1<Chapter Label="Introduction"> 2<Heading>Introduction</Heading> 3 4<Index>Polenta</Index><!-- @Polenta --> 5<Index>Polycyclic</Index><!-- @Polycyclic --> 6 7<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 8<Section Label="The package"> 9<Heading>The package</Heading> 10 11This package provides functions for computation with matrix 12groups. Let <M>G</M> be a subgroup of <M>GL(d,R)</M> where the ring <M>R</M> is 13either equal to <M>&QQ;,&ZZ;</M> or a finite field <M>\mathbb{F}_q</M>. 14Then: 15<List> 16<Item> 17 We can test whether <M>G</M> is solvable. 18</Item> 19<Item> 20 We can test whether <M>G</M> is polycyclic. 21</Item> 22<Item> 23 If <M>G</M> is polycyclic, then we can determine a polycyclic 24 presentation for <M>G</M>. 25</Item> 26</List> 27 28A group <M>G</M> which is given by a polycyclic presentation can be largely 29investigated by algorithms implemented in the &GAP;-package 30<Package>Polycyclic</Package> <Cite Key="Polycyclic"/>. For example 31we can determine if <M>G</M> is torsion-free 32and calculate the torsion subgroup. Further we can compute the derived 33series and the Hirsch length of the group <M>G</M>. Also various methods for 34computations with subgroups, factor groups and extensions are 35available. 36<P/> 37 38As a by-product, the &Polenta; package 39provides some functionality to compute certain module series for 40modules of solvable groups. For example, if 41<M>G</M> is a rational polycyclic matrix group, then we can compute the 42radical series of the natural 43<M>&QQ;[G]</M>-module <M>&QQ;^d</M>. 44 45</Section> 46 47 48<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 49<Section Label="Polycyclic groups"> 50<Heading>Polycyclic groups</Heading> 51 52A group <M>G</M> is called polycyclic if it has a finite subnormal 53series with cyclic 54factors. It is a well-known fact that every polycyclic group is 55finitely presented by a so-called polycyclic presentation (see 56for example Chapter 9 in <Cite Key="Sims"/> or Chapter 2 in <Cite Key="Polycyclic"/> ). 57In &GAP;, groups which are defined by polycyclic 58 presentations are called 59polycyclically presented groups, abbreviated PcpGroups. 60 <P/> 61 62The overall idea of the algorithm implemented in this package was 63first introduced 64by Ostheimer in 1996 <Cite Key="Ostheimer"/>. 65In 2001 Eick presented a more detailed 66version <Cite Key="Eick"/>. This package contains an implementation of Eick's 67algorithm. A description of this implementation together with some 68refinements and extensions can be 69found in <Cite Key="AEi05"/> and <Cite Key="Assmann"/>. 70 71</Section> 72</Chapter> 73 74