<html><head><title>[Polenta] 1 Introduction</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <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> <a name = "I0"></a> <a name = "I1"></a> <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> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>Polenta manual<br>June 2007 </address></body></html>