Sophie

Sophie

distrib > Mandriva > 2010.0 > i586 > media > contrib-release > by-pkgid > 91213ddcfbe7f54821d42c2d9e091326 > files > 1746

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

<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (LAGUNA) - Chapter 1: Introduction</title>
<meta http-equiv="content-type" content="text/html; charset=iso-8859-1" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
</head>
<body>


<div class="pcenter"><table class="chlink"><tr><td class="chlink1">Goto Chapter: </td><td><a href="chap0.html">Top</a></td><td><a href="chap1.html">1</a></td><td><a href="chap2.html">2</a></td><td><a href="chap3.html">3</a></td><td><a href="chap4.html">4</a></td><td><a href="chapBib.html">Bib</a></td><td><a href="chapInd.html">Ind</a></td></tr></table><br /></div>
<p><a id="s0ss0" name="s0ss0"></a></p>

<h3>1. Introduction</h3>

<p><a id="s1ss0" name="s1ss0"></a></p>

<h4>1.1 General aims</h4>

<p><strong class="pkg">LAGUNA</strong> -- <strong class="button">L</strong>ie <strong class="button">A</strong>l<strong class="button">G</strong>ebras and <strong class="button">UN</strong>its of group <strong class="button">A</strong>lgebras -- is the new name of the <strong class="pkg">GAP</strong>4 package <strong class="pkg">LAG</strong>. The <strong class="pkg">LAG</strong> package arose as a byproduct of the third author's PhD thesis <a href="chapBib.html#biBRos97">[R97]</a>. Its first version was ported to <strong class="pkg">GAP</strong>4 and was brought into the standard <strong class="pkg">GAP</strong>4 package format during his visit to St Andrews in September 1998.</p>

<p>The main objective of <strong class="pkg">LAG</strong> is to deal with Lie algebras associated with some associative algebras, and, in particular, Lie algebras of group algebras. Using <strong class="pkg">LAG</strong> it is possible to verify some properties or calculate certain Lie ideals of such Lie algebras very efficiently, due to their special structure. In the current version of <strong class="pkg">LAGUNA</strong> the main part of the Lie algebra functionality is heavily built on the previous <strong class="pkg">LAG</strong> releases.</p>

<p>The <strong class="pkg">GAP</strong>4 package <strong class="pkg">LAGUNA</strong> also extends the <strong class="pkg">GAP</strong> functionality for calculations with units of modular group algebras. In particular, using this package, one can check whether an element of such a group algebra is invertible. <strong class="pkg">LAGUNA</strong> also contains an implementation of an efficient algorithm to calculate the (normalized) unit group of the group algebra of a finite p-group over the field of p elements. Thus, the present version of <strong class="pkg">LAGUNA</strong> provides a part of the functionality of the <strong class="pkg">SISYPHOS</strong> program, which was developed by Martin Wursthorn to study the modular isomorphism problem; see <a href="chapBib.html#biBWursthorn">[W93]</a>.</p>

<p>The corresponding functions of <strong class="pkg">LAGUNA</strong> use the same algorithmic and theoretical approach as those in <strong class="pkg">SISYPHOS</strong>. The reason why we reimplemented the normalised unit group algorithms in the <strong class="pkg">LAGUNA</strong> package is that <strong class="pkg">SISYPHOS</strong> has no interface to <strong class="pkg">GAP</strong>4, and, even in <strong class="pkg">GAP</strong>3, it is cumbersome to use the <strong class="pkg">SISYPHOS</strong> output for further computation with the normalised unit group. For instance, using <strong class="pkg">SISYPHOS</strong> with its <strong class="pkg">GAP</strong>3 interface, it is difficult to embed a finite p-group into the normalized unit group of its group algebra over the field of p elements, but this can easily be done with <strong class="pkg">LAGUNA</strong>.</p>

<p><a id="s2ss0" name="s2ss0"></a></p>

<h4>1.2 General computations in group rings</h4>

<p>The <strong class="pkg">LAGUNA</strong> package provides a set of functions to carry out some basic computations with a group ring and its elements. Among other things, <strong class="pkg">LAGUNA</strong> provides elementary functions to compute such basic notions as support, length, trace and augmentation of an element. For modular group algebras of finite p-groups <strong class="pkg">LAGUNA</strong> is able to calculate the power-structure of the augmentation ideal, which is useful for the construction of the normalised unit group; see Sections <a href="chap4.html#s1ss0"><b>4.1</b></a>--<a href="chap4.html#s3ss0"><b>4.3</b></a> for more details.</p>

<p><a id="s3ss0" name="s3ss0"></a></p>

<h4>1.3 Computations in the normalized unit group</h4>

<p>One of the aims of the <strong class="pkg">LAGUNA</strong> package is to carry out efficient computations in the normalised unit group of the group algebra FG of a finite p-group G over the field F of p elements. If U is the unit group of FG then it is easy to see that U is the direct product of F^* and V(FG), where F^* is the multiplicative group of F, and V(FG) is the group of normalised units. A unit of FG of the form alpha_1 * g_1 + alpha_2 * g_2 + cdots + alpha_k * g_k with alpha_i in F and g_i in G is said to be normalised if the sum alpha_1 + alpha_2 + cdots + alpha_k is equal to 1.</p>

<p>It is well-known that the normalised unit group V has order |F|^|G|-1, and so V is a finite p-group. Thus computing V efficiently means to compute a polycyclic presentation for V. For the theory of polycyclic presentations refer to <a href="chapBib.html#biBSims">[S94, Chapter 9]</a>. For this computation we use an algorithm that was also used in the <strong class="pkg">SISYPHOS</strong> package. For a brief description see Chapter <a href="chap3.html#s0ss0"><b>3.</b></a>. The functions that compute the structure of the normalised unit group are described in Section <a href="chap4.html#s4ss0"><b>4.4</b></a>.</p>

<p><a id="s4ss0" name="s4ss0"></a></p>

<h4>1.4 Computing Lie properties of the group algebra </h4>

<p>The functions that are used to compute Lie properties of p-modular group algebras were already included in the previous versions of <strong class="pkg">LAG</strong>. The bracket operation [*,*] on a p-modular group algebra FG is defined by [a,b]=ab-ba. It is well-known and very easy to check that (FG, +, [*,*]) is a Lie algebra. Then we may ask what kind of Lie algebra properties are satisfied by FG. The results in <a href="chapBib.html#biBLR86">[LR86]</a>, <a href="chapBib.html#biBPPS73">[PPS73]</a>, and <a href="chapBib.html#biBRos00">[R00]</a> give fast, practical algorithms to check whether the Lie algebra FG is abelian, nilpotent, soluble, centre-by-metabelian, etc. The functions that implement these algorithms are described in Section <a href="chap4.html#s5ss0"><b>4.5</b></a>.</p>

<p><a id="s5ss0" name="s5ss0"></a></p>

<h4>1.5 Installation and system requirements</h4>

<p><strong class="pkg">LAGUNA</strong> does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for <strong class="pkg">GAP</strong>4.4 and no compatibility with previous releases of <strong class="pkg">GAP</strong>4 is guaranteed.</p>

<p>To use the <strong class="pkg">LAGUNA</strong> online help it is necessary to install the <strong class="pkg">GAP</strong>4 package <strong class="pkg">GAPDoc</strong> by Frank L\"ubeck and Max Neunh\"offer, which is available from the <strong class="pkg">GAP</strong> site or from <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/">http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/</a>.</p>

<p><strong class="pkg">LAGUNA</strong> is distributed in standard formats (<code class="file">zoo</code>, <code class="file">tar.gz</code>, <code class="file">tar.bz2</code>, <code class="file">-win.zip</code>) and can be obtained from <a href="http://www.cs.st-andrews.ac.uk/~alexk/laguna.htm">http://www.cs.st-andrews.ac.uk/~alexk/laguna.htm</a>. To unpack the archive <code class="file">laguna-3.4.zoo</code> you need the program <code class="file">unzoo</code>, which can be obtained from the <strong class="pkg">GAP</strong> homepage <a href="http://www.gap-system.org/">http://www.gap-system.org/</a> (see section `Distribution'). To install <strong class="pkg">LAGUNA</strong>, copy this archive into the <code class="file">pkg</code> subdirectory of your <strong class="pkg">GAP</strong>4.4 installation. The subdirectory <code class="file">laguna</code> will be created in the <code class="file">pkg</code> directory after the following command:</p>

<p><code class="code">unzoo -x laguna-3.4.zoo</code></p>


<div class="pcenter">
<table class="chlink"><tr><td><a href="chap0.html">Top of Book</a></td><td><a href="chap0.html">Previous Chapter</a></td><td><a href="chap2.html">Next Chapter</a></td></tr></table>
<br />


<div class="pcenter"><table class="chlink"><tr><td class="chlink1">Goto Chapter: </td><td><a href="chap0.html">Top</a></td><td><a href="chap1.html">1</a></td><td><a href="chap2.html">2</a></td><td><a href="chap3.html">3</a></td><td><a href="chap4.html">4</a></td><td><a href="chapBib.html">Bib</a></td><td><a href="chapInd.html">Ind</a></td></tr></table><br /></div>

</div>

<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

Perl :: Catalyst :: PostgreSQL :: RPM Valid HTML 4.01 Transitional