  
  [1X14 [33X[0;0YSubgeometries of projective spaces[133X[101X
  
  [33X[0;0YLet  [22XS=(P,L,I)[122X be a point-line incidence geometry. In case [22XS[122X is a projective
  space  over  a finite field, it is clear that every line (and every subspace
  as  well)  can  be  identified  with  the  set  of  points incident with it.
  Furthermore,  the  incidence  relation [22XI[122X is then symmetrised containment. To
  define  a  subgeometry mathematically, we follow [Dem97]. Let [22XP' ⊂ P[122X and let
  [22XL'[122X be a set of subsets of [22XP'[122X, such that every [22Xl' ∈ L'[122X is a subset of exactly
  one line [22Xl ∈ L[122X. If [22XS'=(P',L',I)[122X is a projective space again, then we call [22XS'[122X
  a [13Xsubgeometry[113X of [22XS[122X. Note that in general the subspaces of [22XS'[122X will be subsets
  of subspaces of [22XS[122X.[133X
  
  [33X[0;0YA typical example of a subgeometry is a Baer subplane of a projective plane.
  In  this  example,  with [22XS'[122X the Baer subplane of the projective plane [22XS[122X, one
  could  say  that  a point of [22XS'[122X is indeed a point of [22XS[122X, but a line of [22XS'[122X, is
  not  a  line  of  [22XS[122X. If one considers a line of [22XS'[122X as a set of points of [22XS'[122X,
  then  a  line of [22XS'[122X is a subset of the set of points on a line of [22XS[122X. Another
  example is the subgeometry of a projective space induced by a subspace [22Xπ[122X. In
  this  example,  clearly, the set of elements of the induced subgeometry can,
  mathematically, be considered as a subset of the set of elements of [22XS[122X.[133X
  
  [33X[0;0YThe   same   considerations   apply   for   classical  polar  spaces.  These
  consideration  have  implications for the behaviour of certain operations in
  [5XFinInG[105X, e.g. when computing the span and meet of different elements.[133X
  
  [33X[0;0YUsing   geometry   morphisms,   and   more   particular   a   function  like
  [11XNaturalEmbeddingBySubField[111X,   one   can   deal   in  an  indirect  way  with
  subgeometries.  However,  using  [11XNaturalEmbeddingBySubField[111X is not flexible,
  and  typical problems such as considering a subgeometry determined by a user
  chosen  frame  and  a  subfield,  cannot be handled easily. Therefore [5XFinInG[105X
  provides  some  functions to naturally construct subgeometries of projective
  spaces.[133X
  
  [33X[0;0YA  subgeometry  in a projective space is completely determined by a frame of
  the  projective  space  and  a  subfield of the base field of the projective
  space.  The  [13Xstandard  frame[113X in an [22Xn[122X-dimensional projective space [22XPG(n,q)[122X is
  the        set        of        [22Xn+2[122X        points       represented       by
  [22X(1,0,...,0),(0,1,...,0),...,(0,0,...,1),(1,1,...,1)[122X.     The     subgeometry
  determined  by  the  standard  frame  will  be  called  [13Xcanonical[113X. Note that
  different frames may determine the same subgeometry (over a fixed subfield).[133X
  
  [33X[0;0YFor a given subfield [22XGF(q') ⊂ GF(q)[122X, the canonical subgeometry determined by
  the  standard  frame  in  [22XPG(n,q)[122X  is mathematically spoken the image of the
  [5XFinInG[105X  geometry morphism [11XNaturalEmbeddingBySubField[111X of the projective space
  [22XPG(n,q')[122X.  The  coordinates  of  the  points  of  the  subgeometry  will  be
  exclusively  over the subfield [22XGF(q')[122X, as are the coordinates of the vectors
  after  normalizing  defining  any  subspace of the subgeometry. Clearly, the
  Frobenius  automorphism  which  maps  [22Xx[122X  to  [22Xx^q'[122X  fixes all elements of the
  subgeometry.[133X
  
  [33X[0;0YFor  an  arbitrary  frame  of  [22XPG(n,q)[122X and a subfield [22XGF(q')[122X, there exists a
  natural  collineation of [22XPG(n,q)[122X which fixes the subgeometry pointwise. This
  collineation  is the conjugation of the Frobenius automorphism by the unique
  collineation  mapping  the defining frame of the subgeometry to the standard
  frame  of  [22XPG(n,q)[122X,  i.e. the frame defining the canonical subsgeometry over
  [22XGF(q')[122X.  Upon  construction  of  a  subgeometry,  both collineations will be
  computed,  and are of use when dealing with the full collineation group of a
  subgeometry.  As for any incidence geometry in [5XFinInG[105X, operations to compute
  this   collineation  group  as  well  as  particular  action  functions  for
  subgeometries are provided.[133X
  
  [33X[0;0YSubgeometries  of  projective  spaces  are  constructed  in a subcategory of
  [10XIsProjectiveSpace[110X,  as such, all operations applicable to projective spaces,
  are  naturally  applicable  to subgeometries. Subspaces of subgeometries are
  constructed   in   a   subcategory  of  [10XIsSubspaceOfProjectiveSpace[110X.  Hence,
  operations  applicable  to  subspaces  of  projective  spaces, are naturally
  applicable to subspaces of subgeometries.[133X
  
  
  [1X14.1 [33X[0;0YParticular Categories[133X[101X
  
  [1X14.1-1 IsSubgeometryOfProjectiveSpace[101X
  
  [33X[1;0Y[29X[2XIsSubgeometryOfProjectiveSpace[102X [32X Category[133X
  
  [33X[0;0YThis  category  is  a  subcategory  of  [10XIsProjectiveSpace[110X,  and contains all
  subgeometries  of projective spaces. Note that mathematically, a subspace of
  a projective space is also a subgeometry. However, in [5XFinInG[105X, subspaces of a
  projective  space are constructed in a category that is not a subcategory of
  [10XIsProjectiveSpace[110X.  Since [10XIsSubgeometryOfProjectiveSpace[110X is a subcategory of
  [10XIsProjectiveSpace[110X,  all  operations  applicable  to  projective  spaces, are
  naturally applicable to subgeometries of projective spaces.[133X
  
  
  [1X14.1-2 [33X[0;0YCategories for elements and collections of elements[133X[101X
  
  [33X[1;0Y[29X[2XIsSubspaceOfSubgeometryOfProjectiveSpace[102X [32X Category[133X
  [33X[1;0Y[29X[2XIsSubspacesOfSubgeometryOfProjectiveSpace[102X [32X Category[133X
  
  [33X[0;0YA     subspace    of    a    subgeometry    belongs    to    the    category
  [10XIsSubspaceOfSubgeometryOfProjectiveSpace[110X.[133X
  
  
  [1X14.2 [33X[0;0YSubgeometries of projective spaces[133X[101X
  
  [1X14.2-1 CanonicalSubgeometryOfProjectiveSpace[101X
  
  [33X[1;0Y[29X[2XCanonicalSubgeometryOfProjectiveSpace[102X( [3Xpg[103X, [3Xsubfield[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCanonicalSubgeometryOfProjectiveSpace[102X( [3Xpg[103X, [3Xq[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya subgeometry of [3Xpg[103X[133X
  
  [33X[0;0YThis  operation  returns the subgeometry of [3Xpg[103X induced by the standard frame
  over  the  subfield  [3Xsubfield[103X. Alternatively, a prime power [3Xq[103X can be used as
  the order of the subfield. It is checked whether the user specified subfield
  is  indeed  a  subfield  of the base field of [3Xpg[103X. If the subfield equals the
  base field of [3Xpg[103X, the projective space [3Xpg[103X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(2,25);[127X[104X
    [4X[28XProjectiveSpace(2, 25)[128X[104X
    [4X[25Xgap>[125X [27Xsub := CanonicalSubgeometryOfProjectiveSpace(pg,GF(5));[127X[104X
    [4X[28XSubgeometry PG(2, 5) of ProjectiveSpace(2, 25)[128X[104X
    [4X[25Xgap>[125X [27XCategoriesOfObject(sub);[127X[104X
    [4X[28X[ "IsIncidenceStructure", "IsIncidenceGeometry", "IsLieGeometry", [128X[104X
    [4X[28X  "IsProjectiveSpace", "IsSubgeometryOfProjectiveSpace" ][128X[104X
    [4X[25Xgap>[125X [27Xpg := PG(3,3^6);[127X[104X
    [4X[28XProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27Xsub := CanonicalSubgeometryOfProjectiveSpace(pg,3^2);[127X[104X
    [4X[28XSubgeometry PG(3, 9) of ProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27Xsub := CanonicalSubgeometryOfProjectiveSpace(pg,3^3);[127X[104X
    [4X[28XSubgeometry PG(3, 27) of ProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27Xsub := CanonicalSubgeometryOfProjectiveSpace(pg,3^6);[127X[104X
    [4X[28XProjectiveSpace(3, 729)[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X14.2-2 RandomFrameOfProjectiveSpace[101X
  
  [33X[1;0Y[29X[2XRandomFrameOfProjectiveSpace[102X( [3Xpg[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  set  of  points  of  [3Xpg[103X,  being a frame. Note that the returned
            object  is  also  a  set  in  the  GAP sense, i.e. an ordered list
            without duplicates.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(1,5^5);[127X[104X
    [4X[28XProjectiveSpace(1, 3125)[128X[104X
    [4X[25Xgap>[125X [27Xframe := RandomFrameOfProjectiveSpace(pg);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(1, 3125)>, <a point in ProjectiveSpace(1, 3125)>[128X[104X
    [4X[28X    , <a point in ProjectiveSpace(1, 3125)> ][128X[104X
    [4X[25Xgap>[125X [27XLength(frame);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27Xpg := PG(6,2^2);[127X[104X
    [4X[28XProjectiveSpace(6, 4)[128X[104X
    [4X[25Xgap>[125X [27Xframe := RandomFrameOfProjectiveSpace(pg);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(6, 4)>, <a point in ProjectiveSpace(6, 4)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(6, 4)>, <a point in ProjectiveSpace(6, 4)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(6, 4)>, <a point in ProjectiveSpace(6, 4)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(6, 4)>, <a point in ProjectiveSpace(6, 4)> ][128X[104X
    [4X[25Xgap>[125X [27XLength(frame);[127X[104X
    [4X[28X8[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X14.2-3 IsFrameOfProjectiveSpace[101X
  
  [33X[1;0Y[29X[2XIsFrameOfProjectiveSpace[102X( [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YWhen  [3Xlist[103X is a list of points of a projective space, this operation returns
  true  if and only if [3Xlist[103X constitutes a frame of the projective space. It is
  checked  as  well  whether  all points in [3Xlist[103X belong to the same projective
  space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(1,7^3);[127X[104X
    [4X[28XProjectiveSpace(1, 343)[128X[104X
    [4X[25Xgap>[125X [27Xp1 := VectorSpaceToElement(pg,[1,1]*Z(7)^0);[127X[104X
    [4X[28X<a point in ProjectiveSpace(1, 343)>[128X[104X
    [4X[25Xgap>[125X [27Xp2 := VectorSpaceToElement(pg,[1,2]*Z(7)^0);[127X[104X
    [4X[28X<a point in ProjectiveSpace(1, 343)>[128X[104X
    [4X[25Xgap>[125X [27Xp3 := VectorSpaceToElement(pg,[1,3]*Z(7)^0);[127X[104X
    [4X[28X<a point in ProjectiveSpace(1, 343)>[128X[104X
    [4X[25Xgap>[125X [27XIsFrameOfProjectiveSpace([p1,p2,p3]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X14.2-4 SubgeometryOfProjectiveSpaceByFrame[101X
  
  [33X[1;0Y[29X[2XSubgeometryOfProjectiveSpaceByFrame[102X( [3Xpg[103X, [3Xlist[103X, [3Xfield[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSubgeometryOfProjectiveSpaceByFrame[102X( [3Xpg[103X, [3Xlist[103X, [3Xq[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya subgeometry of [3Xpg[103X[133X
  
  [33X[0;0YThe argument [3Xpg[103X is a projective space which is not a subgeometry itself, the
  argument  [3Xlist[103X is a list of points of [3Xpg[103X defining a frame of [3Xpg[103X, and finally
  the argument [3Xfield[103X is a subfield of the base field of [3Xpg[103X. Alternatively, the
  argument  [3Xq[103X  is the order of a subfield of the base field of [3Xpg[103X. This method
  returns the subgeometry definied by the frame in [3Xlist[103X and the subfield [3Xfield[103X
  of  the subfield [22XGF([3Xq[103X)[122X. This method checks whether the subfield [3Xfield[103X or the
  field  [22XGF([3Xq[103X)[122X  is  really  a subfield of the base field of [3Xpg[103X and whether the
  list  of points in [3Xlist[103X is a frame of [3Xpg[103X. Note also that it is currently not
  possible  to  construct  subgeometries  recursively,  so  [3Xpg[103X  may  not  be a
  subgeometry  itself.  If the specified subfield equals the base field of [3Xpg[103X,
  then the projective space [3Xpg[103X itself is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(3,3^6);[127X[104X
    [4X[28XProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27Xframe := RandomFrameOfProjectiveSpace(pg);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(3, 729)>, <a point in ProjectiveSpace(3, 729)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(3, 729)>, <a point in ProjectiveSpace(3, 729)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(3, 729)> ][128X[104X
    [4X[25Xgap>[125X [27Xsub1 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,GF(3));[127X[104X
    [4X[28XSubgeometry PG(3, 3) of ProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27Xsub2 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,3^2);[127X[104X
    [4X[28XSubgeometry PG(3, 9) of ProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27Xsub3 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,3^3);[127X[104X
    [4X[28XSubgeometry PG(3, 27) of ProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27Xsub4 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,3^6);[127X[104X
    [4X[28XProjectiveSpace(3, 729)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X14.3 [33X[0;0YBasic operations[133X[101X
  
  
  [1X14.3-1 [33X[0;0YUnderlying vector space and ambient projective space[133X[101X
  
  [33X[1;0Y[29X[2XUnderlyingVectorSpace[102X( [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XAmbientSpace[102X( [3Xsub[103X ) [32X operation[133X
  
  [33X[0;0YLet  [22XP[122X be a projective space over the field [22XF[122X. Let [3Xsub[103X be a subgeometry of [22XP[122X
  over  the  subfield [22XF'[122X. The underlying vector space of [3Xsub[103X is defined as the
  underlying vector space of [22XP[122X (which is a vector space over the field [22XF[122X). The
  ambient space of a subgeometry [3Xsub[103X is the projective space [22XP[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(2,5^6);[127X[104X
    [4X[28XProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27Xsub1 := CanonicalSubgeometryOfProjectiveSpace(pg,5);[127X[104X
    [4X[28XSubgeometry PG(2, 5) of ProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingVectorSpace(pg);[127X[104X
    [4X[28X( GF(5^6)^3 )[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingVectorSpace(sub1);[127X[104X
    [4X[28X( GF(5^6)^3 )[128X[104X
    [4X[25Xgap>[125X [27XAmbientSpace(sub1);[127X[104X
    [4X[28XProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27Xsub2 := CanonicalSubgeometryOfProjectiveSpace(pg,5^3);[127X[104X
    [4X[28XSubgeometry PG(2, 125) of ProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27XAmbientSpace(sub2);[127X[104X
    [4X[28XProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingVectorSpace(sub2);[127X[104X
    [4X[28X( GF(5^6)^3 )[128X[104X
    [4X[25Xgap>[125X [27Xframe := RandomFrameOfProjectiveSpace(pg);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(2, 15625)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 15625)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 15625)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 15625)> ][128X[104X
    [4X[25Xgap>[125X [27Xsub3 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,5^2);[127X[104X
    [4X[28XSubgeometry PG(2, 25) of ProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27XAmbientSpace(sub3);[127X[104X
    [4X[28XProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingVectorSpace(sub3);[127X[104X
    [4X[28X( GF(5^6)^3 )[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X14.3-2 DefiningFrameOfSubgeometry[101X
  
  [33X[1;0Y[29X[2XDefiningFrameOfSubgeometry[102X( [3Xsub[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya set of projective points[133X
  
  [33X[0;0YThis attribute returns a frame of the ambient space of [3Xsub[103X defining it. Note
  that  different frames might define the same subgeometry, but the frame used
  to  constructed [3Xsub[103X is stored at construction, and it is exactly this stored
  object  that  is returned by this attribute. The returned object is a set of
  points,  and it is also a set in the GAP sense, i.e. an ordered list without
  duplicates.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(2,2^4);[127X[104X
    [4X[28XProjectiveSpace(2, 16)[128X[104X
    [4X[25Xgap>[125X [27Xsub := CanonicalSubgeometryOfProjectiveSpace(pg,2);[127X[104X
    [4X[28XSubgeometry PG(2, 2) of ProjectiveSpace(2, 16)[128X[104X
    [4X[25Xgap>[125X [27Xframe := DefiningFrameOfSubgeometry(sub);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)> ][128X[104X
    [4X[25Xgap>[125X [27XList(frame,x->Unpack(UnderlyingObject(x)));[127X[104X
    [4X[28X[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xframe := RandomFrameOfProjectiveSpace(pg);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)> ][128X[104X
    [4X[25Xgap>[125X [27Xsub := SubgeometryOfProjectiveSpaceByFrame(pg,frame,2^2);[127X[104X
    [4X[28XSubgeometry PG(2, 4) of ProjectiveSpace(2, 16)[128X[104X
    [4X[25Xgap>[125X [27Xdef := DefiningFrameOfSubgeometry(sub);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)> ][128X[104X
    [4X[25Xgap>[125X [27XList(def,x->Unpack(UnderlyingObject(x)));[127X[104X
    [4X[28X[ [ Z(2)^0, 0*Z(2), Z(2^4)^9 ], [ Z(2)^0, Z(2^4)^4, Z(2^4)^11 ], [128X[104X
    [4X[28X  [ Z(2)^0, Z(2^4)^2, Z(2^2)^2 ], [ Z(2)^0, Z(2^2)^2, Z(2^4)^14 ] ][128X[104X
    [4X[25Xgap>[125X [27XStandardFrame(sub);[127X[104X
    [4X[28X[ <a point in Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)>, [128X[104X
    [4X[28X  <a point in Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)>, [128X[104X
    [4X[28X  <a point in Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)>, [128X[104X
    [4X[28X  <a point in Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)> ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X14.3-3 [33X[0;0YProjective dimension and rank[133X[101X
  
  [33X[1;0Y[29X[2XProjectiveDimension[102X( [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDimension[102X( [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRank[102X( [3Xsub[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YIf [3Xsub[103X is a subgeometry of a projective space, then it is a projective space
  itself. Therefore, these three operations return the projective dimension of
  [3Xsub[103X, see also [14X4.1-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(7,8^2);[127X[104X
    [4X[28XProjectiveSpace(7, 64)[128X[104X
    [4X[25Xgap>[125X [27Xsub := CanonicalSubgeometryOfProjectiveSpace(pg,8);[127X[104X
    [4X[28XSubgeometry PG(7, 8) of ProjectiveSpace(7, 64)[128X[104X
    [4X[25Xgap>[125X [27XProjectiveDimension(sub);[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27XDimension(sub);[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27XRank(sub);[127X[104X
    [4X[28X7[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X14.3-4 [33X[0;0YUnderlying algebraic structures[133X[101X
  
  [33X[1;0Y[29X[2XUnderlyingVectorSpace[102X( [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XBaseField[102X( [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSubfieldOfSubgeometry[102X( [3Xsub[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  first  operation returns a vector space, the second and third
            operations return a finite field[133X
  
  [33X[0;0YThe   operations   [11XUnderlyingVectorSpace[111X   and  [11XBaseField[111X  are  defined  for
  projective  spaces,  see  [14X4.1-5[114X and [14X4.1-4[114X. For a subgeometry of a projective
  space    [3Xsub[103X    with    ambient    space   [3Xps[103X,   these   operations   return
  [11XUnderlyingVectorSpace(ps)[111X,   [11XBaseField(ps)[111X   respectively.   The   operation
  [11XSubfieldOfSubgeometry[111X returns the subfield over which [3Xsub[103X is defined.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(3,3^6);[127X[104X
    [4X[28XProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27Xsub1 := CanonicalSubgeometryOfProjectiveSpace(pg,3^3);[127X[104X
    [4X[28XSubgeometry PG(3, 27) of ProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27XBaseField(sub1);[127X[104X
    [4X[28XGF(3^6)[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingVectorSpace(sub1);[127X[104X
    [4X[28X( GF(3^6)^4 )[128X[104X
    [4X[25Xgap>[125X [27XSubfieldOfSubgeometry(sub1);[127X[104X
    [4X[28XGF(3^3)[128X[104X
    [4X[25Xgap>[125X [27Xsub2 := CanonicalSubgeometryOfProjectiveSpace(pg,3^2);[127X[104X
    [4X[28XSubgeometry PG(3, 9) of ProjectiveSpace(3, 729)[128X[104X
    [4X[25Xgap>[125X [27XBaseField(sub2);[127X[104X
    [4X[28XGF(3^6)[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingVectorSpace(sub2);[127X[104X
    [4X[28X( GF(3^6)^4 )[128X[104X
    [4X[25Xgap>[125X [27XSubfieldOfSubgeometry(sub2);[127X[104X
    [4X[28XGF(3^2)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X14.3-5 CollineationFixingSubgeometry[101X
  
  [33X[1;0Y[29X[2XCollineationFixingSubgeometry[102X( [3Xsub[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya collineation of the ambient space of [3Xsub[103X[133X
  
  [33X[0;0YLet  [22XGF(q)[122X  be  the  field  over which [3Xsub[103X is defined, this is a subfield of
  [22XGF(q^t)[122X  over  which  the  ambient projective space [22XP[122X is defined. It is well
  known  that there exists a collineation of [22XP[122X of order [22Xt[122X, fixing all elements
  of  [3Xsub[103X,  which  is  returned  by  this  operation. This collineation is the
  collineation  induced  by  the  Frobenius  map  [22Xx↦  x^q[122X,  conjugated  by the
  collineation  of  [22XP[122X mapping the subgeometry [3Xsub[103X to the canonical subgeometry
  of  [22XP[122X  over  [22XGF(q)[122X.  In case of a quadratic field extension (i.e. [22Xt=2[122X), this
  collineation  is  known  in  the  literature  as  the Baer involution of the
  subgeometry.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(2,7^3);[127X[104X
    [4X[28XProjectiveSpace(2, 343)[128X[104X
    [4X[25Xgap>[125X [27Xsub := CanonicalSubgeometryOfProjectiveSpace(pg,GF(7));[127X[104X
    [4X[28XSubgeometry PG(2, 7) of ProjectiveSpace(2, 343)[128X[104X
    [4X[25Xgap>[125X [27Xcoll := CollineationFixingSubgeometry(sub);[127X[104X
    [4X[28X< a collineation: <cmat 3x3 over GF(7,3)>, F^7>[128X[104X
    [4X[25Xgap>[125X [27XOrder(coll);[127X[104X
    [4X[28X3[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X14.4 [33X[0;0YConstructing elements of a subgeometry[133X[101X
  
  [1X14.4-1 VectorSpaceToElement[101X
  
  [33X[1;0Y[29X[2XVectorSpaceToElement[102X( [3Xsub[103X, [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya subspace of a subgeometry[133X
  
  [33X[0;0Y[3Xsub[103X  is  a  subgeometry  of a projective space, and [3Xv[103X is either a row vector
  (for  points)  or  a  matrix (for higher dimensional subspaces). In the case
  that  [3Xv[103X  is  a  matrix,  the  rows represent generators for the subspace. An
  exceptional  case  is  when  [3Xv[103X is the zero-vector, in which case the trivial
  subspace  is returned. This method checks whether [3Xv[103X determines an element of
  [3Xsub[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(2,5^6);[127X[104X
    [4X[28XProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27Xvecs := [ [ Z(5)^0, Z(5^6)^13972, Z(5^6)^11653 ], [127X[104X
    [4X[25X>[125X [27X[ Z(5)^0, Z(5^6)^9384, Z(5^6)^1372 ],[127X[104X
    [4X[25X>[125X [27X[ Z(5)^0, Z(5^6)^14447, Z(5^6)^15032 ], [127X[104X
    [4X[25X>[125X [27X[ Z(5)^0, Z(5^6)^8784, Z(5^6)^10360 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xframe := List(vecs,x->VectorSpaceToElement(pg,x));[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(2, 15625)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 15625)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 15625)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 15625)> ][128X[104X
    [4X[25Xgap>[125X [27Xsub := SubgeometryOfProjectiveSpaceByFrame(pg,frame,5^3);[127X[104X
    [4X[28XSubgeometry PG(2, 125) of ProjectiveSpace(2, 15625)[128X[104X
    [4X[25Xgap>[125X [27XVectorSpaceToElement(sub,[0,0,0]*Z(5)^0);[127X[104X
    [4X[28X< empty subspace >[128X[104X
    [4X[25Xgap>[125X [27Xvec := [ Z(5)^0, Z(5^6)^8584, Z(5^6)^13650 ];[127X[104X
    [4X[28X[ Z(5)^0, Z(5^6)^8584, Z(5^6)^13650 ][128X[104X
    [4X[25Xgap>[125X [27XVectorSpaceToElement(sub,vec);[127X[104X
    [4X[28X<a point in Subgeometry PG(2, 125) of ProjectiveSpace(2, 15625)>[128X[104X
    [4X[25Xgap>[125X [27Xvec := [ [ Z(5)^0, 0*Z(5), Z(5^6)^5740 ], [ 0*Z(5), Z(5)^0, Z(5^6)^15250 ] ];[127X[104X
    [4X[28X[ [ Z(5)^0, 0*Z(5), Z(5^6)^5740 ], [ 0*Z(5), Z(5)^0, Z(5^6)^15250 ] ][128X[104X
    [4X[25Xgap>[125X [27XVectorSpaceToElement(sub,vec);[127X[104X
    [4X[28XError, <obj> does not determine an element in <sub> called from[128X[104X
    [4X[28XVectorSpaceToElementForSubgeometries( geom, v [128X[104X
    [4X[28X ) at ./pkg/fining/lib/subgeometries.gi:400 called from[128X[104X
    [4X[28X<function "unknown">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 19 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
    [4X[25Xgap>[125X [27Xvec := [ [ Z(5)^0, 0*Z(5), Z(5^6)^8268 ], [ 0*Z(5), Z(5)^0, Z(5^6)^1472 ] ];[127X[104X
    [4X[28X[ [ Z(5)^0, 0*Z(5), Z(5^6)^8268 ], [ 0*Z(5), Z(5)^0, Z(5^6)^1472 ] ][128X[104X
    [4X[25Xgap>[125X [27XVectorSpaceToElement(sub,vec);[127X[104X
    [4X[28X<a line in Subgeometry PG(2, 125) of ProjectiveSpace(2, 15625)>[128X[104X
    [4X[25Xgap>[125X [27XVectorSpaceToElement(sub,vecs);[127X[104X
    [4X[28XSubgeometry PG(2, 125) of ProjectiveSpace(2, 15625)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X14.4-2 ExtendElementOfSubgeometry[101X
  
  [33X[1;0Y[29X[2XExtendElementOfSubgeometry[102X( [3Xel[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya subspace of a projective space[133X
  
  [33X[0;0YThe  argument  [3Xel[103X  is an element of a subgeometry [22XP'[122X with ambient projective
  space  [22XP[122X. The projective space is defined over a field [22XF[122X, the subgeometry [22XP'[122X
  is  defined  over  a subfield [22XF'[122X of [22XF[122X. The underlying vectorspace of [3Xel[103X is a
  vector space over [22XF'[122X generated by a set [22XS[122X of vectors. This operation returns
  the  element of [22XP[122X, corresponding to the vector space over [22XF[122X generated by the
  vectors  in  [22XS[122X.  Note that the set [22XS[122X can be otbained using [11XUnderlyingObject[111X,
  see [14X3.2-2[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(3,5^5);[127X[104X
    [4X[28XProjectiveSpace(3, 3125)[128X[104X
    [4X[25Xgap>[125X [27Xframe := RandomFrameOfProjectiveSpace(pg);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(3, 3125)>, <a point in ProjectiveSpace(3, 3125)>[128X[104X
    [4X[28X    , <a point in ProjectiveSpace(3, 3125)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(3, 3125)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(3, 3125)> ][128X[104X
    [4X[25Xgap>[125X [27Xsub := SubgeometryOfProjectiveSpaceByFrame(pg,frame,5);[127X[104X
    [4X[28XSubgeometry PG(3, 5) of ProjectiveSpace(3, 3125)[128X[104X
    [4X[25Xgap>[125X [27Xp := Random(Points(sub));[127X[104X
    [4X[28X<a point in Subgeometry PG(3, 5) of ProjectiveSpace(3, 3125)>[128X[104X
    [4X[25Xgap>[125X [27Xl := Random(Lines(p));[127X[104X
    [4X[28X<a line in Subgeometry PG(3, 5) of ProjectiveSpace(3, 3125)>[128X[104X
    [4X[25Xgap>[125X [27Xp * l;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xq := ExtendElementOfSubgeometry(p);[127X[104X
    [4X[28X<a point in ProjectiveSpace(3, 3125)>[128X[104X
    [4X[25Xgap>[125X [27Xq * l;[127X[104X
    [4X[28XError, <x> and <y> do not belong to the same geometry called from[128X[104X
    [4X[28Xx in y at ./pkg/fining/lib/projectivespace.gi:1658 called from[128X[104X
    [4X[28XIsIncident( b, a ) at ./pkg/fining/lib/geometry.gi:439 called from[128X[104X
    [4X[28X<function "unknown">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 15 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
    [4X[25Xgap>[125X [27Xm := ExtendElementOfSubgeometry(l);[127X[104X
    [4X[28X<a line in ProjectiveSpace(3, 3125)>[128X[104X
    [4X[25Xgap>[125X [27Xq * m;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingObject(q) = UnderlyingObject(p);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingObject(l) = UnderlyingObject(m);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X14.4-3 AmbientGeometry[101X
  
  [33X[1;0Y[29X[2XAmbientGeometry[102X( [3Xel[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan incidence geometry[133X
  
  [33X[0;0YFor [3Xel[103X an element of a subgeometry [22XP[122X, which is also a projective space, this
  operation returns [22XP[122X.[133X
  
  
  [1X14.4-4 [33X[0;0YFlags[133X[101X
  
  [33X[1;0Y[29X[2XFlagOfIncidenceStructure[102X( [3Xsub[103X, [3Xels[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsEmptyFlag[102X( [3Xflag[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsChamberOfIncidenceStructure[102X( [3Xflag[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YThese  operations  are  defined  for  projective spaces and so they are also
  applicable to subgeometries.[133X
  
  
  [1X14.5 [33X[0;0YGroups and actions[133X[101X
  
  [33X[0;0YLet  [22XP'[122X  be a subgeometry of [22XP[122X. Although one could argue that any semilinear
  map  inducing  a  collineation preserving [22XP'[122X can be called a collineation of
  [22XP'[122X,  this  would  cause  problems  with the nice monomorphism functionality,
  since such a collineation does not necessarily have a faithful action on the
  subgeometry. For this reason, we decided to define the collineation group of
  the  subgeometry  [22XP'[122X  as  the  collineation  group  of  the projective space
  isomorphic  to  [22XP'[122X  conjugated  by  the  collineation of [22XP[122X mapping [22XP'[122X on the
  canonical  subgeometry  of  [22XP[122X  over  the  same  field  as [22XP'[122X. Similarly, the
  projectivity  group,  respectively  the special projectivity group, of [22XP'[122X is
  defined  as  the  conjugate  of the projectivity group, respectively special
  projectivity group, of the projective space isomorhpic to [22XP'[122X.[133X
  
  
  [1X14.5-1 [33X[0;0YGroups of collineations[133X[101X
  
  [33X[1;0Y[29X[2XCollineationGroup[102X( [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XProjectivityGroup[102X( [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSpecialProjectivityGroup[102X( [3Xsub[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya group of collineations[133X
  
