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### 1 Preface

Welcome to GAP. This preface serves not only to introduce this manual, "the GAP Tutorial", but also as an introduction to the system as a whole.

GAP stands for Groups, Algorithms and Programming. The name was chosen to reflect the aim of the system, which is introduced in this tutorial manual. Since that choice, the system has become somewhat broader, and you will also find information about algorithms and programming for other algebraic structures, such as semigroups and algebras.

In addition to this manual, there is the GAP Reference Manual (see Reference: Preface) containing detailed documentation of the mathematical functionality of GAP, and the manual called "GAP - Changes from Earlier Versions" (see Changes: Preface) which describes most essential changes from previous GAP releases.

A lot of the functionality of the system and a number of contributed extensions are provided as "GAP packages" and each of these has its own manual.

Subsequent sections of this preface explain the structure of the system and list sources of further information about GAP.

#### 1.1 The GAP System

GAP is a free, open and extensible software package for computation in discrete abstract algebra. The terms "free" and "open" describe the conditions under which the system is distributed -- in brief, it is free of charge (except possibly for the immediate costs of delivering it to you), you are free to pass it on within certain limits, and all of the workings of the system are open for you to examine and change. Details of these conditions can be found in Section Reference: Copyright and License.

The system is "extensible" in that you can write your own programs in the GAP language, and use them in just the same way as the programs which form part of the system (the "library"). Indeed, we actively support the contribution, refereeing and distribution of extensions to the system, in the form of "GAP packages". Further details of this can be found in chapter Reference: Using and Developing GAP Packages, and on our website.

Development of GAP began at Lehrstuhl D für Mathematik, RWTH-Aachen, under the leadership of Joachim Neubüser in 1985. Version 2.4 was released in 1988 and version 3.1 in 1992. In 1997 coordination of GAP development, now very much an international effort, was transferred to St Andrews. A complete internal redesign and almost complete rewrite of the system was completed over the following years and version 4.1 was released in July 1999. A sign of the further internationalization of the project was the GAP 4.4 release in 2004, which has been coordinated from Colorado State University, Fort Collins.

More information on the motivation and development of GAP to date, can be found on our Web pages in a section entitled "Release history and Prefaces".

For those readers who have used an earlier version of GAP, an overview of the changes from GAP 4.4 and a brief summary of changes from earlier versions is given in a separate manual Changes: Changes between GAP 4.4 and GAP 4.5.

The system that you are getting now consists of a "core system" and a number of packages. The core system consists of four main parts.

1. A kernel, written in C, which provides the user with

• automatic dynamic storage management, which the user needn't bother about in his programming;

• a set of time-critical basic functions, e.g. "arithmetic", operations for integers, finite fields, permutations and words, as well as natural operations for lists and records;

• an interpreter for the GAP language, an untyped imperative programming language with functions as first class objects and some extra built-in data types such as permutations and finite field elements. The language supports a form of object-oriented programming, similar to that supported by languages like C++ and Java but with some important differences.

• a small set of system functions allowing the GAP programmer to handle files and execute external programs in a uniform way, regardless of the particular operating system in use.

• a set of programming tools for testing, debugging, and timing algorithms.

• a "read-eval-view" style user interface.

2. A much larger library of GAP functions that implement algebraic and other algorithms. Since this is written entirely in the GAP language, the GAP language is both the main implementation language and the user language of the system. Therefore the user can as easily as the original programmers investigate and vary algorithms of the library and add new ones to it, first for own use and eventually for the benefit of all GAP users.

3. A library of group theoretical data which contains various libraries of groups, including the library of small groups (containing all groups of order at most 2000, except those of order 1024) and others. Large libraries of ordinary and Brauer character tables and Tables of Marks are included as packages.

4. The documentation. This is available as on-line help, as printable files in PDF format and as HTML for viewing with a Web browser.

Also included with the core system are some test files and a few small utilities which we hope you will find useful.

GAP packages are self-contained extensions to the core system. A package contains GAP code and its own documentation and may also contain data files or external programs to which the GAP code provides an interface. These packages may be loaded into GAP using the LoadPackage (Reference: LoadPackage) command, and both the package and its documentation are then available just as if they were parts of the core system. Some packages may be loaded automatically, when GAP is started, if they are present. Some packages, because they depend on external programs, may only be available on the operating systems where those programs are available (usually UNIX). You should note that, while the packages included with this release are the most recent versions ready for release at this time, new packages and new versions may be released at any time and can be easily installed in your copy of GAP.

With GAP there are two packages (the library of ordinary and Brauer character tables, and the library of tables of marks) which contain functionality developed from parts of the GAP core system. These have been moved into packages for ease of maintenance and to allow new versions to be released independently of new releases of the core system. The library of small groups should also be regarded as a package, although it does not currently use the standard package mechanism. Other packages contain functionality which has never been part of the core system, and may extend it substantially, implementing specific algorithms to enhance its capabilities, providing data libraries, interfaces to other computer algebra systems and data sources such as the electronic version of the Atlas of Finite Group Representations; therefore, installation and usage of packages is recommended.

Further details about GAP packages can be found in chapter Reference: Using and Developing GAP Packages, and on the GAP website here: https://www.gap-system.org/Packages/packages.html.

#### 1.2 Further Information about GAP

Information about GAP is best obtained from the GAP website

https://www.gap-system.org

There you will find, amongst other things

• directions to the sites from which you can download the current GAP distribution, all accepted and deposited GAP packages, and a selection of other contributions.

• the GAP manual and an archive of the gap-forum mailing list, formatted for reading with a Web browser, and indexed for searching.

We would particularly ask you to note the following things:

• The GAP Forum – an email discussion forum for comments, discussions or questions about GAP. You must subscribe to the list before you can post to it, see the website for details. In particular we will announce new releases in this mailing list.

• The email address support@gap-system.org to which you are asked to send any questions or bug reports which do not seem likely to be of interest to the whole GAP Forum. Please give a (short, if possible) self-contained excerpt of a GAP session containing both input and output that illustrates your problem (including comments of why you think it is a bug) and state the type of the machine, operating system, (compiler used, if UNIX/Linux) and the version of GAP you are using (the first line after the GAP 4 banner starting GAP, Version 4...).

• We also ask you to send a brief message to support@gap-system.org when you install GAP.

• The correct form of citation of GAP, which we ask you use whenever you publish scientific results obtained using GAP.

It finally remains for us to wish you all pleasure and success in using GAP, and to invite your constructive comment and criticism.

The GAP Group,

01-Nov-2018

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