  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1 [33X[0;0YPhilosophy[133X[101X
  
  [33X[0;0Y[5XFinInG[105X (pronounciation: []) is a package for computation in Finite Incidence
  Geometry. It provides users with the basic tools to work in various areas of
  finite  geometry  from  the realms of projective spaces to the flat lands of
  generalised  polygons.  The algebraic power of GAP is employed, particularly
  in its facility with matrix and permutation groups.[133X
  
  
  [1X1.2 [33X[0;0YHow to cite [5XFinInG[105X[101X[1X[133X[101X
  
  [33X[0;0YThe  development  group  of  [5XFinInG[105X welcomes contact with users. In case you
  have  obtained the package as a deposited package part of archive during the
  installation   of   GAP,   we  call  on  your  beneficence  to  register  at
  [7Xhttp://cage.ugent.be/fining[107X when you use [5XFinInG[105X.[133X
  
  [33X[0;0YPlease  tell us about the use of [5XFinInG[105X in your research or teaching. We are
  very  interested in results obtained using [5XFinInG[105X and we might refer to your
  work  in  the  future.  If your work is published, we ask you to cite [5XFinInG[105X
  like a journal article or book.[133X
  
  [33X[0;0YIf  you  are  using  BibTeX,  you can use the following BibTeX entry for the
  current [5XFinInG[105X version:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X@manual{fining,[128X[104X
    [4X[28X        Author = {Bamberg, John and Betten, Anton and Cara, Philippe and[128X[104X
    [4X[28X            De Beule, Jan and Lavrauw, Michel and Neunh\"offer, Max },[128X[104X
    [4X[28X        Key = {fining},[128X[104X
    [4X[28X        Title = {{FinInG -- Finite Incidence Geometry, Version 1.3}},[128X[104X
    [4X[28X        Url = {\verb+(http://cage.ugent.be/fining)+},[128X[104X
    [4X[28X        Year = 2015}[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHere  is  the  bibliography  entry produced by BibTeX (in bibliography style
  `alpha'),  to be pasted directly inside the bibliography environment of your
  LaTeX document:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X\bibitem[FinInG]{fining}[128X[104X
    [4X[28XJ.~Bamberg, A.~Betten, {Ph}. Cara, J.~De~Beule, M.~Lavrauw, and[128X[104X
    [4X[28X  M.~Neunh\"offer.[128X[104X
    [4X[28X\newblock {\em Finite Incidence Geometry}.[128X[104X
    [4X[28X\newblock FInInG -- a {GAP} package, version 1.3, 2015.[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWhen linking to FinInG from a web page you can use the link[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X<a href="http://cage.ugent.be/fining/">FinInG</a>.[128X[104X
  [4X[32X[104X
  
  
  [1X1.3 [33X[0;0YOverview of this manual[133X[101X
  
  [33X[0;0YThis  chapter  (section  [14X1.4[114X)  describes  the  installation  of the package.
  Chapter [14X2[114X contains some extended examples to introduce the user to the basic
  functionality  and  philosophy to get started. Chapter [14X3[114X contains a rigorous
  discription  of  the  basic structures. This chapter can be omitted at first
  reading,  since  the  set  of  consequent  chapters  is also self contained.
  Chapters  [14X4[114X,  [14X5[114X  and  [14X6[114X  deal  with projective spaces, projective semilinear
  groups  and  polarities of projective spaces, respectively. In Chapter [14X7[114X the
  functionality  for classical polar spaces is treated and in Chapter [14X9[114X affine
  spaces  and  their groups are considered. [13XGeometry morphisms[113X between various
  geometries  that  are available in the package, are introduced and discussed
  in  Chapter  [14X10[114X.  The final three chapters, [14X11[114X, [14X12[114X, and [14X13[114X explain the basic
  functionality  which  is  provided for algebraic varieties (in projective or
  affine  spaces),  generalised  polygons,  of  which  several  models  can be
  constructed, and finally coset geometries and diagrams.[133X
  
  
  [1X1.4 [33X[0;0YInstalling [5XFinInG[105X[101X[1X[133X[101X
  
  [33X[0;0YThis  package  requires  the  GAP  packages [5XGAPDoc[105X, version 1.5.1 or higher,
  [5XForms[105X,  version 1.2.3 or higher, [5XOrb[105X, version 4.7.3 or higher, [5Xcvec[105X, version
  2.5.3 or higher, [5XGenSS[105X, version 1.6.2 or higher, and [5XGRAPE[105X, version 4.6.1 or
  higher.  Currently,  one  function  will  use  the  [5XDesign[105X package, but this
  package  is  not  required  to  load  [5XFinInG[105X. The package [5XGenSS[105X requires the
  package [5XIO[105X. The packages required by [5XFinInG[105X are all part of the standard GAP
  distribution  but some of them have to be compiled. Make sure (at least) [5XIO[105X,
  [5Xorb[105X  and  [5Xcvec[105X have been compiled before trying to load [5XFinInG[105X. You can find
  instructions on how to compile packages for your operating system on the [5XGAP[105X
  webpage.  Here  we  just  show  how  to  compile  the required packages on a
  UNIX-like   system.  In  case  you  don't  have  write  access  to  the  gap
  installation  on  your computer, you should ask your system administrator to
  do this for you. We assume your GAP installation resides in /opt/gap4r7[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xuser@computer:~/$ cd /opt/gap4r7/pkg[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/$ cd orb[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/orb$ ./configure[128X[104X
    [4X[28Xchecking for a BSD-compatible install... /usr/bin/install -c[128X[104X
    [4X[28X...[128X[104X
    [4X[28X...lots of output...[128X[104X
    [4X[28X...[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/orb$ make[128X[104X
    [4X[28Xmake[1]: Entering directory `/opt/gap4r7/pkg/orb'[128X[104X
    [4X[28X...[128X[104X
    [4X[28X...lots of output...[128X[104X
    [4X[28X...[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/orb$ cd ../io[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/io$ ./configure[128X[104X
    [4X[28Xchecking for a BSD-compatible install... /usr/bin/install -c[128X[104X
    [4X[28X...[128X[104X
    [4X[28X...lots of output...[128X[104X
    [4X[28X...[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/io$ make[128X[104X
    [4X[28Xmake[1]: Entering directory `/opt/gap4r7/pkg/io'[128X[104X
    [4X[28X...[128X[104X
    [4X[28X...lots of output...[128X[104X
    [4X[28X...[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/io$ cd ../cvec[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/cvec$ ./configure[128X[104X
    [4X[28Xchecking for a BSD-compatible install... /usr/bin/install -c[128X[104X
    [4X[28X...[128X[104X
    [4X[28X...lots of output...[128X[104X
    [4X[28X...[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/cvec$ make[128X[104X
    [4X[28Xmake[1]: Entering directory `/opt/gap4r7/pkg/cvec'[128X[104X
    [4X[28X...[128X[104X
    [4X[28X...lots of output...[128X[104X
    [4X[28X...[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/cvec$ cd ../grape/[128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/grape$ ./configure [128X[104X
    [4X[28Xuser@computer:/opt/gap4r7/pkg/grape$ make[128X[104X
    [4X[28Xif test ! -d bin;  then mkdir bin;  chmod 755 bin;  fi[128X[104X
    [4X[28X...[128X[104X
    [4X[28X...lots of output...[128X[104X
    [4X[28X...[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  recent  development  stage  of  [5XFinInG[105X is based on GAP4r5. We have done
  testing  using  GAP4r7,  and  no  installation  differences occured. In this
  section,  we  describe  in  detail  the  installation  procedure for [5XFinInG[105X,
  assuming the use of GAP4r7. We have also (succesfully) tested this procedure
  with  some  older  GAP  releases  but  these required manual installation of
  [5XGenSS[105X,  [5XIO[105X  and  [5XOrb[105X.  The installation of [5XFinInG[105X itself is generic for each
  UNIX  like system, including the different flavours of MacOSX. You only need
  a terminal application, and you need acces to the standard unix tools gunzip
  and tar. The installation procedure for [5XFinInG[105X, a standard GAP package, does
  [13Xnot[113X  require  compilation.  Each  GAP  installation  has  a  pkg  directory,
  containing  supplemental GAP packages. If you have acces to this filesystem,
  you can locate it, e.g.[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y/usr/local/gap4r7/pkg/[133X [124X[133X
  
  
  [33X[0;0YDownload  the  [5XFinInG[105X  archive "fining-....tgz" to this location, and unpack
  the archive. This can be done by issuing[133X
  
  
        [33X[1;6Y[24X[33X[0;0Ygunzip fining-....tgz[133X [124X[133X
  
  
  [33X[0;0Ywhich  yields  a  file  "fining-....tar",  in the pkg directory, after which
  issuing the command[133X
  
  
        [33X[1;6Y[24X[33X[0;0Ytar -xf fining-....tar[133X [124X[133X
  
  
  [33X[0;0Yunpacks  the  archive in a subdirectory fining. After successfully unpacking
  the archive, you can locate the directory[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y/usr/local/gap4r7/pkg/fining/.[133X [124X[133X
  
  
  [33X[0;0YThis  directory  contains a subdirectory "./doc", containing an html and pdf
  version  of  the  manual. The html version is accessible by opening the file
  "chap0.html"  in your favorite browser. The pdf version of the manual can be
  found in the file "manual.pdf".[133X
  
  [33X[0;0YPlease  notice  that  you  can  unpack  your  archive in your favorite local
  "./pkg"  directory, e.g. "/home/yourself/pkg/", in case you are using GAP on
  a server on which you have only a restricted access. If you installed [5XFinInG[105X
  in  the  central  GAP  pkg  directory,  you  can startup gap using the usual
  command.  We  suppose  this is gap4r7. If you installed [5XFinInG[105X in your local
  pkg  directory,  e.g.  "/home/yourself/pkg/", then move to "/home/yourself",
  and issue the command[133X
  
  
        [33X[1;6Y[24X[33X[0;0Ygap -l "/usr/local/gap4r7;./"[133X [124X[133X
  
  
  [33X[0;0YThis will cause gap to startup and use as pkg directory both its own central
  pkg  directory,  i.e.  "/usr/local/gap4r7/pkg",  as  well  as your local pkg
  directory,  i.e.  "/home/yourself/pkg/".  You  should see something like the
  following  output.  Notice  that  the  packages  [5XGAPDoc[105X and [5XIO[105X are loaded by
  default.  This is not necessarily the case, but loading fining will force to
  load required packages anyway.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X┌───────┐   GAP,  Version 4.7.8 of 09-Jun-2015 (free software, GPL)[128X[104X
    [4X[28X│  GAP  │   http://www.gap-system.org[128X[104X
    [4X[28X└───────┘   Architecture: x86_64-unknown-linux-gnu-gcc-default64[128X[104X
    [4X[28XLibs used:  gmp[128X[104X
    [4X[28XLoading the library and packages ...[128X[104X
    [4X[28XComponents: trans 1.0, prim 2.1, small* 1.0, id* 1.0[128X[104X
    [4X[28XPackages:   AClib 1.2, Alnuth 3.0.0, AtlasRep 1.5.0, AutPGrp 1.6,[128X[104X
    [4X[28X            Browse 1.8.6, CRISP 1.3.8, Cryst 4.1.12, CrystCat 1.1.6,[128X[104X
    [4X[28X            CTblLib 1.2.2, FactInt 1.5.3, FGA 1.2.0, GAPDoc 1.5.1, IO 4.4.4,[128X[104X
    [4X[28X            IRREDSOL 1.2.4, LAGUNA 3.7.0, Polenta 1.3.2, Polycyclic 2.11,[128X[104X
    [4X[28X            RadiRoot 2.7, ResClasses 3.4.0, Sophus 1.23, SpinSym 1.5,[128X[104X
    [4X[28X            TomLib 1.2.5[128X[104X
    [4X[28XTry '?help' for help. See also  '?copyright' and  '?authors'[128X[104X
    [4X[28Xgap>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo load [5XFinInG[105X, issue[133X
  
  
        [33X[1;6Y[24X[33X[0;0YLoadPackage("fining");[133X [124X[133X
  
  
  [33X[0;0YIf  this  fails,  in most cases, the reason is that either GAP does not find
  the  directory in which [5XFinInG[105X is installed, or one of the required packages
  for  [5XFinInG[105X  is  not installed or not compiled. Make sure (at least) [5XIO[105X, [5Xorb[105X
  and [5Xcvec[105X have been compiled before trying to load [5XFinInG[105X. The easiest way to
  find out what goes wrong is to load each required package before issuing the
  LoadPackage  command to load [5XFinInG[105X. The example below shows this situation,
  the packages [5XForms[105X and [5XFinInG[105X itself are installed in a local pkg directory,
  the  other packages are installed centrally. Starting up gap not pointing to
  the  local  pkg  directory,  causes  the  locally  installed  packages to be
  unloadable.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage("fining");[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage("forms");[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage("grape");[127X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  GRAPE 4.6.1 (GRaph Algorithms using PErmutation groups)[128X[104X
    [4X[28Xby Leonard H. Soicher (http://www.maths.qmul.ac.uk/~leonard/).[128X[104X
    [4X[28XHomepage: http://www.maths.qmul.ac.uk/~leonard/grape/[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage("orb");[127X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  orb 4.7.3 (Methods to enumerate orbits)[128X[104X
    [4X[28Xby Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller),[128X[104X
    [4X[28X   Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and[128X[104X
    [4X[28X   Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske).[128X[104X
    [4X[28XHomepage: http://gap-system.github.io/orb/[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage("genss");[127X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  genss 1.6.2 (Generic Schreier-Sims)[128X[104X
    [4X[28Xby Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef) and[128X[104X
    [4X[28X   Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske).[128X[104X
    [4X[28XHomepage: http://gap-system.github.io/genss/[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  the  installation  was  successful, then, the following output should be
  visible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage("fining"); [127X[104X
    [4X[28X---------------------------------------------------------------------[128X[104X
    [4X[28XLoading 'Forms' 1.2.3 (26/10/2015)[128X[104X
    [4X[28Xby John Bamberg (http://school.maths.uwa.edu.au/~bamberg/)[128X[104X
    [4X[28X   Jan De Beule (http://www.debeule.eu)[128X[104X
    [4X[28XFor help, type: ?Forms [128X[104X
    [4X[28X---------------------------------------------------------------------[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  orb 4.7.3 (Methods to enumerate orbits)[128X[104X
    [4X[28Xby Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller),[128X[104X
    [4X[28X   Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and[128X[104X
    [4X[28X   Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske).[128X[104X
    [4X[28XHomepage: http://gap-system.github.io/orb/[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  cvec 2.5.3 (Compact vectors over finite fields)[128X[104X
    [4X[28Xby Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef).[128X[104X
    [4X[28XHomepage: http://gap-system.github.io/cvec/[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  genss 1.6.2 (Generic Schreier-Sims)[128X[104X
    [4X[28Xby Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef) and[128X[104X
    [4X[28X   Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske).[128X[104X
    [4X[28XHomepage: http://gap-system.github.io/genss/[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  GRAPE 4.6.1 (GRaph Algorithms using PErmutation groups)[128X[104X
    [4X[28Xby Leonard H. Soicher (http://www.maths.qmul.ac.uk/~leonard/).[128X[104X
    [4X[28XHomepage: http://www.maths.qmul.ac.uk/~leonard/grape/[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28Xloading: geometry, liegeometry, group, projectivespace, correlations, polarspace[128X[104X
    [4X[28X/morphisms, enumerators, diagram, varieties, affinespace/affinegroup, gpolygons,[128X[104X
    [4X[28X orbits+stabilisers[128X[104X
    [4X[28X-------------------------------------------------------------------------------[128X[104X
    [4X[28X         ______________       ________      _________   ______________         [128X[104X
    [4X[28X         ___  ____/__(_)__________  _/________  ____/   __<  /__|__  /[128X[104X
    [4X[28X         __  /_   __  /__  __ \__  / __  __ \  / __     __  / ___/_ <[128X[104X
    [4X[28X         _  __/   _  / _  / / /_/ /  _  / / / /_/ /     _  /______/ /          [128X[104X
    [4X[28X         /_/      /_/  /_/ /_//___/  /_/ /_/\____/      /_/_(_)____/           [128X[104X
    [4X[28X-------------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  FinInG 1.3 (Finite Incidence Geometry) [128X[104X
    [4X[28Xby John Bamberg (http://school.maths.uwa.edu.au/~bamberg/)[128X[104X
    [4X[28X   Anton Betten (http://www.math.colostate.edu/~betten)[128X[104X
    [4X[28X   Jan De Beule (http://www.debeule.eu)[128X[104X
    [4X[28X   Philippe Cara (http://homepages.vub.ac.be/~pcara)[128X[104X
    [4X[28X   Michel Lavrauw (http://cage.ugent.be/~ml)[128X[104X
    [4X[28X   Max Neunhoeffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef/)[128X[104X
    [4X[28XFor help, type: ?FinInG [128X[104X
    [4X[28X---------------------------------------------------------------------[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.5 [33X[0;0YThe Development Team[133X[101X
  
  [33X[0;0YThis  is  the  development  team  (without Anton), meeting in St. Andrews in
  september  2008,  from  left  to  right:  Philippe Cara, Michel Lavrauw, Max
  Neunhöffer, Jan De Beule and John Bamberg[133X
  
  [33X[0;0YThe  development  team meeting again (without Anton and Max), now in Vicenza
  in  april  2011.  from left to right: Michel Lavrauw, John Bamberg, Philippe
  Cara, Jan De Beule.[133X
  
  [33X[0;0YSurvivors  of  the  first  version  of  [5XFinInG[105X, enjoying a trip to Chioggia,
  december 2011.[133X
  
  [33X[0;0YThe same survivors, staring at the destiny.[133X
  
  [33X[0;0YAnton Betten, during a milestone meeting at the finite geometries conference
  in Irsee, september 2014.[133X
  
