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<p><a id="X874E1D45845007FE" name="X874E1D45845007FE"></a></p>
<div class="ChapSects"><a href="chap1.html#X874E1D45845007FE">1 <span class="Heading">Preface</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X869E3CF37EC69123">1.1 <span class="Heading">The <strong class="pkg">GAP</strong> System</span></a>
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</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X877A62A1781C2147">1.2 <span class="Heading">Authors and Maintainers</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X82A988D47DFAFCFA">1.3 <span class="Heading">Acknowledgements</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7950EFA183E3F666">1.4 <span class="Heading">Copyright and License</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7B488D2E8492AB6A">1.5 <span class="Heading">Further Information about <strong class="pkg">GAP</strong></span></a>
</span>
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</div>
<h3>1 <span class="Heading">Preface</span></h3>
<p>Welcome to <strong class="pkg">GAP</strong>. This is one of three manuals documenting the core part of <strong class="pkg">GAP</strong>, the other being the <em><strong class="pkg">GAP</strong> Tutorial</em> (see <a href="../../doc/tut/chap1.html#X874E1D45845007FE"><span class="RefLink">Tutorial: Preface</span></a>). and the document called <em>"<strong class="pkg">GAP</strong> - Changes from Earlier Versions"</em> (see <a href="../../doc/changes/chap1.html#X874E1D45845007FE"><span class="RefLink">Changes: Preface</span></a>).</p>
<p>This preface serves not only to introduce "The <strong class="pkg">GAP</strong> Reference Manual", but also as an introduction to the whole system.</p>
<p><strong class="pkg">GAP</strong> stands for <em>Groups, Algorithms and Programming</em>. The name was chosen to reflect the aim of the system, which is introduced in this reference 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.</p>
<p>This manual, the <em><strong class="pkg">GAP</strong> reference manual</em> contains the official definitions of <strong class="pkg">GAP</strong> functions. It should contain all the information needed to use <strong class="pkg">GAP</strong>, and is not intended to be read cover-to-cover.</p>
<p>To get started a new user may first look at parts of the <em><strong class="pkg">GAP</strong> Tutorial</em> (see <a href="../../doc/tut/chap1.html#X874E1D45845007FE"><span class="RefLink">Tutorial: Preface</span></a>).</p>
<p>A lot of the functionality of the system and a number of contributed extensions are provided as "<strong class="pkg">GAP</strong> packages" which are developed independently of the core part of <strong class="pkg">GAP</strong> and can be loaded into a <strong class="pkg">GAP</strong> session. Each package comes with a its own manual which is also available through the <strong class="pkg">GAP</strong> help system.</p>
<p>This manual is divided into chapters, sections and subsections. Chapter <a href="chap2.html#X8755A2C67B197C63"><span class="RefLink">2</span></a> describes the <em>help system</em>, which provides access to all the manuals from a running <strong class="pkg">GAP</strong> session. Chapter <a href="chap3.html#X79CCD3A6821E5A37"><span class="RefLink">3</span></a> gives technical advice for <em>running</em> <strong class="pkg">GAP</strong>. Chapter <a href="chap4.html#X7FE7C0C17E1ED118"><span class="RefLink">4</span></a> introduces the <strong class="pkg">GAP</strong> language, and the next chapters deal with the <em>environment</em> provided by <strong class="pkg">GAP</strong> for the user. These are followed by the main bulk of chapters which are devoted to the various mathematical structures that <strong class="pkg">GAP</strong> can handle.</p>
<p>Subsequent sections of this preface explain the structure of the system and provide copyright and licensing information.</p>
<p><a id="X869E3CF37EC69123" name="X869E3CF37EC69123"></a></p>
<h4>1.1 <span class="Heading">The <strong class="pkg">GAP</strong> System</span></h4>
<p><strong class="pkg">GAP</strong> is a <em>free</em>, <em>open</em> and <em>extensible</em> 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 <em>free of charge</em> (except possibly for the immediate costs of delivering it to you), you are <em>free to pass it on</em> within certain limits, and all of the workings of the system are <em>open for you to examine and change</em>. Details of these conditions can be found in Section <a href="../../doc/ref/chap1.html#X7950EFA183E3F666"><span class="RefLink">Reference: Copyright and License</span></a>.</p>
<p>The system is "extensible" in that you can write your own programs in the <strong class="pkg">GAP</strong> 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 "<strong class="pkg">GAP</strong> packages". Further details of this can be found in chapter <a href="../../doc/ref/chap76.html#X7B6CD527825945CD"><span class="RefLink">Reference: Using GAP Packages</span></a>, and on our website.</p>
<p>Development of <strong class="pkg">GAP</strong> 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 <strong class="pkg">GAP</strong> 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 <strong class="pkg">GAP</strong> 4.4 release in 2004, which has been coordinated from Colorado State University, Fort Collins.</p>
<p>More information on the motivation and development of <strong class="pkg">GAP</strong> to date, can be found on our Web pages in a section entitled "Release history and Prefaces".</p>
<p>For those readers who have used an earlier version of <strong class="pkg">GAP</strong>, an overview of the changes from <strong class="pkg">GAP</strong> 4.4 and a brief summary of changes from earlier versions is given in a separate manual <a href="../../doc/changes/chap4.html#X837AADDA7EA9A541"><span class="RefLink">Changes: Changes between GAP 4.4 and GAP 4.5</span></a>.</p>
<p>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.</p>
<ol>
<li><p>A <em>kernel</em>, written in C, which provides the user with</p>
<ul>
<li><p>automatic dynamic storage management, which the user needn't bother about in his programming;</p>
</li>
<li><p>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;</p>
</li>
<li><p>an interpreter for the <strong class="pkg">GAP</strong> 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.</p>
</li>
<li><p>a small set of system functions allowing the <strong class="pkg">GAP</strong> programmer to handle files and execute external programs in a uniform way, regardless of the particular operating system in use.</p>
</li>
<li><p>a set of programming tools for testing, debugging, and timing algorithms.</p>
</li>
<li><p>a "read-eval-view" style user interface.</p>
</li>
</ul>
</li>
<li><p>A much larger <em>library of <strong class="pkg">GAP</strong> functions</em> that implement algebraic and other algorithms. Since this is written entirely in the <strong class="pkg">GAP</strong> language, the <strong class="pkg">GAP</strong> 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 <strong class="pkg">GAP</strong> users.</p>
</li>
<li><p>A <em>library of group theoretical data</em> 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.</p>
</li>
<li><p>The <em>documentation</em>. This is available as on-line help, as printable files in PDF format and as HTML for viewing with a Web browser.</p>
</li>
</ol>
<p>Also included with the core system are some test files and a few small utilities which we hope you will find useful.</p>
<p><strong class="pkg">GAP</strong> <em>packages</em> are self-contained extensions to the core system. A package contains <strong class="pkg">GAP</strong> code and its own documentation and may also contain data files or external programs to which the <strong class="pkg">GAP</strong> code provides an interface. These packages may be loaded into <strong class="pkg">GAP</strong> using the <code class="func">LoadPackage</code> (<a href="../../doc/ref/chap76.html#X79B373A77B29D1F5"><span class="RefLink">Reference: LoadPackage</span></a>) 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 <strong class="pkg">GAP</strong> 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 <strong class="pkg">GAP</strong>.</p>
<p>With <strong class="pkg">GAP</strong> 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 <strong class="pkg">GAP</strong> 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.</p>
<p>Further details about <strong class="pkg">GAP</strong> packages can be found in chapter <a href="../../doc/ref/chap76.html#X7B6CD527825945CD"><span class="RefLink">Reference: Using GAP Packages</span></a>, and on the <strong class="pkg">GAP</strong> website here: <span class="URL"><a href="http://www.gap-system.org/Packages/packages.html">http://www.gap-system.org/Packages/packages.html</a></span>.</p>
<p><a id="X877A62A1781C2147" name="X877A62A1781C2147"></a></p>
<h4>1.2 <span class="Heading">Authors and Maintainers</span></h4>
<p><strong class="pkg">GAP</strong> is the work of very many people, many of whom still maintain parts of the system. A complete list of authors, and an approximation to the current list of maintainers can be found on the <strong class="pkg">GAP</strong> World Wide Web site at <span class="URL"><a href="http://www.gap-system.org/Contacts/People/authors.html">http://www.gap-system.org/Contacts/People/authors.html</a></span> and <span class="URL"><a href="http://www.gap-system.org/Contacts/People/modules.html">http://www.gap-system.org/Contacts/People/modules.html</a></span>. All <strong class="pkg">GAP</strong> packages have their own authors and maintainers. It should however be noted that some packages provide interfaces between <strong class="pkg">GAP</strong> and an external program, a copy of which is included for convenience, and that, in these cases, we do not claim that the package authors or maintainers wrote, or maintain, this external program. Similarly, the system and some packages include large data libraries that may have been computed by many people. We try to make clear in each case what credit is attributable to whom.</p>
<p>We have, for some time, operated a refereeing system for contributed packages, both to ensure the quality of the software we distribute, and to provide recognition for the authors. We now consider this to be a refereeing system for modules, and we would note, in particular that, although it does not use the standard package interface, the library of small groups has been refereed and accepted on exactly the same basis as the accepted packages.</p>
<p>We also include with this distribution a number of packages which have not (yet) gone through our refereeing process. Some may be accepted in the future, in other cases the authors have chosen not to submit them. More information can be found on our World Wide Web site (see Section <a href="chap1.html#X7B488D2E8492AB6A"><span class="RefLink">1.5</span></a>).</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h4>1.3 <span class="Heading">Acknowledgements</span></h4>
<p>Very many people have worked on, and contributed to, <strong class="pkg">GAP</strong> over the years since its inception. On our Web site you will find the prefaces to the previous releases, each of which acknowledges people who have made special contributions to that release. Even so, it is appropriate to mention here Joachim Neubüser whose vision of a free, open and extensible system for computational algebra inspired <strong class="pkg">GAP</strong> in the first place, and Martin Schönert, who was the technical architect of <strong class="pkg">GAP</strong> 3 and <strong class="pkg">GAP</strong> 4.</p>
<p><a id="X7950EFA183E3F666" name="X7950EFA183E3F666"></a></p>
<h4>1.4 <span class="Heading">Copyright and License</span></h4>
<p>Copyright © (1987-2014) by the <strong class="pkg">GAP</strong> Group,</p>
<p>incorporating the Copyright © 1999, 2000 by School of Mathematical and Computational Sciences, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland</p>
<p>being the Copyright © 1992 by Lehrstuhl D für Mathematik, RWTH, 52056 Aachen, Germany, transferred to St Andrews on July 21st, 1997.</p>
<p>except for files in the distribution, which have an explicit different copyright statement. In particular, the copyright of packages distributed with <strong class="pkg">GAP</strong> is usually with the package authors or their institutions.</p>
<p><strong class="pkg">GAP</strong> is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the file <code class="file">GPL</code> in the <code class="file">etc</code> directory of the <strong class="pkg">GAP</strong> distribution or see <span class="URL"><a href="http://www.gnu.org/licenses/gpl.html">http://www.gnu.org/licenses/gpl.html</a></span>.</p>
<p>If you obtain <strong class="pkg">GAP</strong> please send us a short notice to that effect, e.g., an e-mail message to the address <span class="URL"><a href="mailto:support@gap-system.org">support@gap-system.org</a></span>. This helps us to keep track of the number of <strong class="pkg">GAP</strong> users.</p>
<p>If you publish a mathematical result that was partly obtained using <strong class="pkg">GAP</strong>, please cite <strong class="pkg">GAP</strong>, just as you would cite another paper that you used (see below for sample citation). Also we would appreciate if you could inform us about such a paper, which we will add to the <strong class="pkg">GAP</strong> <span class="URL"><a href="http://www.gap-system.org/Doc/Bib/bib.html">bibliography</a></span>.</p>
<p>Specifically, please refer to</p>
<div class="example"><pre>
[GAP] The GAP Group, GAP - Groups, Algorithms, and Programming,
Version 4.7.5; 2014 (http://www.gap-system.org)
</pre></div>
<p>You are permitted to modify and redistribute <strong class="pkg">GAP</strong>, but you are not allowed to restrict further redistribution. That is to say proprietary modifications will not be allowed. We want all versions of <strong class="pkg">GAP</strong> to remain free.</p>
<p>If you modify any part of <strong class="pkg">GAP</strong> and redistribute it, you must supply a <code class="file">README</code> document. This should specify what modifications you made in which files. We do not want to take credit or be blamed for your modifications.</p>
<p>Of course we are interested in all of your modifications. In particular we would like to see bug-fixes, improvements and new functions. So again we would appreciate it if you would inform us about all modifications you make.</p>
<p>In addition to the general copyright for <strong class="pkg">GAP</strong> set forth above, the following terms apply to the versions of <strong class="pkg">GAP</strong> for Windows.</p>
<p>The executable of <strong class="pkg">GAP</strong> for Windows that we distribute was compiled with the <code class="code">gcc</code> compiler supplied with <strong class="pkg">Cygwin</strong> installation (<span class="URL"><a href="http://cygwin.com/">http://cygwin.com/</a></span>).</p>
<p>The GNU C compiler is</p>
<p><em>Copyright © 2010 Free Software Foundation, Inc.</em></p>
<p>under the terms of the GNU General Public License (GPL).</p>
<p>The <strong class="pkg">Cygwin</strong> API library is also covered by the GNU GPL. The executable we provide is linked against this library (and in the process includes GPL'd <strong class="pkg">Cygwin</strong> glue code). This means that the executable falls under the GPL too, which it does anyhow.</p>
<p>The <code class="file">cyggcc_s-1.dll</code>, <code class="file">cygncurses-10.dll</code>, <code class="file">cygncursesw-10.dll</code>, <code class="file">cygpanel-10.dll</code>, <code class="file">cygpopt-0.dll</code>, <code class="file">cygreadline7.dll</code>, <code class="file">cygstart.exe</code>, <code class="file">cygwin1.dll</code>, <code class="file">libW11.dll</code>, <code class="file">mintty.exe</code>, <code class="file">rxvt.exe</code> and <code class="file">regtool.exe</code> are taken unmodified from the <strong class="pkg">Cygwin</strong> distribution. They are copyright by RedHat Software and released under the GPL. For more information on <strong class="pkg">Cygwin</strong>, see <span class="URL"><a href="http://www.cygwin.com">http://www.cygwin.com</a></span>.</p>
<p>Please contact <span class="URL"><a href="mailto:support@gap-system.org">support@gap-system.org</a></span> if you need further information.</p>
<p><a id="X7B488D2E8492AB6A" name="X7B488D2E8492AB6A"></a></p>
<h4>1.5 <span class="Heading">Further Information about <strong class="pkg">GAP</strong></span></h4>
<p>Information about <strong class="pkg">GAP</strong> is best obtained from the <strong class="pkg">GAP</strong> website</p>
<p><span class="URL"><a href="http://www.gap-system.org">http://www.gap-system.org</a></span></p>
<p>There you will find, amongst other things</p>
<ul>
<li><p>directions to the sites from which you can download the current <strong class="pkg">GAP</strong> distribution, all accepted and deposited <strong class="pkg">GAP</strong> packages, and a selection of other contributions.</p>
</li>
<li><p>the <strong class="pkg">GAP</strong> manual and an archive of the <code class="code">gap-forum</code> mailing list, formatted for reading with a Web browser, and indexed for searching.</p>
</li>
<li><p>information about <strong class="pkg">GAP</strong> developers, and about the email addresses available for comment, discussion and support.</p>
</li>
</ul>
<p>We would particularly ask you to note the following things:</p>
<ul>
<li><p>The <strong class="pkg">GAP</strong> Forum – an email discussion forum for comments, discussions or questions about <strong class="pkg">GAP</strong>. 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.</p>
</li>
<li><p>The email address <span class="URL"><a href="mailto:support@gap-system.org">support@gap-system.org</a></span> to which you are asked to send any questions or bug reports which do not seem likely to be of interest to the whole <strong class="pkg">GAP</strong> Forum. Please give a (short, if possible) self-contained excerpt of a <strong class="pkg">GAP</strong> 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 <strong class="pkg">GAP</strong> you are using (the first line after the <strong class="pkg">GAP</strong> 4 banner starting <code class="code">GAP, Version 4...</code>).</p>
</li>
<li><p>We also ask you to send a brief message to <span class="URL"><a href="mailto:support@gap-system.org">support@gap-system.org</a></span> when you install <strong class="pkg">GAP</strong>.</p>
</li>
<li><p>The correct form of citation of <strong class="pkg">GAP</strong>, which we ask you use whenever you publish scientific results obtained using <strong class="pkg">GAP</strong>.</p>
</li>
</ul>
<p>It finally remains for us to wish you all pleasure and success in using <strong class="pkg">GAP</strong>, and to invite your constructive comment and criticism.</p>
<p>The GAP Group,</p>
<p>24-May-2014</p>
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