GAP (LAGUNA) - Chapter 1: Introduction

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1 Introduction
1.1 General aims
1.2 General computations in group rings
1.3 Computations in the normalized unit group
1.4 Computing Lie properties of the group algebra
1.5 Installation and system requirements

### 1 Introduction

#### 1.1 General aims

LAGUNA -- Lie AlGebras and UNits of group Algebras -- is the new name of the GAP4 package LAG. The LAG package arose as a byproduct of the third author's PhD thesis [Ros97]. Its first version was ported to GAP4 and was brought into the standard GAP4 package format during his visit to St Andrews in September 1998.

The main objective of LAG is to deal with Lie algebras associated with some associative algebras, and, in particular, Lie algebras of group algebras. Using LAG 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 LAGUNA the main part of the Lie algebra functionality is heavily built on the previous LAG releases.

The GAP4 package LAGUNA also extends the GAP 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. LAGUNA 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 LAGUNA provides a part of the functionality of the SISYPHOS program, which was developed by Martin Wursthorn to study the modular isomorphism problem; see [Wur93].

The corresponding functions of LAGUNA use the same algorithmic and theoretical approach as those in SISYPHOS. The reason why we reimplemented the normalised unit group algorithms in the LAGUNA package is that SISYPHOS has no interface to GAP4, and, even in GAP3, it is cumbersome to use the SISYPHOS output for further computation with the normalised unit group. For instance, using SISYPHOS with its GAP3 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 LAGUNA.

#### 1.2 General computations in group rings

The LAGUNA package provides a set of functions to carry out some basic computations with a group ring and its elements. Among other things, LAGUNA 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 LAGUNA is able to calculate the power-structure of the augmentation ideal, which is useful for the construction of the normalised unit group; see Sections 4.1--4.3 for more details.

#### 1.3 Computations in the normalized unit group

One of the aims of the LAGUNA 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 \(\alpha_1 \cdot g_1 + \alpha_2 \cdot g_2 + \cdots + \alpha_k \cdot g_k\) 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 \(1\).

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 [Sim94, Chapter 9]. For this computation we use an algorithm that was also used in the SISYPHOS package. For a brief description see Chapter 3. The functions that compute the structure of the normalised unit group are described in Section 4.4.

#### 1.4 Computing Lie properties of the group algebra

The functions that are used to compute Lie properties of \(p\)-modular group algebras were already included in the previous versions of LAG. The bracket operation \([\cdot,\cdot]\) on a \(p\)-modular group algebra \(FG\) is defined by \([a,b]=ab-ba\). It is well-known and very easy to check that \((FG, +, [\cdot,\cdot])\) is a Lie algebra. Then we may ask what kind of Lie algebra properties are satisfied by \(FG\). The results in [LR86], [PPS73], and [Ros00] 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 4.5.

#### 1.5 Installation and system requirements

LAGUNA does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for GAP4.4 and no compatibility with previous releases of GAP4 is guaranteed.

To use the LAGUNA online help it is necessary to install the GAP4 package GAPDoc by Frank Lübeck and Max Neunhöffer, which is available from the GAP site or from http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/.

LAGUNA is distributed in standard formats (`zoo`, `tar.gz`, `tar.bz2`, `-win.zip`) and can be obtained from http://www.cs.st-andrews.ac.uk/~alexk/laguna/. To unpack the archive `laguna-X.X.X.zoo` you need the program `unzoo`, which can be obtained from the GAP homepage http://www.gap-system.org/ (see section `Distribution'). To install LAGUNA, copy this archive into the `pkg` subdirectory of your GAP4.4 installation. The subdirectory `laguna` will be created in the `pkg` directory after the following command:

`unzoo -x laguna-X.X.X.zoo`

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