Sophie: gap-system-4.4.12-5mdv2010.0 i586

gap-system-4.4.12-5mdv2010.0.i586.rpm

<html><head><title>[Polenta] 1 Introduction</title></head>
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<h1>1 Introduction</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP001.htm#SECT001">The package</a>
<li> <A HREF="CHAP001.htm#SECT002">Polycyclic groups</a>
</ol><p>
<p>
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<p>
<h2><a name="SECT001">1.1 The package</a></h2>
<p><p>
This package provides functions for computation with matrix
groups. Let <var>G</var> be a subgroup of <var>GL(d,R)</var> where the ring <var>R</var> is
either equal to <var><font face="helvetica,arial">Q</font>,<font face="helvetica,arial">Z</font></var> or a finite field <var><font face="helvetica,arial">F</font><sub>q</sub></var>.
Then: 
<dl compact>
<dt>--<dd>
    We can test whether <var>G</var> is solvable.
<dt>--<dd>
    We can test whether <var>G</var> is polycyclic.
<dt>--<dd>
    If <var>G</var> is polycyclic, then we can determine a polycyclic
    presentation for <var>G</var>. 
</dl>
<p>
A group <var>G</var> which is given by a polycyclic presentation can be largely
investigated by algorithms implemented in the <font face="Gill Sans,Helvetica,Arial">GAP</font>-package
Polycyclic <a href="biblio.htm#Polycyclic"><cite>Polycyclic</cite></a>. For example 
we can determine if <var>G</var> is torsion-free
and calculate the torsion subgroup. Further we can compute the derived
series and the Hirsch length of the group <var>G</var>. Also various methods for
computations with subgroups, factor groups and extensions are
available.
<p>
As a by-product, the <font face="Gill Sans,Helvetica,Arial">Polenta</font> package 
provides some functionality to compute certain module series for
modules of solvable groups. For example, if
<var>G</var> is a rational polycyclic matrix group, then we can compute the 
radical series of the natural
<var><font face="helvetica,arial">Q</font>[G]</var>-module <var><font face="helvetica,arial">Q</font><sup>d</sup></var>.  
<p>
<p>
<h2><a name="SECT002">1.2 Polycyclic groups</a></h2>
<p><p>
A group <var>G</var> 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 <a href="biblio.htm#Sims"><cite>Sims</cite></a> or Chapter 2 in <a href="biblio.htm#Polycyclic"><cite>Polycyclic</cite></a> ). 
In <font face="Gill Sans,Helvetica,Arial">GAP</font>, 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 <a href="biblio.htm#Ostheimer"><cite>Ostheimer</cite></a>. 
In 2001 Eick presented a more detailed
version <a href="biblio.htm#Eick"><cite>Eick</cite></a>. This package contains an implementation of Eick's
algorithm. A description of this implementation together with some
refinements and extensions can be
found in <a href="biblio.htm#AEi05"><cite>AEi05</cite></a> and <a href="biblio.htm#Assmann"><cite>Assmann</cite></a>. 
<p>
<p>
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<address>Polenta manual<br>June 2007
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