  
  [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
  
