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1 Introduction

Sections

  1. The package
  2. Polycyclic groups

1.1 The package

This package provides functions for the computation with matrix groups. Let G be a subgroup of GL(d,R) where the ring R is either equal to Q,Z or a finite field Fq. Then: smallskip parindent=25pt

  • We can test whether G is solvable. smallskip
  • If G is polycyclic, then we can determine a polycyclic presentation for G. smallskip smallskip

    A group G which is given by a polycyclic presentation can be largely investigated by algorithms implemented in the GAP-package Polycyclic polycyclic. For example we can determine if G is torsion-free and calculate the torsion subgroup. Further we can compute the derived series and the Hirsch length of the group G. Also various methods for computations with subgroups, factor groups and extensions are available.

    In the case that the matrix group G is a subgroup of GL(d,Fq) or GL(d,Z), the group G is solvable if and only if G is polycyclic (see Chapter 1 in Segal). Therefore, in this case, we can test if G is polycyclic.

    As a by-product, the Polenta package provides some functionality to compute certain module series for modules of solvable groups. For example, if G is a rational polycyclic matrix group, then we can compute the radical series of the natural Q[G]-module Qd.

    1.2 Polycyclic groups

    A group G is called polycyclic if it has a finite subnormal series with cyclic factors. It is a well-known fact that every polycyclic group is finitely presented by a so-called polycyclic presentation (see for example Chapter 9 in Sims or Chapter 2 in polycyclic ). In GAP groups which are defined by polycyclic presentations are called polycyclically presented groups, short PcpGroups.

    The overall idea of the algorithm implemented in this package have first been introduced by Ostheimer in 1996 Ostheimer. In 2001 Eick presented a more detailed version Eick. This package contains an implementation of this algorithm. A description of this implementation together with some refinements and extensions can be found in Assmann.

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    Polenta manual
    November 2003