This package provides functions for computation with matrix
groups. Let <M>G</M> be a subgroup of <M>GL(d,R)</M> where the ring <M>R</M> is
either equal to <M>&QQ;,&ZZ;</M> or a finite field <M>\mathbb{F}_q</M>.
Then:
<List>
<Item>
We can test whether <M>G</M> is solvable.
</Item>
<Item>
We can test whether <M>G</M> is polycyclic.
</Item>
<Item>
If <M>G</M> is polycyclic, then we can determine a polycyclic
presentation for <M>G</M>.
</Item>
</List>
A group <M>G</M> which is given by a polycyclic presentation can be largely
investigated by algorithms implemented in the &GAP;-package
<Package>Polycyclic</Package> <Cite Key="Polycyclic"/>. For example
we can determine if <M>G</M> is torsion-free
and calculate the torsion subgroup. Further we can compute the derived
series and the Hirsch length of the group <M>G</M>. Also various methods for
computations with subgroups, factor groups and extensions are
available.
<P/>
As a by-product, the &Polenta; package
provides some functionality to compute certain module series for
modules of solvable groups. For example, if
<M>G</M> is a rational polycyclic matrix group, then we can compute the
radical series of the natural
<M>&QQ;[G]</M>-module <M>&QQ;^d</M>.
A group <M>G</M> 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 <Cite Key="Sims"/> or Chapter 2 in <Cite Key="Polycyclic"/> ).
In &GAP;, groups which are defined by polycyclic
presentations are called
polycyclically presented groups, abbreviated PcpGroups.
<P/>
The overall idea of the algorithm implemented in this package was
first introduced
by Ostheimer in 1996 <Cite Key="Ostheimer"/>.
In 2001 Eick presented a more detailed version <Cite Key="Eick"/>. This package contains an implementation of Eick's
algorithm. A description of this implementation together with some
refinements and extensions can be
found in <Cite Key="AEi05"/> and <Cite Key="Assmann"/>.
</Section>
</Chapter>
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