ [ 4, 6 ], [ 8, 4 ], [ 16, 3 ] ]
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> decs := List(stabelm,DecreasingOn);
  [ 0(4), 3(4), 0(4), 3(4), 2(4), 0(4), 4(8), 2(4), 2(4), 0(4), 1(4), 0(8), 
    3(8) ]
  gap> Union(decs);
  Integers
  
\end{Verbatim}
 Similar as in the previous section, it remains to check that the integers with ``small'' absolute value all lie in the orbit containing~1 under the action of the stabilizer of~0: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Maximum(List(stabelm,f->Maximum(List(Coefficients(f),c->AbsInt(c[2])))));
  21
  gap> S := [1];;
  gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od;
  gap> IsSubset(S,Difference([-21..21],[0])); # Not yet ..
  false
  gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od;
  gap> IsSubset(S,Difference([-21..21],[0])); # ... but now!
  true
  
\end{Verbatim}
 Now we have to check for 3-transitivity. Since we cannot find for every
residue class an element of the pointwise stabilizer of $\{0,1\}$ which properly divides its elements, we also have to take additions and
subtractions into consideration. Since the moduli of all of our stabilizer
elements are quite small, simply looking at sets of representatives is cheap: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> tups := Concatenation(List([1..10],k->Tuples([-1,0,1],k)));;
  gap> tups := Filtered(tups,tup->ForAll([[0,0],[-1,1],[1,-1]],
  >                                      l->PositionSublist(tup,l)=fail));;
  gap> Length(tups);
  3069
  gap> stab := [];
  [  ]
  gap> for tup in tups do
  >      l := [0,1];
  >      for i in tup do l := List(l,n->n^gens[i+2]); od;
  >      if l = [0,1] then Add(stab,tup); fi;
  >    od;
  gap> Length(stab);
  10
  gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i+2])));;
  gap> Maximum(List(stabelm,Modulus));
  8
  gap> Maximum(List(stabelm,f->Maximum(List(Coefficients(f),c->AbsInt(c[2])))));
  8
  
\end{Verbatim}
 \pagebreak[4] 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> decsp := List(stabelm,elm->Filtered([9..16],n->n^elm<n));
  [ [ 9, 13 ], [ 10, 12, 14, 16 ], [ 12, 16 ], [ 9, 13 ], [ 12, 16 ], 
    [ 9, 11, 13, 15 ], [ 9, 11, 13, 15 ], [ 12, 16 ], [ 12, 16 ], 
    [ 9, 11, 13, 15 ] ]
  gap> Union(decsp);
  [ 9, 10, 11, 12, 13, 14, 15, 16 ]
  gap> decsm := List(stabelm,elm->Filtered([-16..-9],n->n^elm>n));
  [ [ -15, -13, -11, -9 ], [ -16, -12 ], [ -16, -12 ], [ -15, -11 ], 
    [ -16, -14, -12, -10 ], [ -15, -11 ], [ -15, -11 ], [ -16, -14, -12, -10 ], 
    [ -16, -14, -12, -10 ], [ -15, -11 ] ]
  gap> Union(decsm);
  [ -16, -15, -14, -13, -12, -11, -10, -9 ]
  gap> S := [2];;
  gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od;
  gap> IsSubset(S,Difference([-8..8],[0,1]));
  true
  
\end{Verbatim}
 At this point we have established 3-transitivity. It remains to check that the
group $G$ does not act 4-transitively. We do this by checking that it is not transitive
on 4-tuples (mod~4). Since $n$~mod~8 determines the image of $n$ under a generator of $G$ (mod~4), it suffices to compute (mod~8): 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> orb := [[0,1,2,3]];;
  gap> extend := function ()
  >      local gen;
  >      for gen in gens do
  >        orb := Union(orb,List(orb,l->List(l,n->n^gen) mod 8));
  >      od;
  >    end;;
  gap> repeat
  >      old := ShallowCopy(orb);
  >      extend(); Print(Length(orb),"\n");
  >    until orb = old;
  7
  27
  97
  279
  573
  916
  1185
  1313
  1341
  1344
  1344
  gap> Length(Set(List(orb,l->l mod 4)));
  120
  gap> last < 4^4;
  true
  
\end{Verbatim}
 This shows that $G$ is not 4-transitive on $\mathbb{Z}$. The corresponding calculation for 3-tuples looks as follows: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> orb := [[0,1,2]];;
  gap> repeat
  >      old := ShallowCopy(orb);
  >      extend(); Print(Length(orb),"\n");
  >    until orb = old;
  7
  27
  84
  207
  363
  459
  503
  512
  512
  gap> Length(Set(List(orb,l->l mod 4)));
  64
  gap> last = 4^3;
  true
  
\end{Verbatim}
 Needless to say that the latter kind of argumentation is not suitable for
proving, but only for disproving $k$-transitivity. }

  
\section{\textcolor{Chapter }{ Grigorchuk groups }}\label{sec:GrigorchukGroups}
\logpage{[ 4, 17, 0 ]}
{
  In this section, we show how construct finite quotients of the two infinite
periodic groups introduced by Rostislav Grigorchuk in~\cite{Grigorchuk80} with the help of \textsf{RCWA}. The first of these, nowadays known as ``Grigorchuk group'', is investigated in an example given on the \textsf{GAP} website -- see \href{http://www.gap-system.org/Doc/Examples/grigorchuk.html}{\texttt{http://www.gap-system.org/Doc/Examples/grigorchuk.html}}. The \textsf{RCWA} package permits a simpler and more elegant construction of the finite
quotients of this group: The function \texttt{TopElement} given on the mentioned webpage gets unnecessary, and the function \texttt{SequenceElement} can be simplified as follows: 
\begin{Verbatim}[fontsize=\small,frame=single,label=]
  
  SequenceElement := function ( r, level )
  
    return Permutation(Product(Filtered([1..level-1],k->k mod 3 <> r),
                               k->ClassTransposition(    2^(k-1)-1, 2^(k+1),
                                                     2^k+2^(k-1)-1, 2^(k+1))),
                       [0..2^level-1]);
  end;
  
\end{Verbatim}
 \pagebreak[4] The actual constructors for the generators are modified as follows: 
\begin{Verbatim}[fontsize=\small,frame=single,label=]
  
  a := level -> Permutation(ClassTransposition(0,2,1,2),[0..2^level-1]);
  b := level -> SequenceElement(0,level);
  c := level -> SequenceElement(2,level);
  d := level -> SequenceElement(1,level);
  
\end{Verbatim}
 All computations given on the webpage can now be done just as with the ``original'' construction of the quotients of the Grigorchuk group. In the sequel, finite
quotients of the second group introduced in~\cite{Grigorchuk80} are constructed: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> FourCycle := RcwaMapping((4,5,6,7),[4..7]);
  <bijective rcwa mapping of Z with modulus 4, of order 4>
  gap> GrigorchukGroup2Generator := function ( level )
  >      if level = 1 then return FourCycle; else
  >        return   Restriction(FourCycle, RcwaMapping([[4,1,1]]))
  >               * Restriction(FourCycle, RcwaMapping([[4,3,1]]))
  >               * Restriction(GrigorchukGroup2Generator(level-1),
  >                             RcwaMapping([[4,0,1]]));
  >      fi;
  >    end;;
  gap> GrigorchukGroup2 := level -> Group(FourCycle,
  >                                       GrigorchukGroup2Generator(level));;
  
\end{Verbatim}
 We can do similar things as shown in the example on the \textsf{GAP} webpage for the ``first'' Grigorchuk group: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> G := List([1..4],lev->GrigorchukGroup2(lev)); # The first 4 quotients.
  [ <rcwa group over Z with 2 generators>, <rcwa group over Z with 2 generators>
      , <rcwa group over Z with 2 generators>, 
    <rcwa group over Z with 2 generators> ]
  gap> H := List([1..4],lev->Action(G[lev],[0..4^lev-1])); # Isomorphic perm.-gps.
  [ Group([ (1,2,3,4), (1,2,3,4) ]), 
    Group([ (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), 
        (1,5,9,13)(2,6,10,14)(4,8,12,16) ]), 
    <permutation group with 2 generators>, 
    <permutation group with 2 generators> ]
  gap> List(H,Size);
  [ 4, 1024, 4294967296, 1329227995784915872903807060280344576 ]
  gap> List(last,n->Collected(Factors(n)));
  [ [ [ 2, 2 ] ], [ [ 2, 10 ] ], [ [ 2, 32 ] ], [ [ 2, 120 ] ] ]
  gap> List(H,NilpotencyClassOfGroup);      
  [ 1, 6, 14, 40 ]
  
\end{Verbatim}
 }

  
\section{\textcolor{Chapter }{ Forward orbits of a monoid with 2 generators }}\label{sec:ForwardOrbit}
\logpage{[ 4, 18, 0 ]}
{
  The $3n+1$ Conjecture asserts that the forward orbit of any positive integer under the
Collatz mapping $T$ contains~1. In contrast, it seems likely that ``most'' trajectories of the two mappings  
\[ T_5^\pm: \ \mathbb{Z} \longrightarrow \mathbb{Z}, \ \ \ \ n \ \longmapsto \
\begin{cases} \frac{n}{2} & \text{if} \ n \ \text{even}, \\ \frac{5n \pm 1}{2}
& \text{if} \ n \ \text{odd} \end{cases} \]
   diverge. However we can show by means of computation that the forward orbit of
any positive integer under the action of the monoid generated by the two
mappings $T_5^-$ and~$T_5^+$ indeed contains~1. First of all, we enter the generators: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> T5m := RcwaMapping([[1,0,2],[5,-1,2]]);;
  gap> T5p := RcwaMapping([[1,0,2],[5, 1,2]]);;
  
\end{Verbatim}
 We look for a number $k$ such that for any residue class $r(2^k)$ there is a product~$f$ of $k$~mappings $T_5^\pm$ whose restriction to $r(2^k)$ is given by $n \mapsto (an+b)/c$ where $c>a$: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> k := 1;;
  gap> repeat
  >      maps := List(Tuples([T5m,T5p],k),Product);
  >      decr := List(maps,DecreasingOn);
  >      decreasable := Union(decr);
  >      Print(k,": "); View(decreasable); Print("\n");
  >      k := k + 1;
  >    until decreasable = Integers;
  1: 0(2)
  2: 0(4)
  3: Z \ 1(8) U 7(8)
  4: 0(4) U 3(16) U 6(16) U 10(16) U 13(16)
  5: Z \ 7(32) U 25(32)
  6: <union of 48 residue classes (mod 64)>
  7: Integers
  
\end{Verbatim}
 Thus $k=7$ serves our purposes. To be sure that for any positive integer $n$ our monoid contains a mapping $f$ such that $n^f<n$, we still need to check this condition for ``small''~$n$. Since in case $c>a$ we have $(an+b)/c \geq n$ if only if $n \leq b/(c-a)$, we only need to check those $n$ which are not larger than the largest coefficient $b_{r(m)}$ occuring in any of the products under consideration: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> maxb := Maximum(List(maps,f->Maximum(List(Coefficients(f),t->t[2]))));
  25999
  gap> small := Filtered([1..maxb],n->ForAll(maps,f->n^f>=n));
  [ 1, 7, 9, 11 ]
  
\end{Verbatim}
 This means that except of~1, only for $n \in \{{7,9,11\}}$ there is no product of 7 mappings $T_5^\pm$ which maps $n$ to a smaller integer. We check that also the forward orbits of these three
integers contain~1 by successively computing preimages of~1: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> S := [1];; k := 0;;
  gap> repeat
  >      S := Union(S,PreImage(T5m,S),PreImage(T5p,S));
  >      k := k+1;
  >    until IsSubset(S,small);
  gap> k;
  17
  
\end{Verbatim}
 }

  
\section{\textcolor{Chapter }{ Representations of the free group of rank 2 }}\label{sec:FreeGroup}
\logpage{[ 4, 19, 0 ]}
{
  The free group of rank~2 embeds in RCWA($\Z$) -- in fact it embeds even in the subgroup which is generated by all class
transpositions. An explicit embedding can be constructed by transferring the
construction of the so-called ``Schottky groups'' (cp.~\cite{LaHarpe00}, page~27) from PSL(2,$\mathbb{C}$) to RCWA($\mathbb{Z}$) (we use the notation from the cited book): 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> D := AllResidueClassesModulo(4);
  [ 0(4), 1(4), 2(4), 3(4) ]
  gap> gamma1 := RepresentativeAction(RCWA(Integers),Difference(Integers,D[1]),D[2]);;
  gap> gamma2 := RepresentativeAction(RCWA(Integers),Difference(Integers,D[3]),D[4]);;
  gap> F2 := Group(gamma1,gamma2);
  <rcwa group over Z with 2 generators>
  
\end{Verbatim}
 We can do some checks: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> X1 := Union(D{[1,2]});; X2 := Union(D{[3,4]});;
  gap>     IsSubset(X1,X2^gamma1) and IsSubset(X1,X2^(gamma1^-1))
  >    and IsSubset(X2,X1^gamma2) and IsSubset(X2,X1^(gamma2^-1));
  true
  
\end{Verbatim}
 The generators are products of 3 class transpositions, each: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Factorization(gamma1);
  [ ClassTransposition(0,2,1,2), ClassTransposition(3,4,5,8),
    ClassTransposition(0,2,1,8) ]
  gap> Factorization(gamma2);
  [ ClassTransposition(0,2,1,2), ClassTransposition(1,4,7,8),
    ClassTransposition(0,2,3,8) ]
  
\end{Verbatim}
 The above construction is used by \texttt{IsomorphismRcwaGroup} (\ref{IsomorphismRcwaGroup:G}) to embed free groups of any rank~$\geq 2$. 

 We give another only slightly different representation of the free group of
rank~2. We verify that it really is one by applying the so-called \emph{Table-Tennis Lemma} (see e.g.~\cite{LaHarpe00}, Section~II.B.) to the infinite cyclic groups generated by the two generators and to
the same two sets \texttt{X1} and \texttt{X2} as above: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> r1 := ClassTransposition(0,2,1,2)*ClassTransposition(0,2,1,4);;
  gap> r2 := ClassTransposition(0,2,1,2)*ClassTransposition(0,2,3,4);;
  gap> F2 := Group(r1^2,r2^2);; SetName(F2,"F_2");
  gap> List(GeneratorsOfGroup(F2),IsTame);
  [ false, false ]
  gap>     IsSubset(X1,X2^F2.1) and IsSubset(X1,X2^(F2.1^-1))
  >    and IsSubset(X2,X1^F2.2) and IsSubset(X2,X1^(F2.2^-1));
  true
  gap> [Sources(r1),Sinks(r1),Loops(r1)]; # compare with X1
  [ [ 0(4) ], [ 1(4) ], [ 0(4), 1(4) ] ]
  gap> [Sources(r2),Sinks(r2),Loops(r2)]; # compare with X2
  [ [ 2(4) ], [ 3(4) ], [ 2(4), 3(4) ] ]
  gap>     IsSubset(X1,Union(Sinks(r1))) and IsSubset(X1,Union(Sinks(r1^-1)))
  >    and IsSubset(X2,Union(Sinks(r2))) and IsSubset(X2,Union(Sinks(r2^-1)));
  true
  gap> IsSubset(Union(Sinks(r1)),X2^F2.1) and
  >    IsSubset(Union(Sinks(r1^-1)),X2^(F2.1^-1));
  true
  gap> IsSubset(Union(Sinks(r2)),X1^F2.2) and
  >    IsSubset(Union(Sinks(r2^-1)),X1^(F2.2^-1));
  true
  
\end{Verbatim}
 Drawing the transition graphs of \texttt{r1} and \texttt{r2} for modulus~4 may help understanding what is actually done in this calculation. It is easy
to see that the group generated by \texttt{r1} and~\texttt{r2} is \emph{not} free: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Order(r1/r2);
  3
  
\end{Verbatim}
 }

  
\section{\textcolor{Chapter }{ Representations of the modular group PSL(2,Z) }}\label{sec:ModularGroup}
\logpage{[ 4, 20, 0 ]}
{
  The modular group PSL(2,$\mathbb{Z}$) embeds in the group generated by all class transpositions as well. We give
an explicit embedding, and check that it really is one by applying the Table
Tennis Lemma as in the previous section: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> PSL2Z := Group(ClassTransposition(0,3,1,3) * ClassTransposition(0,3,2,3),
  >                   ClassTransposition(1,3,0,6) * ClassTransposition(2,3,3,6));;
  gap> List(GeneratorsOfGroup(PSL2Z),Order);
  [ 3, 2 ]
  
\end{Verbatim}
 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> X1 := Difference(Integers,ResidueClass(0,3));
  Z \ 0(3)
  gap> X2 := ResidueClass(0,3);
  0(3)
  gap> IsSubset(X1,X2^PSL2Z.1) and IsSubset(X1,X2^(PSL2Z.1^2));
  true
  gap> IsSubset(X2,X1^PSL2Z.2);
  true
  
\end{Verbatim}
 A slightly different representation of PSL(2,$\mathbb{Z}$) can be obtained by using \textsf{RCWA}'s general method for \texttt{IsomorphismRcwaGroup} for free products of finite groups: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> Display(Image(IsomorphismRcwaGroup(FreeProduct(CyclicGroup(3),
  >                                                   CyclicGroup(2)))));
  
  Wild rcwa group over Z, generated by
  
  [
  
  Bijective rcwa mapping of Z with modulus 4
  
                  n mod 4                |                 n^f
  ---------------------------------------+--------------------------------------
    0                                    | n + 2
    1 3                                  | 2n - 2
    2                                    | n/2
  
  
  Bijective rcwa mapping of Z with modulus 2
  
                  n mod 2                |                 n^f
  ---------------------------------------+--------------------------------------
    0                                    | n + 1
    1                                    | n - 1
  
  ]
  
  
\end{Verbatim}
  }

  }

         
\chapter{\textcolor{Chapter }{The Algorithms Implemented in RCWA}}\label{ch:algorithms}
\logpage{[ 5, 0, 0 ]}
{
  Apart from the method for factoring residue class-wise affine permutations of~$\Z$ into class transpositions, class shifts and class reflections, most
mathematically interesting algorithms implemented in this package are
described in the author's thesis~\cite{Kohl05} in the form of constructive proofs. This chapter provides references to the
corresponding theorems, and lists short descriptions of the other algorithms
and methods implemented in this package. The word ``trivial'' as a description means that essentially nothing is done except of storing or
recalling one or several values, and ``straightforward'' means that no sophisticated algorithm is used. The descriptions are kept very
informal and short. They are listed in alphabetical order. 
\begin{description}
\item[{ \texttt{ActionOnRespectedPartition(\mbox{\texttt{G}})} }]  ``Straightforward'' after having computed a respected partition by \texttt{RespectedPartition}. One only needs to know how to compute images of residue classes under affine
mappings. 
\item[{ \texttt{Ball(\mbox{\texttt{G}},\mbox{\texttt{g}},\mbox{\texttt{d}})} }]  ``Straightforward''. 
\item[{ \texttt{Ball(\mbox{\texttt{G}},\mbox{\texttt{p}},\mbox{\texttt{d}},\mbox{\texttt{act}})} }]  ``Straightforward''. 
\item[{ \texttt{ClassReflection(\mbox{\texttt{r}},\mbox{\texttt{m}})} }]  ``Trivial''. 
\item[{ \texttt{ClassShift(\mbox{\texttt{r}},\mbox{\texttt{m}})} }]  ``Trivial''. 
\item[{ \texttt{ClassTransposition(\mbox{\texttt{r1}},\mbox{\texttt{m1}},\mbox{\texttt{r2}},\mbox{\texttt{m2}})} }]  See Remark~2.9.2 in~\cite{Kohl05}. 
\item[{ \texttt{Coefficients(\mbox{\texttt{f}})} }]  ``Trivial''. 
\item[{ \texttt{CoefficientsOnTrajectory(\mbox{\texttt{f}},\mbox{\texttt{n}},\mbox{\texttt{val}},\mbox{\texttt{cond}},\mbox{\texttt{all}})} }]  Iterated application of an rcwa mapping, and composition of affine mappings. 
\item[{ \texttt{CommonRightInverse(\mbox{\texttt{l}},\mbox{\texttt{r}})} }]  ``Straightforward'' if one knows how to compute images of residue classes under affine mappings,
and how to compute inverses of affine mappings. 
\item[{ \texttt{DecreasingOn(\mbox{\texttt{f}})} }]  Form the union of the residue classes which are determined by the coefficients
as indicated. 
\item[{ \texttt{Determinant(\mbox{\texttt{sigma}})} }]  Evaluation of the given expression. For the mathematical meaning
(epimorphism!), see Theorem~2.11.9 in~\cite{Kohl05}. 
\item[{ \texttt{DirectProduct(\mbox{\texttt{G1}},\mbox{\texttt{G2}}, ... )} }]  Restrict the groups \mbox{\texttt{G1}}, \mbox{\texttt{G2}}, ... to disjoint residue classes. See \texttt{Restriction} and Corollary~2.3.3 in~\cite{Kohl05}. 
\item[{ \texttt{Display(\mbox{\texttt{f}})} }]  ``Trivial''. 
\item[{ \texttt{Divisor(\mbox{\texttt{f}})} }]  Lcm of coefficients, as indicated. 
\item[{ \texttt{FactorizationIntoCSCRCT(\mbox{\texttt{g}})} }]  This uses a rather sophisticated method which will likely some time be
published elsewhere. At the moment termination is not guaranteed, but in case
of termination the result is certain. The strategy is roughly first to make
the mapping class-wise order-preserving and balanced, and then to remove all
prime factors from multiplier and divisor one after the other in decreasing
order by dividing by appropriate class transpositions. The remaining integral
mapping can be factored almost similarly easily as a permutation of a finite
set can be factored into transpositions. 
\item[{ \texttt{FactorizationOnConnectedComponents(\mbox{\texttt{f}},\mbox{\texttt{m}})} }]  Call \textsf{GRAPE} to get the connected components of the transition graph, and then compute a
partition of the suitably ``blown up'' coefficient list corresponding to the connected components. 
\item[{ \texttt{FixedPointsOfAffinePartialMappings(\mbox{\texttt{f}})} }]  ``Straightforward''. 
\item[{ \texttt{GuessedDivergence(\mbox{\texttt{f}})} }]  Numerical computation of the limit of some series, which seems to converge ``often''. Caution!!! 
\item[{ \texttt{Image(\mbox{\texttt{f}})}, \texttt{Image(\mbox{\texttt{f}},\mbox{\texttt{S}})} }]  ``Straightforward'' if one can compute images of residue classes under affine mappings and unite
and intersect residue classes (Chinese Remainder Theorem). See Lemma~1.2.1 in~\cite{Kohl05}. 
\item[{ \texttt{ImageDensity(\mbox{\texttt{f}})} }]  Evaluation of the given expression. 
\item[{ \texttt{\mbox{\texttt{g}} in \mbox{\texttt{G}}} (membership test) }]  Test whether the mapping \mbox{\texttt{g}} or its inverse is in the list of generators of~\mbox{\texttt{G}}. If it is, return \texttt{true}. Test whether its prime set is a subset of the prime set of~\mbox{\texttt{G}}. If not, return \texttt{false}. Test if \mbox{\texttt{G}} is class-wise order-preserving, and \mbox{\texttt{g}} is not. If so, return \texttt{false}. Test if the sign of \mbox{\texttt{g}} is~-1 and all generators of~\mbox{\texttt{G}} have sign~1. If yes, return \texttt{false}. Test if \mbox{\texttt{G}} is class-wise order-preserving, all generators of~\mbox{\texttt{G}} have determinant~0 and \mbox{\texttt{g}} has determinant~$\neq 0$. If yes, return \texttt{false}. Test whether the support of~\mbox{\texttt{g}} is a subset of the support of~\mbox{\texttt{G}}. If not, return \texttt{false}. 

 If \mbox{\texttt{G}} is not tame, try to factor \mbox{\texttt{g}} into generators of~\mbox{\texttt{G}} using \texttt{PreImagesRepresentative}. If successful, return \texttt{true}. If \mbox{\texttt{g}} is in \mbox{\texttt{G}}, this terminates after a finite number of steps. Both runtime and memory
requirements are exponential in the word length. If \mbox{\texttt{g}} is not in \mbox{\texttt{G}}, it runs into an infinite loop. If \mbox{\texttt{G}} is tame, proceed as follows: 

 Test whether the modulus of \mbox{\texttt{g}} divides the modulus of \mbox{\texttt{G}}. If not, return \texttt{false}. Test whether \mbox{\texttt{G}} is finite and \mbox{\texttt{g}} has infinite order. If so, return \texttt{false}. Test whether \mbox{\texttt{g}} is tame. If not, return \texttt{false}. Compute a respected partition \texttt{P} of \mbox{\texttt{G}} and the finite permutation group \texttt{H} induced by \mbox{\texttt{G}} on it (see \texttt{RespectedPartition}). Check whether \mbox{\texttt{g}} permutes \texttt{P}. If not, return \texttt{false}. Let \texttt{h} be the permutation induced by \mbox{\texttt{g}} on \texttt{P}. Check whether \texttt{h} lies in \texttt{H}. If not, return \texttt{false}. 

 If \mbox{\texttt{G}} is class-wise order-preserving, do the following: Compute an element \texttt{g1} of \mbox{\texttt{G}} which acts on \texttt{P} like \mbox{\texttt{g}}. For this purpose, factor \mbox{\texttt{h}} into generators of \texttt{H} using \texttt{PreImagesRepresentative}. Compute the corresponding product of generators of \mbox{\texttt{G}}. Set \texttt{k := g/g1}. The mapping \texttt{k} is necessarily integral. Compute the kernel \texttt{K} of the action of \mbox{\texttt{G}} on \texttt{P} using \texttt{KernelOfActionOnRespectedPartition}. Check whether \texttt{k} lies in the kernel of the action of \mbox{\texttt{G}} on \texttt{P} by using \texttt{SolutionIntMat} to decide membership of the coefficient vector (second entry of each triple)
of \texttt{k} in the lattice spanned by the rows of the matrix \texttt{KernelOfActionOnRespectedPartitionHNFMat(\mbox{\texttt{G}})}. If it is contained, return \texttt{true}. 

 If membership still has not been decided yet, try to factor \mbox{\texttt{g}} into generators of~\mbox{\texttt{G}} using \texttt{PreImagesRepresentative}. If successful, return \texttt{true}. If \mbox{\texttt{g}} is in \mbox{\texttt{G}}, this terminates after a finite number of steps. Both runtime and memory
requirements are exponential in the word length. If \mbox{\texttt{g}} is not in \mbox{\texttt{G}}, the method runs into an infinite loop. 
\item[{ \texttt{IncreasingOn(\mbox{\texttt{f}})} }]  Form the union of the residue classes which are determined by the coefficients
as indicated. 
\item[{ \texttt{IntegralConjugate(\mbox{\texttt{f}})}, \texttt{IntegralConjugate(\mbox{\texttt{G}})} }]  Uses the algorithm described in the proof of Theorem~2.5.14 in~\cite{Kohl05}. 
\item[{ \texttt{IntegralizingConjugator(\mbox{\texttt{f}})}, \texttt{IntegralizingConjugator(\mbox{\texttt{G}})} }]  Uses the algorithm described in the proof of Theorem~2.5.14 in~\cite{Kohl05}. 
\item[{ \texttt{Inverse(\mbox{\texttt{f}})} }]  Essentially inversion of affine mappings. See Lemma~1.3.1, Part~(b) in~\cite{Kohl05}. 
\item[{ \texttt{IsClassWiseOrderPreserving(\mbox{\texttt{f}})} }]  Test whether the first entry of all coefficient triples is positive. 
\item[{ \texttt{IsConjugate(RCWA(Integers),\mbox{\texttt{f}},\mbox{\texttt{g}})} }]  Test whether \mbox{\texttt{f}} and \mbox{\texttt{g}} have the same order, and whether either both or none of them is tame. If not,
return \texttt{false}. 

 If the mappings are wild, use \texttt{ShortCycles} to search for finite cycles not belonging to an infinite series, until their
numbers for a particular length differ. This may run into an infinite loop. If
it terminates, return \texttt{false}. 

 If the mappings are tame, use the method described in the proof of Theorem~2.5.14 in~\cite{Kohl05} to construct integral conjugates of \mbox{\texttt{f}} and \mbox{\texttt{g}}. Then essentially use the algorithm described in the proof of Theorem~2.6.7 in~\cite{Kohl05} to compute ``standard representatives'' of the conjugacy classes which the integral conjugates of \mbox{\texttt{f}} and \mbox{\texttt{g}} belong to. Finally compare these standard representatives, and return \texttt{true} if they are equal and \texttt{false} if not. 
\item[{ \texttt{IsInjective(\mbox{\texttt{f}})} }]  See \texttt{Image}. 
\item[{ \texttt{IsIntegral(\mbox{\texttt{f}})} }]  ``Trivial''. 
\item[{ \texttt{IsomorphismMatrixGroup(\mbox{\texttt{G}})} }]  Use the algorithm described in the proof of Theorem~2.6.3 in~\cite{Kohl05}. 
\item[{ \texttt{IsomorphismPermGroup(\mbox{\texttt{G}})} }]  If \mbox{\texttt{G}} is wild or \texttt{KernelOfActionOnRespectedPartition} is not trivial, return \texttt{fail}. Otherwise use \texttt{ActionOnRespectedPartition}. 
\item[{ \texttt{IsomorphismRcwaGroup(\mbox{\texttt{G}})} }]  The method for finite groups uses \texttt{RcwaMapping}, Part~(d). 

 The method for free products of finite groups uses the Table-Tennis Lemma
(which is also known as \emph{Ping-Pong Lemma}, cp. e.g. Section~II.B. in~\cite{LaHarpe00}). It uses regular permutation representations of the factors $G_r$ ($r = 0, \dots ,m-1$) of the free product on residue classes modulo $n_r := |G_r|$. The basic idea is that since point stabilizers in regular permutation groups
are trivial, all non-identity elements map any of the permuted residue classes
into their complements. To get into a situation where the Table-Tennis Lemma
is applicable, the method computes conjugates of the images of the mentioned
permutation representations under bijective rcwa mappings $\sigma_r$ which satisfy $0(n_r)^{\sigma_r} = \mathbb{Z} \setminus r(m)$. 

 The method for free groups uses an adaptation of the construction given on
page~27 in~\cite{LaHarpe00} from PSL(2,$\mathbb{C}$) to RCWA($\mathbb{Z}$). As an equivalent for the closed discs used there, the method takes the
residue classes modulo two times the rank of the free group. 
\item[{ \texttt{IsSurjective(\mbox{\texttt{f}})} }]  See \texttt{Image}. 
\item[{ \texttt{IsTame(\mbox{\texttt{G}})} }]  Checks whether the modulus of the group is non-zero. 
\item[{ \texttt{IsTame(\mbox{\texttt{f}})} }]  Application of the criteria given in Corollary~2.5.10 and~2.5.12 and Theorem~A.8 and~A.11 in~\cite{Kohl05}. For applying the last-mentioned criterium (existence of weakly-connected
components of the transition graph which are not strongly-connected), \textsf{GRAPE} is needed. 

 In addition, some probabilistic methods are used. If the result depends on one
of these, a warning is displayed. 
\item[{ \texttt{IsTransitive(\mbox{\texttt{G}},Integers)} }]  Look for finite orbits, using \texttt{ShortOrbits} on a couple of intervals. If a finite orbit is found, return \texttt{false}. Test if \mbox{\texttt{G}} is finite. If yes, return \texttt{false}. 

 Search for an element \texttt{g} and a residue class $r(m)$ such that the restriction of \texttt{g} to $r(m)$ is given by $n \mapsto n + m$. Then the cyclic group generated by \texttt{g} acts transitively on $r(m)$. The element \texttt{g} is searched among the generators of \mbox{\texttt{G}}, its powers, its commutators, powers of its commutators and products of few
different generators. The search for such an element may run into an infinite
loop, as there is no guarantee that the group has a suitable element. 

 If suitable \texttt{g} and $r(m)$ are found, proceed as follows: 

 Set $S := r(m)$. Set $S := S \cup S^g$ for all generators $g$ of \mbox{\texttt{G}}, and repeat this until $S$ remains constant. This may run into an infinite loop. 

 If it terminates: If $S = \mathbb{Z}$, return \texttt{true}, otherwise return \texttt{false}. 
\item[{ \texttt{KernelOfActionOnRespectedPartition(\mbox{\texttt{G}})} }]  Use a random walk through the group \mbox{\texttt{G}}. Compute powers of elements encountered along the way which fix the respected
partition of \mbox{\texttt{G}} which has been computed by \texttt{RespectedPartition}. Get vectors from these powers by taking the second entry of each coefficient
triple. Form a lattice out of these vectors. Stop if for a while all found
vectors already belong to this lattice (this is probabilistic). Bring the
lattice to Hermite Normal Form, and transform the rows of the resulting matrix
back to rcwa mappings generating the kernel. 
\item[{ \texttt{KernelOfActionOnRespectedPartitionHNFMat(\mbox{\texttt{G}})} }]  This is a ``spin-off'' of \texttt{KernelOfActionOnRespectedPartition}. 
\item[{ \texttt{LargestSourcesOfAffineMappings(\mbox{\texttt{f}})} }]  Form unions of residue classes modulo the modulus of the mapping, whose
corresponding coefficient triples are equal. 
\item[{ \texttt{LaTeXObj(\mbox{\texttt{f}})} }]  Collect residue classes those corresponding coefficient triples are equal. 
\item[{ \texttt{LikelyContractionCentre(\mbox{\texttt{f}},\mbox{\texttt{maxn}},\mbox{\texttt{bound}})} }]  Compute trajectories with starting values from a given interval, until a cycle
is reached. Abort if the trajectory exceeds the prescribed bound. Form the
union of the detected cycles. 
\item[{ \texttt{LocalizedRcwaMapping(\mbox{\texttt{f}},\mbox{\texttt{p}})} }]  ``Trivial''. 
\item[{ \texttt{mKnot(\mbox{\texttt{m}})} }]  ``Straightforward'', following the definition given in \cite{Keller99}. 
\item[{ \texttt{Modulus(\mbox{\texttt{G}})} }]  Searches for a wild element in the group. If unsuccessful, tries to construct
a respected partition (see \texttt{RespectedPartition}). 
\item[{ \texttt{Modulus(\mbox{\texttt{f}})} }]  ``Trivial''. 
\item[{ \texttt{MovedPoints(\mbox{\texttt{G}})} }]  Needs only forming unions of residue classes and determining fixed points of
affine mappings. 
\item[{ \texttt{Multiplier(\mbox{\texttt{f}})} }]  Lcm of coefficients, as indicated. 
\item[{ \texttt{Multpk(\mbox{\texttt{f}},\mbox{\texttt{p}},\mbox{\texttt{k}})} }]  Form the union of the residue classes modulo the modulus of the mapping, which
are determined by the given divisibility criteria for the coefficients of the
corresponding affine mapping. 
\item[{ \texttt{NrConjugacyClassesOfRCWAZOfOrder(\mbox{\texttt{ord}})} }]  The class numbers are taken from Corollary~2.7.1 in~\cite{Kohl05}. 
\item[{ \texttt{OrbitsModulo(\mbox{\texttt{f}},\mbox{\texttt{m}})} }]  Use \textsf{GRAPE} to compute the connected components of the transition graph. 
\item[{ \texttt{OrbitsModulo(\mbox{\texttt{G}},\mbox{\texttt{m}})} }]  ``Straightforward''. 
\item[{ \texttt{Order(\mbox{\texttt{f}})} }]  Test for \texttt{IsTame}. If the mapping is not tame, then return \texttt{infinity}. Otherwise use Corollary~2.5.10 in~\cite{Kohl05}. 
\item[{ \texttt{PreImage(\mbox{\texttt{f}},\mbox{\texttt{S}})} }]  See \texttt{Image}. 
\item[{ \texttt{PreImagesRepresentative(\mbox{\texttt{phi}},\mbox{\texttt{g}})}, \texttt{PreImagesRepresentatives(\mbox{\texttt{phi}},\mbox{\texttt{g}})} }]  As indicated in the documentation of these methods. The underlying idea to
successively compute two balls around~1 and~\mbox{\texttt{g}} until they intersect non-trivially is standard in computational group theory.
For rcwa groups it would mean wasting both memory and runtime to actually
compute group elements. Thus only images of tuples of points are computed and
stored. 
\item[{ \texttt{PrimeSet(\mbox{\texttt{f}})}, \texttt{PrimeSet(\mbox{\texttt{G}})} }]  ``Straightforward''. 
\item[{ \texttt{PrimeSwitch(\mbox{\texttt{p}})} }]  Multiplication of rcwa mappings as indicated. 
\item[{ \texttt{Print(\mbox{\texttt{f}})} }]  ``Trivial''. 
\item[{ \texttt{\mbox{\texttt{f}}*\mbox{\texttt{g}}} }]  Essentially composition of affine mappings. See Lemma~1.3.1, Part~(a) in~\cite{Kohl05}. 
\item[{ \texttt{Random(RCWA(Integers))} }]  Computes a product of ``randomly'' chosen class shifts, class reflections and class transpositions. This seems to
be suitable for generating reasonably good examples. 
\item[{ \texttt{RankOfKernelOfActionOnRespectedPartition(\mbox{\texttt{G}})} }]  This is a ``spin-off'' of \texttt{KernelOfActionOnRespectedPartition}. 
\item[{ \texttt{RCWA(\mbox{\texttt{R}})} }]  Attributes are set according to Theorem~2.1.1, Theorem~2.1.2, Corollary~2.1.6 and Theorem~2.12.8 in~\cite{Kohl05}. 
\item[{ \texttt{RcwaGroupByPermGroup(\mbox{\texttt{G}})} }]  Uses \texttt{RcwaMapping}, Part~(d). 
\item[{ \texttt{RcwaMapping} }]  (a)-(c): ``trivial'', (d): \texttt{n{\textasciicircum}perm - n} for determining the coefficients, (e): ``affine mappings by values at two given points'', (f) and (g): ``trivial'', (h) and (i): correspond to Lemma~2.1.4 in~\cite{Kohl05}. 
\item[{ \texttt{RepresentativeAction(\mbox{\texttt{G}},\mbox{\texttt{src}},\mbox{\texttt{dest}},\mbox{\texttt{act}})}, \texttt{RepresentativeActionPreImage} }]  As indicated in the documentation of these methods. The underlying idea to
successively compute two balls around \mbox{\texttt{src}} and \mbox{\texttt{dest}} until they intersect non-trivially is standard in computational group theory.
Words standing for products of generators of \mbox{\texttt{G}} are stored for any image of \mbox{\texttt{src}} or \mbox{\texttt{dest}}. 
\item[{ \texttt{RepresentativeAction(\mbox{\texttt{G}},\mbox{\texttt{P1}},\mbox{\texttt{P2}})} }]  Arbitrary mapping: see Lemma~2.1.4 in~\cite{Kohl05}. Tame mapping: see proof of Theorem~2.8.9 in~\cite{Kohl05}. The former is almost trivial, while the latter is a bit complicate and takes
usually also much more time. 
\item[{ \texttt{RepresentativeAction(RCWA(Integers),\mbox{\texttt{f}},\mbox{\texttt{g}})} }]  The algorithm used by \texttt{IsConjugate} constructs actually also an element \texttt{x} such that \texttt{\mbox{\texttt{f}}{\textasciicircum}x = \mbox{\texttt{g}}}. 
\item[{ \texttt{RespectedPartition(\mbox{\texttt{f}})}, \texttt{RespectedPartition(\mbox{\texttt{G}})} }]  Uses the algorithm described in the proof of Theorem~2.5.8 in~\cite{Kohl05}. 
\item[{ \texttt{Restriction(\mbox{\texttt{g}},\mbox{\texttt{f}})} }]  Computes images \texttt{(n{\textasciicircum}\mbox{\texttt{g}}){\textasciicircum}\mbox{\texttt{f}}} and preimages \texttt{n{\textasciicircum}\mbox{\texttt{f}}} for sufficiently many integers \texttt{n} under the image of \mbox{\texttt{g}} under the restriction monomorphism associated to~\mbox{\texttt{f}}. Then it constructs the desired mapping by \texttt{RcwaMapping(\mbox{\texttt{m}},\mbox{\texttt{values}})}. Finally, the result is checked by a direct verification that the diagram in
Definition~2.3.1 in~\cite{Kohl05} commutes. 
\item[{ \texttt{Restriction(\mbox{\texttt{G}},\mbox{\texttt{f}})} }]  Get a set of generators by applying \texttt{Restriction(\mbox{\texttt{g}},\mbox{\texttt{f}})} to the generators \mbox{\texttt{g}} of~\mbox{\texttt{G}}. 
\item[{ \texttt{Root(\mbox{\texttt{f}},\mbox{\texttt{k}})} }]  If \mbox{\texttt{f}} is bijective, class-wise order-preserving and has finite order: 

 Find a conjugate of \mbox{\texttt{f}} which is a product of class transpositions. Slice cycles $\prod_{i=2}^l \tau_{r_1(m_1),r_i(m_i)}$ of~\mbox{\texttt{f}} a respected partition $\mathcal{P}$ into cycles $\prod_{i=1}^l \prod_{j=0}^{k-1} \tau_{r_1(km_1),r_i+jm_i(km_i)}$ of the \mbox{\texttt{k}}-fold length on the refined partition which one gets from $\mathcal{P}$ by decomposing any $r_i(m_i) \in \mathcal{P}$ into residue classes (mod~$km_i$). Finally conjugate the resulting permutation back. 

 Other cases seem to be more difficult and are currently not covered. 
\item[{ \texttt{SemilocalizedRcwaMapping(\mbox{\texttt{f}},\mbox{\texttt{pi}})} }]  ``Trivial''. 
\item[{ \texttt{ShortCycles(\mbox{\texttt{f}},\mbox{\texttt{maxlng}})} }]  Look for fixed points of affine partial mappings of powers of~\mbox{\texttt{f}}. 
\item[{ \texttt{ShortOrbits(\mbox{\texttt{G}},\mbox{\texttt{S}},\mbox{\texttt{maxlng}})} }]  ``Straightforward''. 
\item[{ \texttt{SetOnWhichMappingIsClassWiseOrderPreserving(\mbox{\texttt{f}})}, etc. }]  Form the union of the residue classes modulo the modulus of the mapping, in
whose corresponding coefficient triple the first entry is positive, zero resp.
negative. 
\item[{ \texttt{Sign(\mbox{\texttt{sigma}})} }]  Evaluation of the given expression. For the mathematical meaning
(epimorphism!), see Theorem~2.12.8 in~\cite{Kohl05}. 
\item[{ \texttt{Size(\mbox{\texttt{G}})} }]  Test whether the group \mbox{\texttt{G}} is tame. If not, return \texttt{infinity}. Otherwise use \texttt{ActionOnRespectedPartition} to compute the permutation group \texttt{H} induced by \mbox{\texttt{G}} on a respected partition \texttt{P}, and \texttt{KernelOfActionOnRespectedPartition} to compute the kernel \texttt{K} of the action of \mbox{\texttt{G}} on \texttt{P}. The group \texttt{K} is infinite if and only if one of its generators has infinite order. Return
the product of the order of \texttt{H} and the order of~\texttt{K}. 
\item[{ \texttt{\mbox{\texttt{f}}+\mbox{\texttt{g}}} }]  Pointwise addition of affine mappings. 
\item[{ \texttt{Trajectory(\mbox{\texttt{f}},\mbox{\texttt{n}},...)} }]  Iterated application of an rcwa mapping. In the methods computing ``accumulated coefficients'' additionally composition of affine mappings. 
\item[{ \texttt{TransitionGraph(\mbox{\texttt{f}},\mbox{\texttt{m}})} }]  ``Straightforward'' -- just check a sufficiently long interval. 
\item[{ \texttt{TransitionMatrix(\mbox{\texttt{f}},\mbox{\texttt{m}})} }]  Evaluation of the given expression. 
\item[{ \texttt{ViewObj(\mbox{\texttt{f}})} }]  ``Trivial''. 
\item[{ \texttt{WreathProduct(\mbox{\texttt{G}},\mbox{\texttt{P}})} }]  Uses \texttt{DirectProduct} to embed the \texttt{DegreeAction(\mbox{\texttt{P}})}th direct power of~\mbox{\texttt{G}}, and \texttt{RcwaMapping}, Part~(d) to embed the finite permutation group~\mbox{\texttt{P}}. 
\item[{ \texttt{WreathProduct(\mbox{\texttt{G}},\mbox{\texttt{Z}})} }]  Restricts \mbox{\texttt{G}} to the residue class~3(4), and encodes the generator of~\mbox{\texttt{Z}} as $\tau_{0(2),1(2)} \cdot \tau_{0(2),1(4)}$. It is used that the images of~3(4) under powers of this mapping are pairwise disjoint residue classes. 
\end{description}
 }

         
\chapter{\textcolor{Chapter }{Installation and auxiliary functions}}\label{ch:InstallationAndAuxiliaryFunctions}
\logpage{[ 6, 0, 0 ]}
{
   
\section{\textcolor{Chapter }{Requirements}}\label{sec:Requirements}
\logpage{[ 6, 1, 0 ]}
{
  The \textsf{RCWA} package needs at least \textsf{GAP}~4.4.7, \textsf{ResClasses}~2.2.2, \textsf{GRAPE}~4.0~\cite{GRAPE} and \textsf{GAPDoc}~0.999~\cite{GAPDoc}. It can be used under UNIX, under Windows and on the MacIntosh. \textsf{RCWA} is completely written in the \textsf{GAP} language and does neither contain nor require external binaries. In
particular, warnings concerning missing binaries when \textsf{GRAPE} is loaded can savely be ignored. }

  
\section{\textcolor{Chapter }{Installation}}\label{sec:Installation}
\logpage{[ 6, 2, 0 ]}
{
  Like any other \textsf{GAP} package, \textsf{RCWA} must be installed in the \texttt{pkg} subdirectory of the \textsf{GAP} distribution. This is accomplished by extracting the distribution file in this
directory. If you have done this, you can load the package as usual via \texttt{LoadPackage( "rcwa" );}. }

  
\section{\textcolor{Chapter }{The Info class of the package}}\label{sec:TheInfoClass}
\logpage{[ 6, 3, 0 ]}
{
  

\subsection{\textcolor{Chapter }{InfoRCWA}}
\logpage{[ 6, 3, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{InfoRCWA\index{InfoRCWA@\texttt{InfoRCWA}}
\label{InfoRCWA}
}\hfill{\scriptsize (info class)}}\\


 This is the Info class of the \textsf{RCWA} package. See section \emph{Info Functions} in the \textsf{GAP} Reference Manual for a description of the Info mechanism. For convenience: \texttt{RCWAInfo(}\mbox{\texttt{n}}\texttt{)} is a shorthand for \texttt{SetInfoLevel(InfoRCWA},\mbox{\texttt{n}}\texttt{)}. }

 }

  
\section{\textcolor{Chapter }{The testing routine}}\label{sec:TheTestingRoutine}
\logpage{[ 6, 4, 0 ]}
{
  

\subsection{\textcolor{Chapter }{RCWATest}}
\logpage{[ 6, 4, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RCWATest(  )\index{RCWATest@\texttt{RCWATest}}
\label{RCWATest}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 Nothing. 



 Performs tests of the \textsf{RCWA} package. Errors, i.e. differences to the correct results of the test
computations, are reported. The processed test files are in the directory \texttt{pkg/rcwa/tst}. }

 }

 }

  
\section{\textcolor{Chapter }{Building the manual}}\label{sec:BuildingTheManual}
\logpage{[ 6, 5, 0 ]}
{
  The following routine is a development tool. As all files it generates are
included in the distribution file anyway, users will not need it. 

\subsection{\textcolor{Chapter }{RCWABuildManual}}
\logpage{[ 6, 5, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{RCWABuildManual(  )\index{RCWABuildManual@\texttt{RCWABuildManual}}
\label{RCWABuildManual}
}\hfill{\scriptsize (function)}}\\
\textbf{\indent Returns:\ }
 Nothing. 



 This function builds the manual of the \textsf{RCWA} package in the file formats {\LaTeX}, DVI, Postscript, PDF, HTML and ASCII text. This is accomplished using the \textsf{GAPDoc} package by Frank Lübeck and Max Neunhöffer. Building the manual is possible
only on UNIX systems and requires {\LaTeX}, PDF{\LaTeX} and \textsf{dvips}. }

 }

  \def\bibname{References\logpage{[ "Bib", 0, 0 ]}}

\bibliographystyle{alpha}
\bibliography{rcwa.bib}

\def\indexname{Index\logpage{[ "Ind", 0, 0 ]}}


\printindex

\newpage
\immediate\write\pagenrlog{["End"], \arabic{page}];}
\immediate\closeout\pagenrlog
\end{document}
