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