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\Chapter{Introduction to polycyclic presentations}

Let $G$ be a polycyclic group and let $G = C_1 \rhd C_2 \ldots C_n\rhd
C_{n+1} = 1$ be a *polycyclic series*, that  is, a subnormal series of
$G$ with non-trivial cyclic factors.  For $1  \leq i \leq n$ we choose
$g_i \in C_i$ such  that $C_i =  \langle  g_i, C_{i+1}  \rangle$.  Then
the
sequence $(g_1,   \ldots, g_n)$  is   called a  *polycyclic generating
sequence of  $G$*.  Let $I$ be  the set of those  $i  \in \{1, \ldots,
n\}$ with $r_i := [C_i : C_{i+1}]$ finite.  Each element of $G$ can be
written <uniquely> as  $g_1^{e_1}\cdots g_n^{e_n}$ with $e_i\in\Z$ for
$1\leq i\leq n$ and $0\leq e_i\<r_i$ for $i\in I$.

Each polycyclic    generating sequence   of  $G$   gives  rise  to a
*power-conjugate  (pc-)   presentation*  for  $G$  with  the  conjugate
relations
$$g_j^{g_i} = g_{i+1}^{e(i,j,i+1)} \cdots g_n^{e(i,j,n)}
                    \hbox{ for } 1 \leq i \< j \leq n,$$
$$g_j^{g_i^{-1}} = g_{i+1}^{f(i,j,i+1)} \cdots g_n^{f(i,j,n)}
                    \hbox{ for } 1 \leq i \< j \leq n,$$
and the power relations
$$g_i^{r_i} = g_{i+1}^{l(i,i+1)} \cdots g_n^{l(i,n)}
                    \hbox{ for } i \in I\.$$
\bigskip

Vice versa, we say that a group $G$ is defined by a pc-presentation if
$G$  is  given by  a  presentation of  the  form  above on  generators
$g_1,\ldots,g_n$.  These generators are  the *defining  generators* of
$G$.  Here, $I$  is the set of  $1\leq i\leq n$ such that  $g_i$ has a
power relation.  The positive integer $r_i$ for $i\in I$ is called the
relative order  of $g_i$. If $G$  is given by a  pc-presentation, then
$G$ is polycyclic. The subgroups $C_i = \langle g_i, \ldots, g_n
\rangle$ form a subnormal series $G = C_1 \geq \ldots \geq C_{n+1} = 1$
with cyclic factors and we have that $g_i^{r_i}\in C_{i+1}$. However,
some of the factors of this series may be smaller than $r_i$ for $i\in
I$ or finite if $i\not\in I.$ 

If $G$ is defined by a pc-presentation, then
each  element of  $G$    can be described   by   a word of   the  form
$g_1^{e_1}\cdots g_n^{e_n}$ in the defining generators with $e_i\in\Z$
for $1\leq i\leq n$ and $0\leq e_i\<r_i$ for $i\in I$.  Such a word is
said to be in *collected  form*.  In general, an  element of the group
can  be  represented by   more  than   one   collected word.    If the
pc-presentation  has  the property    that each element  of  $G$   has
precisely one word in collected form, then  the presentation is called
*confluent* or *consistent*.  If that is the case, the generators with
a power  relation correspond precisely to  the  finite factors  in the
polycyclic series and $r_i$ is the order of $C_i/C_{i+1}$.

The GAP 4 package {\sf polycyclic} is designed for computations with
polycyclic groups which are given by consistent pc-presentations. In
particular, all the functions described below assume that we compute
with a group defined by a consistent pc-presentation. See Section 
"Collectors" for a routine that checks the consistency of a
pc-presentation. 

