GAP (UnitLib) - Chapter 1: Introduction

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1 Introduction
1.1 General aims
1.2 Theoretical background
1.3 Installation and system requirements

### 1 Introduction

#### 1.1 General aims

Let \(KG\) be a group algebra of a finite \(p\)-group \(G\) over the field \(K\) of characteristic \(p\), and let \(V(KG)\) be the normalized unit group of \(KG\). The pc-presentation of the group \(V(KG)\) can be computed using the GAP package LAGUNA (http://www.cs.st-andrews.ac.uk/~alexk/laguna/), but for groups of orders 64 and more such computation will already take a lot of time.

The UnitLib package is an extension of the LAGUNA package that is focused on this problem. It contains the library of normalized unit groups of modular group algebras of finite \(p\)-groups over the field of \(p\) elements. This allows the user to retrieve the pre-computed group from the library instead of the time-consuming computation. The group created with UnitLib will have the same properties and attributes as the one computed with LAGUNA.

The version UnitLib 3.0.0 released in May 2009 also contained a parallel implementation of the computation of the normalized unit group of a modular group algebra of a finite \(p\)-group over the field of \(p\) elements, which works for groups from the GAP small groups library. It is developed on the base of the sequential version of this algorithm (which works for any \(p\)-group with no limitations) from the LAGUNA package. Parallelisation is implemented using the SCSCP package that is capable of connecting several local or remote GAP instances using the SCSCP protocol.

In April 2012, UnitLib 3.1.0 was updated to comply with GAP 4.5.

The current version of UnitLib provides the library of normalized unit groups \(V(KG)\) for all \(p\)-groups of order not greater than 243.

If you need to work with groups of bigger orders, please write to the maintainers, because they may be already computed or we can compute them for you.

#### 1.2 Theoretical background

Since the UnitLib package is an extension of the LAGUNA package [BKRS], we refer to the LAGUNA: LAGUNA package manual for the theoretical backround. In particular, Chapter 3 (The basic theory behind LAGUNA) of that manual contains definitions of the modular group algebra and its normalized unit group, the power-commutator presentation of the group, and also more details about the algorithm for the computation of the pc-presentation of the normalized unit group of a modular group algebra of a finite \(p\)-group.

#### 1.3 Installation and system requirements

UnitLib 3.2.0 is designed for GAP 4.5 and no compatibility with previous releases of GAP 4 is guaranteed.

Libraries of normalized unit groups of groups of orders less than 243, except for the order 128, will be available in any operating system.

The library for groups of order 128 was compressed using the `gzip` program and, therefore, will be available only in UNIX-type systems (including UNIX-installation in Mac OS X and Cygwin installation in Windows). If you need to work with groups of order 128 in Windows, please write to the maintainer to request a version of UnitLib with locally stored non-compressed data.

To work with the library for groups of order 243 you will also need the IO package by Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef/Computer/Software/Gap/io.html) to retrieve the data from the UnitLib homepage.

Because the UnitLib is an extension of the LAGUNA package, you must have the LAGUNA package installed. You can obtain it from the GAP homepage or from its homepage http://www.cs.st-andrews.ac.uk/~alexk/laguna.htm.

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

UnitLib is distributed in standard formats (`tar.gz`, `tar.bz2`, `.zip`, `-win.zip`) and can be obtained from the GAP homepage or from http://www.cs.st-andrews.ac.uk/~alexk/unitlib/. To install UnitLib, unpack its archive into the `pkg` subdirectory of your GAP4.5 installation. When you don't have access to the directory of your main GAP installation, you can also install the package outside the GAP main directory by unpacking it inside a directory `MYGAPDIR/pkg`. Then to be able to load UnitLib you need to call GAP with the `-l ";MYGAPDIR"` option.

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