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<Chapter Label="Introduction to polycyclic presentations">
<Heading>Introduction to polycyclic presentations</Heading>
Let <M>G</M> be a polycyclic group and let <M>G = C_1 \rhd C_2 \ldots C_n\rhd
C_{n+1} = 1</M> be a <E>polycyclic series</E>, that is, a subnormal series of
<M>G</M> with non-trivial cyclic factors. For <M>1 \leq i \leq n</M> we choose
<M>g_i \in C_i</M> such that <M>C_i = \langle g_i, C_{i+1} \rangle</M>. Then
the
sequence <M>(g_1, \ldots, g_n)</M> is called a <E>polycyclic generating
sequence of <M>G</M></E>. Let <M>I</M> be the set of those <M>i \in \{1, \ldots,
n\}</M> with <M>r_i := [C_i : C_{i+1}]</M> finite. Each element of <M>G</M> can be
written <A>uniquely</A> as <M>g_1^{e_1}\cdots g_n^{e_n}</M> with <M>e_i\in &ZZ;</M> for
<M>1\leq i\leq n</M> and <M>0\leq e_i < r_i</M> for <M>i\in I</M>.
<P/>
Each polycyclic generating sequence of <M>G</M> gives rise to a
<E>power-conjugate (pc-) presentation</E> for <M>G</M> with the conjugate
relations
<Display>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,</Display>
<Display>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,</Display>
and the power relations
<Display>g_i^{r_i} = g_{i+1}^{l(i,i+1)} \cdots g_n^{l(i,n)}
\hbox{ for } i \in I.</Display>
<P/>
Vice versa, we say that a group <M>G</M> is defined by a pc-presentation if
<M>G</M> is given by a presentation of the form above on generators
<M>g_1,\ldots,g_n</M>. These generators are the <E>defining generators</E> of
<M>G</M>. Here, <M>I</M> is the set of <M>1\leq i\leq n</M> such that <M>g_i</M> has a
power relation. The positive integer <M>r_i</M> for <M>i\in I</M> is called the
<E>relative order</E> of <M>g_i</M>. If <M>G</M> is given by a pc-presentation, then
<M>G</M> is polycyclic. The subgroups <M>C_i = \langle g_i, \ldots, g_n
\rangle</M> form a subnormal series <M>G = C_1 \geq \ldots \geq C_{n+1} =
1</M> with cyclic factors and we have that <M>g_i^{r_i}\in C_{i+1}</M>.
However, some of the factors of this series may be smaller than <M>r_i</M>
for <M>i\in I</M> or finite if <M>i\not\in I</M>.
<P/>
If <M>G</M> is defined by a pc-presentation, then
each element of <M>G</M> can be described by a word of the form
<M>g_1^{e_1}\cdots g_n^{e_n}</M> in the defining generators with <M>e_i\in &ZZ;</M>
for <M>1\leq i\leq n</M> and <M>0\leq e_i < r_i</M> for <M>i\in I</M>. Such a word is
said to be in <E>collected form</E>. 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 <M>G</M> has
precisely one word in collected form, then the presentation is called
<E>confluent</E> or <E>consistent</E>. If that is the case, the generators with
a power relation correspond precisely to the finite factors in the
polycyclic series and <M>r_i</M> is the order of <M>C_i/C_{i+1}</M>.
<P/>
The &GAP; package &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 Chapter
<Ref Chap="Collectors" Style="Text"/> for a routine that checks the consistency of a
pc-presentation.
<P/>
A pc-presentation can be interpreted as a <E>rewriting system</E> in the
following way. One needs to add a new generator <M>G_i</M> for each
generator <M>g_i</M> together with the relations <M>g_iG_i = 1</M> and <M>G_ig_i =
1</M>. Any occurrence in a relation of an inverse generator <M>g_i^{-1}</M>
is replaced by <M>G_i</M>. In this way one obtains a monoid presentation
for the group <M>G</M>. With respect to a particular ordering on the set
of monoid words in the generators <M>g_1,\ldots g_n,G_1,\ldots G_n</M>, the
<E>wreath product ordering</E>, this monoid presentation is a rewriting
system. If the pc-presentation is consistent, the
rewriting system is confluent.
<P/>
In this package we do not address this aspect of pc-presentations
because it is of little relevance for the algorithms implemented here.
For the definition of rewriting systems and confluence in this context
as well as further details on the connections between pc-presentations
and rewriting systems we recommend the book <Cite Key="Sims94"/>.
</Chapter>
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