1 Preface
  
  Welcome to GAP. This is one of the manuals documenting the core part of GAP,
  the others being the GAP Tutorial and HPC-GAP Reference Manual.
  
  There  is also a document CHANGES.md on the changes from earlier versions in
  the  root  directory.  This  preface  serves  not  only to introduce The GAP
  Reference Manual, but also as an introduction to the whole system.
  
  GAP  stands  for  Groups, Algorithms and Programming. The name was chosen to
  reflect  the  original  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.
  
  This  manual,  the GAP Reference Manual contains the official definitions of
  GAP  functions. It should contain all the information needed to use GAP, and
  is not intended to be read cover-to-cover.
  
  To get started a new user may first look at parts of the GAP Tutorial.
  
  A  lot  of  the  functionality  of  the  system  and a number of contributed
  extensions are provided as GAP packages which are developed independently of
  the  core  part  of  GAP  and can be loaded into a GAP session. Each package
  comes  with  a  its  own manual which is also available through the GAP help
  system.
  
  This  manual  is  divided into chapters, sections and subsections. Chapter 2
  describes  the  help system, which provides access to all the manuals from a
  running  GAP  session.  Chapter 3  gives  technical  advice for running GAP.
  Chapter 4  introduces  the GAP language, and the next chapters deal with the
  environment  provided  by  GAP  for the user. These are followed by the main
  bulk  of  chapters  which are devoted to the various mathematical structures
  that GAP can handle.
  
  Subsequent  sections of this preface explain the structure of the system and
  provide copyright and licensing information.
  
  
  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  website  in  a  section  entitled  Some  History  of  GAP:
  https://www.gap-system.org/Doc/History/history.html.
  
  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 CHANGES.md file in the main directory.
  
  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 when 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  a  user  can,  as easily as the
        original  programmers,  investigate and vary algorithms of the library
        and add new ones to it, first for their 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 Authors and Maintainers
  
  GAP  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   GAP   website   at
  https://www.gap-system.org/Contacts/People/authors.html                  and
  https://www.gap-system.org/Contacts/People/modules.html.  All  GAP  packages
  have their own authors and maintainers. It should however be noted that some
  packages  provide  interfaces between GAP 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.
  
  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.
  
  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 website (see Section 1.5).
  
  
  1.3 Acknowledgements
  
  Very  many  people  have  worked  on, and contributed to, GAP over the years
  since  its  inception.  On  our  website  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 GAP in the first place, and Martin Schönert,
  who was the technical architect of GAP 3 and GAP 4.
  
  
  1.4 Copyright and License
  
  Copyright © (1987-2022) by the GAP Group,
  
  incorporating  the  Copyright  ©  1999,  2000  by School of Mathematical and
  Computational  Sciences,  University of St Andrews, North Haugh, St Andrews,
  Fife KY16 9SS, Scotland
  
  being  the  Copyright  ©  1992  by  Lehrstuhl  D für Mathematik, RWTH, 52056
  Aachen, Germany, transferred to St Andrews on July 21st, 1997.
  
  except  for  files  in  the  distribution,  which have an explicit different
  copyright  statement.  In  particular, the copyright of packages distributed
  with GAP is usually with the package authors or their institutions.
  
  GAP  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 LICENSE in the root directory of the GAP
  distribution or see https://www.gnu.org/licenses/gpl.html.
  
  If  you  obtain  GAP  please send us a short notice to that effect, e.g., an
  e-mail  message  to the address mailto:support@gap-system.org. This helps us
  to keep track of the number of GAP users.
  
  If  you  publish  a  mathematical result that was partly obtained using GAP,
  please  cite  GAP,  just  as you would cite another paper that you used (see
  below  for  a sample citation). Also we would appreciate if you could inform
  us  about  such  a  paper,  which  we  will  add  to  the  GAP  bibliography
  (https://www.gap-system.org/Doc/Bib/bib.html).
  
  Specifically, please refer to
  
  
    [GAP] The GAP Group, GAP - Groups, Algorithms, and Programming,
          Version 4.12.1; 2022 (https://www.gap-system.org)
  
  
  You are permitted to modify and redistribute GAP, but you are not allowed to
  restrict  further  redistribution.  That is to say proprietary modifications
  will not be allowed. We want all versions of GAP to remain free.
  
  If  you modify any part of GAP and redistribute it, you must supply a README
  document. This should specify what modifications you made in which files. We
  do not want to take credit or be blamed for your modifications.
  
  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.
  
  In  addition to the general copyright for GAP set forth above, the following
  terms apply to the versions of GAP for Windows.
  
  GAP   for   Windows   is   packaged   with   several  packages  from  Cygwin
  (https://cygwin.com).  Files from Cygwin are provided under their respective
  licenses,  which  are  included  in  the  installation.  The GAP for Windows
  executables  that  we  distribute are themselves released under the terms of
  the  GNU  General Public License (GNU GPL); either version 2 of the License,
  or (at your option) any later version.
  
  Please   contact   mailto:support@gap-system.org   if   you   need   further
  information.
  
  
  1.5 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.
  
      information  about  GAP  developers,  and  about  the  email addresses
        available for comment, discussion and support.
  
  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 mailto: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
        mailto: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,
  
  2022-10-20