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<Chapter Label="Preface">
<Heading>Preface</Heading>
A group <M>G</M> is called <E>polycyclic</E> if there exists a subnormal series
in <M>G</M> with cyclic factors. Every polycyclic group is soluble and
every supersoluble group is polycyclic. The class of polycyclic
groups is closed with respect to forming subgroups, factor groups and
extensions. Polycyclic groups can also be characterised as those
soluble groups in which each subgroup is finitely generated.
<P/>
K. A. Hirsch has initiated the investigation of polycyclic groups in
1938, see <Cite Key="Hir38a"/>, <Cite Key="Hir38b"/>, <Cite Key="Hir46"/>, <Cite Key="Hir52"/>,
<Cite Key="Hir54"/>, and their central position in infinite group theory has
been recognised since.
<P/>
A well-known result of Hirsch asserts that each polycyclic group is
finitely presented. In fact, a polycyclic group has a presentation
which exhibits its polycyclic structure: a <E>pc-presentation</E> as defined
in the Chapter <Ref Chap="Introduction to polycyclic presentations" Style="Text"/>. Pc-presentations
allow efficient computations with the groups they define. In particular,
the word problem is efficiently solvable in a group given by a
pc-presentation. Further, subgroups and factor groups of groups given
by a pc-presentation can be handled effectively.
<P/>
The &GAP; 4 package &Polycyclic; is designed for computations
with polycyclic groups which are given by a pc-presentation. The
package contains methods to solve the word problem in such groups and
to handle subgroups and factor groups of polycyclic groups. Based on
these basic algorithms we present a collection of methods to construct
polycyclic groups and to investigate their structure.
<P/>
In <Cite Key="BCRS91"/> and <Cite Key="Seg90"/> the theory of problems which are
decidable in polycyclic-by-finite groups has been started. As a result
of these investigation we know that a large number of group theoretic
problems are decidable by algorithms in polycyclic groups. However,
practical algorithms which are suitable for computer implementations
have not been obtained by this study. We have developed a new set of
practical methods for groups given by pc-presentations, see for example
<Cite Key="Eic00"/>, and this package is a collection of implementations for these
and other methods.
<P/>
We refer to <Cite Key="Rob82"/>, page 147ff, and <Cite Key="Seg83"/> for background on
polycyclic groups. Further, in <Cite Key="Sims94"/> a variation of the basic
methods for groups with pc-presentation is introduced. Finally, we note
that the main GAP library contains many practical algorithms to compute
with finite polycyclic groups. This is described in the Section on
polycyclic groups in the reference manual.
</Chapter>
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