  
  
                              [1XGAP 4 Package [5XFinInG[105X[101X
  
  
                           [1XFinite Incidence Geometry[101X
  
  
                                     1.5.4
  
  
                                   13/12/2022
  
  
                                  John Bamberg
  
                                  Anton Betten
  
                                  Jan De Beule
  
                                 Philippe Cara
  
                                 Michel Lavrauw
  
                                 Max Neunhöffer
  
  
  
  
  
      Email:    [7Xmailto:support@fining.org[107X
      Homepage: [7Xhttp://www.fining.org[107X
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2014-2022 by the authors[133X
  
  [33X[0;0YThis  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2 or higher.[133X
  
  [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://www.fining.org[107X  when  you  use  [5XFinInG[105X,  or  to tell us by sending an
  e-mail to [7Xmailto:council@fining.org[107X.[133X
  
  [33X[0;0YPlease also 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. We provide the necessary BibTex and
  LaTeX code in Section [14X1.2[114X.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThe development phase of [5XFinInG[105X started roughly in 2005. The idea to write a
  package  for  projective geometry in GAP had emerged before, and resulted in
  [5Xpg[105X,  a  relic that still can be found in the undeposited packages section of
  the  GAP  website. One of the main objectives was to develop the new package
  was  to  create  a  tighter connection between finite geometry and the group
  theoretical functions present in GAP.[133X
  
  [33X[0;0YThe  authors  thank  Michael  Pauley, Maska Law and Sven Reichard, for their
  contributions during the early days of the project.[133X
  
  [33X[0;0YJan  De  Beule  and  Michel  Lavrauw  have  been  supported  by the Research
  Foundation Flanders -- Belgium (FWO), and John Bamberg has been supported by
  a  Marie  Curie  grant  and an ARC grant during almost the whole development
  phase of [5XFinInG[105X. The authors are grateful for this support.[133X
  
  [33X[0;0YJohn  Bamberg,  Philippe  Cara  and Jan De Beule have spent several weeks in
  Vicenza  while  developing  [5XFinInG[105X. We acknowledge the hospitality of Michel
  Lavrauw  and  Corrado Zanella. Our stays in Vicenza were always fruitful and
  very  enjoyable.  During  or  daily  morning and afternoon coffee breaks, we
  discussed several topics, while enjoying coffee in [13XCaffe Pigaffeta[113X. Although
  one  cannot  acknowledge  everybody, such as the bakery who provided us with
  the sandwiches, we acknowledge very much the hospitality of Carla and Luigi,
  who  introduced  us  to  many  different  kinds of coffees from all over the
  world.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (FinInG)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YPhilosophy[133X
    1.2 [33X[0;0YHow to cite [5XFinInG[105X[133X
    1.3 [33X[0;0YOverview of this manual[133X
    1.4 [33X[0;0YGetting and installing [5XFinInG[105X[133X
      1.4-1 [33X[0;0YInstallation procedure under UNIX like systems[133X
      1.4-2 [33X[0;0YCompiling packages[133X
      1.4-3 [33X[0;0YUpdating [5XFinInG[105X[133X
    1.5 [33X[0;0YThe Development Team[133X
  2 [33X[0;0YExamples[133X
    2.1 [33X[0;0YElementary examples[133X
      2.1-1 [33X[0;0Ysubspaces of projective spaces[133X
      2.1-2 [33X[0;0YSubspaces of classical polar spaces[133X
      2.1-3 [33X[0;0YUnderlying objects[133X
      2.1-4 [33X[0;0YConstructing polar spaces[133X
      2.1-5 [33X[0;0YSome collineation groups[133X
    2.2 [33X[0;0YSome objects with interesting combinatorial properties[133X
      2.2-1 [33X[0;0YThe Tits ovoid[133X
      2.2-2 [33X[0;0YLines meeting a hermitian curve[133X
      2.2-3 [33X[0;0YThe Patterson ovoid[133X
      2.2-4 [33X[0;0YA hyperoval[133X
    2.3 [33X[0;0YGeometry morphisms[133X
      2.3-1 [33X[0;0YIsomorphic polar spaces[133X
      2.3-2 [33X[0;0YIntertwiners[133X
      2.3-3 [33X[0;0YKlein correspondence[133X
      2.3-4 [33X[0;0YEmbedding in a subspace[133X
      2.3-5 [33X[0;0YSubgeometries[133X
      2.3-6 [33X[0;0YEmbedding by field reduction[133X
    2.4 [33X[0;0YSome geometrical objects[133X
      2.4-1 [33X[0;0YSpreads of [22XW(5,3)[122X[133X
      2.4-2 [33X[0;0YDistance-6 spread of the split Cayley hexagon[133X
    2.5 [33X[0;0YSome particular incidence geometries[133X
      2.5-1 [33X[0;0YThe split Cayley hexagon[133X
      2.5-2 [33X[0;0YAn (apartment of) a building of type [22XE_6[122X[133X
      2.5-3 [33X[0;0YA rank 4 geometry for [22XPSL(2,11)[122X[133X
      2.5-4 [33X[0;0YThe Ree-Tits octagon of order [22X[2,4][122X as coset geometry[133X
    2.6 [33X[0;0YElation generalised quadrangles[133X
      2.6-1 [33X[0;0YThe classical q-clan[133X
      2.6-2 [33X[0;0YTwo ways to construct a flock generalised quadrangle from a
      Kantor-Knuth semifield q-clan[133X
    2.7 [33X[0;0YAlgebraic varieties[133X
      2.7-1 [33X[0;0YA projective variety[133X
  3 [33X[0;0YIncidence Geometry[133X
    3.1 [33X[0;0YIncidence structures[133X
      3.1-1 IsIncidenceStructure
      3.1-2 IsIncidenceGeometry
      3.1-3 IncidenceStructure
      3.1-4 [33X[0;0YMain categories in [10XIsIncidenceGeometry[110X[133X
      3.1-5 [33X[0;0YExamples of categories of incidence geometries[133X
      3.1-6 TypesOfElementsOfIncidenceStructure
      3.1-7 Rank
      3.1-8 IncidenceGraph
    3.2 [33X[0;0YElements of incidence structures[133X
      3.2-1 [33X[0;0YMain categories for individual elements of incidence structures[133X
      3.2-2 UnderlyingObject
      3.2-3 Type
      3.2-4 ObjectToElement
      3.2-5 [33X[0;0YMain categories for collections of all the elements of a given
      type of an incidence structure[133X
      3.2-6 ElementsOfIncidenceStructure
      3.2-7 ElementsOfIncidenceStructure
      3.2-8 [33X[0;0YShort names for ElementsOfIncidenceStructure[133X
      3.2-9 NrElementsOfIncidenceStructure
      3.2-10 Random
      3.2-11 IsIncident
      3.2-12 AmbientGeometry
    3.3 [33X[0;0YFlags of incidence structures[133X
      3.3-1 FlagOfIncidenceStructure
      3.3-2 IsChamberOfIncidenceStructure
      3.3-3 IsEmptyFlag
      3.3-4 ElementsOfFlag
      3.3-5 Rank
      3.3-6 Size
      3.3-7 AmbientGeometry
      3.3-8 Type
      3.3-9 IsIncident
    3.4 [33X[0;0YShadow of elements[133X
      3.4-1 ShadowOfElement
      3.4-2 ElementsIncidentWithElementOfIncidenceStructure
      3.4-3 ShadowOfFlag
      3.4-4 ResidueOfFlag
      3.4-5 [33X[0;0YShort names for ElementsIncidentWithElementOfIncidenceStructure[133X
    3.5 [33X[0;0YEnumerating elements of an incidence structure[133X
      3.5-1 Iterator
      3.5-2 Enumerator
      3.5-3 List
      3.5-4 AsList
    3.6 [33X[0;0YLie geometries[133X
      3.6-1 [33X[0;0YMain categories in [10XIsLieGeometry[110X[133X
      3.6-2 AmbientSpace
      3.6-3 UnderlyingVectorSpace
      3.6-4 ProjectiveDimension
      3.6-5 IsEmptySubspace
    3.7 [33X[0;0YElements of Lie geometries[133X
      3.7-1 VectorSpaceToElement
      3.7-2 UnderlyingObject
      3.7-3 \in
      3.7-4 [33X[0;0YMore short names for
      [11XElementsIncidentWithElementOfIncidenceStructure[111X[133X
    3.8 [33X[0;0YChanging the ambient geometry of elements of a Lie geometry[133X
      3.8-1 ElementToElement
  4 [33X[0;0YProjective Spaces[133X
    4.1 [33X[0;0YProjective Spaces and basic operations[133X
      4.1-1 IsProjectiveSpace
      4.1-2 ProjectiveSpace
      4.1-3 ProjectiveDimension
      4.1-4 BaseField
      4.1-5 UnderlyingVectorSpace
      4.1-6 AmbientSpace
    4.2 [33X[0;0YSubspaces of projective spaces[133X
      4.2-1 VectorSpaceToElement
      4.2-2 EmptySubspace
      4.2-3 ProjectiveDimension
      4.2-4 ElementsOfIncidenceStructure
      4.2-5 [33X[0;0YShort names for ElementsOfIncidenceStructure[133X
      4.2-6 [33X[0;0YIncidence and containment[133X
      4.2-7 StandardFrame
      4.2-8 Coordinates
      4.2-9 DualCoordinatesOfHyperplane
      4.2-10 HyperplaneByDualCoordinates
      4.2-11 EquationOfHyperplane
      4.2-12 AmbientSpace
      4.2-13 BaseField
      4.2-14 Random
      4.2-15 RandomSubspace
      4.2-16 Span
      4.2-17 Meet
      4.2-18 FlagOfIncidenceStructure
      4.2-19 IsEmptyFlag
      4.2-20 IsChamberOfIncidenceStructure
    4.3 [33X[0;0YShadows of Projective Subspaces[133X
      4.3-1 ShadowOfElement
      4.3-2 ShadowOfFlag
      4.3-3 ElementsIncidentWithElementOfIncidenceStructure
      4.3-4 [33X[0;0YShort names for [11XElementsIncidentWithElementOfIncidenceStructure[111X[133X
    4.4 [33X[0;0YEnumerating subspaces of a projective space[133X
      4.4-1 Iterator
      4.4-2 Enumerator
      4.4-3 List
  5 [33X[0;0YProjective Groups[133X
    5.1 [33X[0;0YProjectivities, collineations and correlations of projective spaces.[133X
      5.1-1 [33X[0;0YCategories for group elements[133X
      5.1-2 [33X[0;0YRepresentations for group elements[133X
      5.1-3 [33X[0;0YProjectivities[133X
      5.1-4 [33X[0;0YCollineations of projective spaces[133X
      5.1-5 [33X[0;0YProjective strictly semilinear maps[133X
      5.1-6 [33X[0;0YCorrelations and collineations[133X
    5.2 [33X[0;0YConstruction of projectivities, collineations and correlations.[133X
      5.2-1 Projectivity
      5.2-2 CollineationOfProjectiveSpace
      5.2-3 ProjectiveSemilinearMap
      5.2-4 IdentityMappingOfElementsOfProjectiveSpace
      5.2-5 StandardDualityOfProjectiveSpace
      5.2-6 CorrelationOfProjectiveSpace
    5.3 [33X[0;0YBasic operations for projectivities, collineations and correlations of
    projective spaces[133X
      5.3-1 Representative
      5.3-2 MatrixOfCollineation
      5.3-3 MatrixOfCorrelation
      5.3-4 BaseField
      5.3-5 FieldAutomorphism
      5.3-6 ProjectiveSpaceIsomorphism
      5.3-7 Order
    5.4 [33X[0;0YThe groups [22XPΓL[122X, [22XPGL[122X, and [22XPSL[122X in [5XFinInG[105X[133X
      5.4-1 ProjectivityGroup
      5.4-2 CollineationGroup
      5.4-3 SpecialProjectivityGroup
      5.4-4 IsProjectivityGroup
      5.4-5 IsCollineationGroup
      5.4-6 CorrelationCollineationGroup
    5.5 [33X[0;0YBasic operations for projective groups[133X
      5.5-1 BaseField
      5.5-2 Dimension
    5.6 [33X[0;0YNatural embedding of a collineation group in a
    correlation/collineation group[133X
      5.6-1 Embedding
    5.7 [33X[0;0YBasic action of projective group elements[133X
      5.7-1 \^
    5.8 [33X[0;0YProjective group actions[133X
      5.8-1 OnProjSubspaces
      5.8-2 ActionOnAllProjPoints
      5.8-3 OnProjSubspacesExtended
    5.9 [33X[0;0YSpecial subgroups of the projectivity group[133X
      5.9-1 ElationOfProjectiveSpace
      5.9-2 ProjectiveElationGroup
      5.9-3 HomologyOfProjectiveSpace
      5.9-4 ProjectiveHomologyGroup
    5.10 [33X[0;0YNice Monomorphisms[133X
      5.10-1 NiceMonomorphism
      5.10-2 NiceObject
      5.10-3 FINING
      5.10-4 CanComputeActionOnPoints
      5.10-5 NiceMonomorphismByDomain
      5.10-6 NiceMonomorphismByOrbit
  6 [33X[0;0YPolarities of Projective Spaces[133X
    6.1 [33X[0;0YCreating polarities of projective spaces[133X
      6.1-1 PolarityOfProjectiveSpace
      6.1-2 PolarityOfProjectiveSpace
      6.1-3 PolarityOfProjectiveSpace
      6.1-4 PolarityOfProjectiveSpace
    6.2 [33X[0;0YOperations, attributes and properties for polarities of projective
    spaces[133X
      6.2-1 SesquilinearForm
      6.2-2 BaseField
      6.2-3 GramMatrix
      6.2-4 CompanionAutomorphism
      6.2-5 IsHermitianPolarityOfProjectiveSpace
      6.2-6 IsSymplecticPolarityOfProjectiveSpace
      6.2-7 IsOrthogonalPolarityOfProjectiveSpace
      6.2-8 IsPseudoPolarityOfProjectiveSpace
    6.3 [33X[0;0YPolarities, absolute points, totally isotropic elements and finite
    classical polar spaces[133X
      6.3-1 GeometryOfAbsolutePoints
      6.3-2 AbsolutePoints
      6.3-3 PolarSpace
    6.4 [33X[0;0YCommuting polarities[133X
  7 [33X[0;0YFinite Classical Polar Spaces[133X
    7.1 [33X[0;0YFinite Classical Polar Spaces[133X
      7.1-1 IsClassicalPolarSpace
      7.1-2 PolarSpace
    7.2 [33X[0;0YCanonical and standard Polar Spaces[133X
      7.2-1 SymplecticSpace
      7.2-2 HermitianPolarSpace
      7.2-3 ParabolicQuadric
      7.2-4 HyperbolicQuadric
      7.2-5 EllipticQuadric
      7.2-6 IsCanonicalPolarSpace
      7.2-7 CanonicalPolarSpace
      7.2-8 StandardPolarSpace
    7.3 [33X[0;0YBasic operations for finite classical polar spaces[133X
      7.3-1 UnderlyingVectorSpace
      7.3-2 AmbientSpace
      7.3-3 ProjectiveDimension
      7.3-4 Rank
      7.3-5 BaseField
      7.3-6 IsHyperbolicQuadric
      7.3-7 IsEllipticQuadric
      7.3-8 IsParabolicQuadric
    7.4 [33X[0;0YSubspaces of finite classical polar spaces[133X
      7.4-1 VectorSpaceToElement
      7.4-2 EmptySubspace
      7.4-3 ProjectiveDimension
      7.4-4 ElementsOfIncidenceStructure
      7.4-5 AmbientSpace
      7.4-6 Coordinates
    7.5 [33X[0;0YBasic operations for polar spaces and subspaces of projective spaces[133X
      7.5-1 [33X[0;0YIncidence and containment[133X
      7.5-2 Span
      7.5-3 Meet
      7.5-4 IsCollinear
      7.5-5 PolarityOfProjectiveSpace
      7.5-6 TypeOfSubspace
      7.5-7 TangentSpace
      7.5-8 Pole
      7.5-9 EvaluateForm
    7.6 [33X[0;0YShadow of elements[133X
      7.6-1 ShadowOfElement
      7.6-2 ElementsIncidentWithElementOfIncidenceStructure
    7.7 [33X[0;0YProjective Orthogonal/Unitary/Symplectic groups in [5XFinInG[105X[133X
      7.7-1 SpecialIsometryGroup
      7.7-2 IsometryGroup
      7.7-3 SimilarityGroup
      7.7-4 CollineationGroup
    7.8 [33X[0;0YEnumerating subspaces of polar spaces[133X
      7.8-1 [33X[0;0YEnumerators for polar spaces[133X
      7.8-2 Enumerator
      7.8-3 [33X[0;0YIterators for polar spaces[133X
      7.8-4 Iterator
      7.8-5 AsList
  8 [33X[0;0YOrbits, stabilisers and actions[133X
    8.1 [33X[0;0YOrbits[133X
      8.1-1 FiningOrbit
      8.1-2 FiningOrbits
      8.1-3 FiningOrbitsDomain
    8.2 [33X[0;0YStabilisers[133X
      8.2-1 FiningStabiliser
      8.2-2 FiningStabiliserOrb
      8.2-3 FiningSetwiseStabiliser
      8.2-4 StabiliserGroupOfSubspace
      8.2-5 ProjectiveStabiliserGroupOfSubspace
      8.2-6 SpecialProjectiveStabiliserGroupOfSubspace
    8.3 [33X[0;0YActions and nice monomorphisms revisited[133X
      8.3-1 [33X[0;0YAction functions[133X
      8.3-2 [33X[0;0YGeneric GAP functions[133X
      8.3-3 NiceMonomorphism
      8.3-4 NiceObject
      8.3-5 [33X[0;0YDifferent behaviour for different collineation groups[133X
      8.3-6 SetParent
  9 [33X[0;0YAffine Spaces[133X
    9.1 [33X[0;0YAffine spaces and basic operations[133X
      9.1-1 IsAffineSpace
      9.1-2 AffineSpace
      9.1-3 Dimension
      9.1-4 BaseField
      9.1-5 UnderlyingVectorSpace
      9.1-6 AmbientSpace
    9.2 [33X[0;0YSubspaces of affine spaces[133X
      9.2-1 AffineSubspace
      9.2-2 ElementsOfIncidenceStructure
      9.2-3 [33X[0;0YShort names for ElementsOfIncidenceStructure[133X
      9.2-4 [33X[0;0YIncidence and containment[133X
      9.2-5 AmbientSpace
      9.2-6 BaseField
      9.2-7 Span
      9.2-8 Meet
      9.2-9 IsParallel
      9.2-10 ParallelClass
    9.3 [33X[0;0YShadows of Affine Subspaces[133X
      9.3-1 ShadowOfElement
      9.3-2 ShadowOfFlag
    9.4 [33X[0;0YIterators and enumerators[133X
      9.4-1 Iterator
      9.4-2 Enumerator
    9.5 [33X[0;0YAffine groups[133X
      9.5-1 AffineGroup
      9.5-2 CollineationGroup
      9.5-3 OnAffineSpaces
    9.6 [33X[0;0YLow level operations[133X
      9.6-1 IsVectorSpaceTransversal
      9.6-2 VectorSpaceTransversal
      9.6-3 VectorSpaceTransversalElement
      9.6-4 ComplementSpace
  10 [33X[0;0YGeometry Morphisms[133X
    10.1 [33X[0;0YGeometry morphisms in FinInG[133X
      10.1-1 IsGeometryMorphism
      10.1-2 Intertwiner
    10.2 [33X[0;0YType preserving bijective geometry morphisms[133X
      10.2-1 IsomorphismPolarSpaces
    10.3 [33X[0;0YKlein correspondence and derived dualities[133X
      10.3-1 PluckerCoordinates
      10.3-2 KleinCorrespondence
      10.3-3 KleinCorrespondence
      10.3-4 KleinCorrespondenceExtended
      10.3-5 NaturalDuality
      10.3-6 SelfDuality
    10.4 [33X[0;0YEmbeddings of projective spaces[133X
      10.4-1 NaturalEmbeddingBySubspace
      10.4-2 NaturalEmbeddingBySubField
      10.4-3 [33X[0;0YEmbedding of projective spaces by field reduction[133X
      10.4-4 BlownUpSubspaceOfProjectiveSpace
      10.4-5 NaturalEmbeddingByFieldReduction
    10.5 [33X[0;0YEmbeddings of polar spaces[133X
      10.5-1 NaturalEmbeddingBySubspace
      10.5-2 NaturalEmbeddingBySubField
      10.5-3 [33X[0;0YEmbedding of polar spaces by field reduction[133X
      10.5-4 NaturalEmbeddingByFieldReduction
      10.5-5 NaturalEmbeddingByFieldReduction
    10.6 [33X[0;0YProjections[133X
      10.6-1 NaturalProjectionBySubspace
    10.7 [33X[0;0YProjective completion[133X
      10.7-1 ProjectiveCompletion
  11 [33X[0;0YAlgebraic Varieties[133X
    11.1 [33X[0;0YAlgebraic Varieties[133X
      11.1-1 AlgebraicVariety
      11.1-2 DefiningListOfPolynomials
      11.1-3 AmbientSpace
      11.1-4 PointsOfAlgebraicVariety
      11.1-5 Iterator
      11.1-6 \in
    11.2 [33X[0;0YProjective Varieties[133X
      11.2-1 ProjectiveVariety
    11.3 [33X[0;0YQuadrics and Hermitian varieties[133X
      11.3-1 HermitianVariety
      11.3-2 QuadraticVariety
      11.3-3 QuadraticForm
      11.3-4 SesquilinearForm
      11.3-5 PolarSpace
    11.4 [33X[0;0YAffine Varieties[133X
      11.4-1 AffineVariety
    11.5 [33X[0;0YGeometry maps[133X
      11.5-1 Source
      11.5-2 Range
      11.5-3 ImageElm
      11.5-4 ImagesSet
      11.5-5 \^
    11.6 [33X[0;0YSegre Varieties[133X
      11.6-1 SegreVariety
      11.6-2 PointsOfSegreVariety
      11.6-3 SegreMap
      11.6-4 Source
    11.7 [33X[0;0YVeronese Varieties[133X
      11.7-1 VeroneseVariety
      11.7-2 PointsOfVeroneseVariety
      11.7-3 VeroneseMap
      11.7-4 Source
    11.8 [33X[0;0YGrassmann Varieties[133X
      11.8-1 GrassmannVariety
      11.8-2 PointsOfGrassmannVariety
      11.8-3 GrassmannMap
      11.8-4 Source
  12 [33X[0;0YGeneralised Polygons[133X
    12.1 [33X[0;0YCategories[133X
      12.1-1 IsGeneralisedPolygon
      12.1-2 [33X[0;0YSubcategories in [10XIsGeneralisedPolygon[110X[133X
      12.1-3 IsWeakGeneralisedPolygon
      12.1-4 [33X[0;0YSubcategories in [10XIsProjectivePlaneCategory[110X[133X
      12.1-5 [33X[0;0YSubcategories in [10XIsGeneralisedQuadrangle[110X[133X
      12.1-6 IsClassicalGeneralisedHexagon
    12.2 [33X[0;0YGeneric functions to create generalised polygons[133X
      12.2-1 GeneralisedPolygonByBlocks
      12.2-2 GeneralisedPolygonByIncidenceMatrix
      12.2-3 GeneralisedPolygonByElements
    12.3 [33X[0;0YAttributes and operations for generalised polygons[133X
      12.3-1 Order
      12.3-2 IncidenceGraphAttr
      12.3-3 IncidenceGraph
      12.3-4 IncidenceMatrixOfGeneralisedPolygon
      12.3-5 CollineationGroup
      12.3-6 CollineationAction
      12.3-7 BlockDesignOfGeneralisedPolygon
    12.4 [33X[0;0YElements of generalised polygons[133X
      12.4-1 [33X[0;0YCollections of elements of generalised polygons[133X
      12.4-2 Size
      12.4-3 [33X[0;0YCreating elements from objects and retrieving objects from
      elements[133X
      12.4-4 [33X[0;0YIncidence[133X
      12.4-5 Span
      12.4-6 Meet
      12.4-7 [33X[0;0YShadow elements[133X
      12.4-8 DistanceBetweenElements
    12.5 [33X[0;0YThe classical generalised hexagons[133X
      12.5-1 [33X[0;0YTrialities of the hyperbolic quadric and generalised hexagons[133X
      12.5-2 IsLieGeometry
      12.5-3 SplitCayleyHexagon
      12.5-4 TwistedTrialityHexagon
      12.5-5 vector spaceToElement
      12.5-6 ObjectToElement
      12.5-7 UnderlyingObject
      12.5-8 \in
      12.5-9 [33X[0;0YSpan and meet of elements[133X
      12.5-10 CollineationGroup
    12.6 [33X[0;0YElation generalised quadrangles[133X
      12.6-1 [33X[0;0YElation generalised quadrangles and Kantor families[133X
      12.6-2 [33X[0;0YCategories[133X
      12.6-3 [33X[0;0YKantor families[133X
      12.6-4 EGQByKantorFamily
      12.6-5 [33X[0;0YRepresentation of elements and underlying objects[133X
      12.6-6 [33X[0;0YElation group and natural action on elements[133X
      12.6-7 [33X[0;0YKantor families, q-clans, and elation generalised quadrangles[133X
      12.6-8 qClan
      12.6-9 [33X[0;0YParticular q-clans[133X
      12.6-10 KantorFamilyByqClan
      12.6-11 EGQByqClan
      12.6-12 [33X[0;0YBLT-sets, flocks, q-clans, and elation generalised quadrangles[133X
      12.6-13 IsEGQByBLTSet
      12.6-14 BLTSetByqClan
      12.6-15 EGQByBLTSet
      12.6-16 DefiningPlanesOfEGQByBLTSet
      12.6-17 [33X[0;0YRepresentation of elements and underlying objects[133X
      12.6-18 CollineationSubgroup
  13 [33X[0;0YCoset Geometries and Diagrams[133X
    13.1 [33X[0;0YCoset Geometries[133X
      13.1-1 IsCosetGeometry
      13.1-2 CosetGeometry
      13.1-3 IsIncident
      13.1-4 ParabolicSubgroups
      13.1-5 AmbientGroup
      13.1-6 Borelsubgroup
      13.1-7 RandomElement
      13.1-8 RandomFlag
      13.1-9 RandomChamber
      13.1-10 IsFlagTransitiveGeometry
      13.1-11 OnCosetGeometryElement
      13.1-12 \^
      13.1-13 \^
      13.1-14 IsFirmGeometry
      13.1-15 IsThickGeometry
      13.1-16 IsThinGeometry
      13.1-17 IsConnected
      13.1-18 IsResiduallyConnected
      13.1-19 StandardFlagOfCosetGeometry
      13.1-20 FlagToStandardFlag
      13.1-21 CanonicalResidueOfFlag
      13.1-22 ResidueOfFlag
      13.1-23 IncidenceGraph
      13.1-24 Rk2GeoGonality
      13.1-25 Rk2GeoDiameter
      13.1-26 GeometryOfRank2Residue
      13.1-27 Rank2Parameters
    13.2 [33X[0;0YAutomorphisms, Correlations and Isomorphisms[133X
      13.2-1 AutGroupIncidenceStructureWithNauty
      13.2-2 CorGroupIncidenceStructureWithNauty
      13.2-3 IsIsomorphicIncidenceStructureWithNauty
    13.3 [33X[0;0YDiagrams[133X
      13.3-1 DiagramOfGeometry
      13.3-2 GeometryOfDiagram
      13.3-3 DrawDiagram
      13.3-4 DrawDiagramWithNeato
  14 [33X[0;0YSubgeometries of projective spaces[133X
    14.1 [33X[0;0YParticular Categories[133X
      14.1-1 IsSubgeometryOfProjectiveSpace
      14.1-2 [33X[0;0YCategories for elements and collections of elements[133X
    14.2 [33X[0;0YSubgeometries of projective spaces[133X
      14.2-1 CanonicalSubgeometryOfProjectiveSpace
      14.2-2 RandomFrameOfProjectiveSpace
      14.2-3 IsFrameOfProjectiveSpace
      14.2-4 SubgeometryOfProjectiveSpaceByFrame
    14.3 [33X[0;0YBasic operations[133X
      14.3-1 [33X[0;0YUnderlying vector space and ambient projective space[133X
      14.3-2 DefiningFrameOfSubgeometry
      14.3-3 [33X[0;0YProjective dimension and rank[133X
      14.3-4 [33X[0;0YUnderlying algebraic structures[133X
      14.3-5 CollineationFixingSubgeometry
    14.4 [33X[0;0YConstructing elements of a subgeometry[133X
      14.4-1 VectorSpaceToElement
      14.4-2 ExtendElementOfSubgeometry
      14.4-3 AmbientGeometry
      14.4-4 [33X[0;0YFlags[133X
    14.5 [33X[0;0YGroups and actions[133X
      14.5-1 [33X[0;0YGroups of collineations[133X
  A [33X[0;0YThe structure of [5XFinInG[105X[133X
    A.1 [33X[0;0YThe different components[133X
    A.2 [33X[0;0YThe complete inventory[133X
      A.2-1 [33X[0;0YDeclarations[133X
      A.2-2 [33X[0;0YFunctions/Methods[133X
  B [33X[0;0YThe finite classical groups in [5XFinInG[105X[133X
    B.1 [33X[0;0YStandard forms used to produce the finite classical groups.[133X
      B.1-1 CanonicalGramMatrix
      B.1-2 CanonicalQuadraticForm
    B.2 [33X[0;0YDirect commands to construct the projective classical groups in [5XFinInG[105X[133X
      B.2-1 SOdesargues
      B.2-2 GOdesargues
      B.2-3 SUdesargues
      B.2-4 GUdesargues
      B.2-5 Spdesargues
      B.2-6 GeneralSymplecticGroup
      B.2-7 GSpdesargues
      B.2-8 GammaSp
      B.2-9 DeltaOminus
      B.2-10 DeltaOplus
      B.2-11 GammaOminus
      B.2-12 GammaO
      B.2-13 GammaOplus
      B.2-14 GammaU
      B.2-15 G2fining
      B.2-16 3D4fining
    B.3 [33X[0;0YBasis of the collineation groups[133X
      B.3-1 FindBasePointCandidates
  C [33X[0;0YLow level functions for morphisms[133X
    C.1 [33X[0;0YField reduction and vector spaces[133X
      C.1-1 ShrinkVec
      C.1-2 ShrinkMat
      C.1-3 BlownUpProjectiveSpace
      C.1-4 BlownUpProjectiveSpaceBySubfield
      C.1-5 BlownUpSubspaceOfProjectiveSpace
      C.1-6 BlownUpSubspaceOfProjectiveSpaceBySubfield
      C.1-7 IsDesarguesianSpreadElement
    C.2 [33X[0;0YField reduction and forms[133X
      C.2-1 QuadraticFormFieldReduction
      C.2-2 BilinearFormFieldReduction
      C.2-3 HermitianFormFieldReduction
    C.3 [33X[0;0YLow level functions[133X
      C.3-1 PluckerCoordinates
      C.3-2 IsomorphismPolarSpacesProjectionFromNucleus
      C.3-3 IsomorphismPolarSpacesNC
      C.3-4 NaturalEmbeddingBySubspaceNC
      C.3-5 NaturalProjectionBySubspaceNC
  
  
  [32X
