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[1X1 [33X[0;0YPreface[133X[101X
[33X[0;0YA group [22XG[122X is called [13Xpolycyclic[113X if there exists a subnormal series in [22XG[122X 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.[133X
[33X[0;0YK. A. Hirsch has initiated the investigation of polycyclic groups in 1938,
see [Hir38a], [Hir38b], [Hir46], [Hir52], [Hir54], and their central
position in infinite group theory has been recognised since.[133X
[33X[0;0YA 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 [13Xpc-presentation[113X as defined in the Chapter
[14X'[33X[0;0YIntroduction to polycyclic presentations[133X'[114X. 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.[133X
[33X[0;0YThe [5XGAP[105X 4 package [5XPolycyclic[105X 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.[133X
[33X[0;0YIn [BCRS91] and [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 [Eic00], and this package is a
collection of implementations for these and other methods.[133X
[33X[0;0YWe refer to [Rob82], page 147ff, and [Seg83] for background on polycyclic
groups. Further, in [Sim94] 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.[133X
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