<p>This package provides functions for computation with matrix groups. Let <span class="SimpleMath">G</span> be a subgroup of <span class="SimpleMath">GL(d,R)</span> where the ring <span class="SimpleMath">R</span> is either equal to <span class="SimpleMath">ℚ,ℤ</span> or a finite field <span class="SimpleMath">F_q</span>. Then:</p>
<ul>
<li><p>We can test whether <span class="SimpleMath">G</span> is solvable.</p>
</li>
<li><p>We can test whether <span class="SimpleMath">G</span> is polycyclic.</p>
</li>
<li><p>If <span class="SimpleMath">G</span> is polycyclic, then we can determine a polycyclic presentation for <span class="SimpleMath">G</span>.</p>
</li>
</ul>
<p>A group <span class="SimpleMath">G</span> which is given by a polycyclic presentation can be largely investigated by algorithms implemented in the <strong class="pkg">GAP</strong>-package <strong class="pkg">Polycyclic</strong> <a href="chapBib.html#biBPolycyclic">[EN00]</a>. For example we can determine if <span class="SimpleMath">G</span> is torsion-free and calculate the torsion subgroup. Further we can compute the derived series and the Hirsch length of the group <span class="SimpleMath">G</span>. Also various methods for computations with subgroups, factor groups and extensions are available.</p>
<p>As a by-product, the <strong class="pkg">Polenta</strong> package provides some functionality to compute certain module series for modules of solvable groups. For example, if <span class="SimpleMath">G</span> is a rational polycyclic matrix group, then we can compute the radical series of the natural <span class="SimpleMath">ℚ[G]</span>-module <span class="SimpleMath">ℚ^d</span>.</p>
<p>A group <span class="SimpleMath">G</span> 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="chapBib.html#biBSims">[Sim94]</a> or Chapter 2 in <a href="chapBib.html#biBPolycyclic">[EN00]</a> ). In <strong class="pkg">GAP</strong>, groups which are defined by polycyclic presentations are called polycyclically presented groups, abbreviated PcpGroups.</p>
<p>The overall idea of the algorithm implemented in this package was first introduced by Ostheimer in 1996 <a href="chapBib.html#biBOstheimer">[Ost96]</a>. In 2001 Eick presented a more detailed version <a href="chapBib.html#biBEick">[Eic01]</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 [AE05] and [Ass03].
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