Construct the collineation group of the plane P acting on the points of P.
Construct the collineation group of the plane P in its action on the points of P.
Construct the collineation group of the plane P in its action on the lines of P.
> Bundle := function(q)
>
> K<w> := GF(q^3);
> P, V, L := ProjectivePlane(q^3);
> G := CollineationGroup(P);
> S := Support(P); // normalized vectors
> sig := sub< G | [V | [S[i][1], S[i][2]^q, S[i][3]^q] : i in [1..#V]] >;
> // group of planar collineations of order 3
> p := V ! [1, w^2, w];
> T := p^sig;
> e2 := V![0, 1, 0];
> e3 := V![0, 0, 1];
> S := Support(SubfieldSubplane(P, GF(q)));
> c23 := Conic(T join {e2, e3}) meet S;
> e1 := Rep(S diff c23);
> c12 := (Conic(T join {e1, e2}) meet S) diff { e1 };
> c13 := (Conic(T join {e1, e3}) meet S) diff { e1 };
> bundle := [ Conic(T join {e1, e}) meet S : e in c23 ] cat
> [ Conic(T join {v1, v2}) meet S : v2 in c13, v1 in c12 ];
> return ProjectivePlane< S | bundle >;
>
> end function;
>
> PB := Bundle(3);
> print PB;
Projective plane of order 3
The group of (p, l)-central collineations of a projective plane P.
The group of central collineations with centre p of a projective plane P.
The group of central collineations with axis l of a projective plane P.
True iff the collineation g of a projective plane P is a central collineation; if true, also returns the centre and axis of g.
The image of the point u of the plane P under the action of the collineation a of P (returns a point).
The image of the set S of points of the plane P under the action of the collineation a of P (returns a set of points).
The image of the sequence S of points of the plane P under the action of the collineation a of P (returns a sequence of points).
The image of the line b of the plane P under the action of the collineation a of P (returns a line).
The image of the set T of lines of the plane P under the action of the collineation a of P (returns a set of lines).
The image of the sequence U of lines of the plane P under the action of the collineation a of P (returns a sequence of lines).
The image of the point u of the plane P under the action of the subgroup B of the collineation group of P (returns a set of points).
The image of the set S of points of the plane P under the action of the subgroup B of the collineation group of P (returns a set of sets of points).
The image of the sequence Q of lines of the plane P under the action of the subgroup B of the collineation group of P (returns a set of sequences of lines).
The image of the line b of plane P under the action of the subgroup B of the collineation group of P (returns a set of lines).
The image of the set T of lines of the plane P under the action of the subgroup B of the collineation group of P (returns a set of sets of lines).
The image of the sequence U of lines of the plane P under the action of the subgroup B of the collineation group of P (returns a set of sequences of lines).
Given an plane P on n points and a permutation a of Sym(n), construct the image of the plane P under a (returns a plane).
Given an plane P on n points and a subgroup A of Sym(n), construct the orbit of the plane P under A (returns a set of planes).
Given a point u belonging to the point set of the plane P, construct the stabilizer of u in the subgroup A of Aut(P).
Given a sequence Q of points belonging to the point set of the plane P, construct the pointwise stabilizer of Q in the subgroup A of Aut(P).
Given a set S of points belonging to the point set of the plane P, construct the stabilizer of S in the subgroup A of Aut(P).
Given a line b belonging to the line set of the plane P, construct the stabilizer of b in the subgroup A of Aut(P).
Given a sequence Q of lines belonging to the line set of the plane P, construct the pointwise stabilizer of Q in the subgroup A of Aut(P).
> BaerDerivation := function(q)
> //---------------------------------
>
> // Construct an affine plane by the technique of derivation using
> // Baer subplanes
>
> Fq2< w > := FiniteField(q^2);
> V := VectorSpace(Fq2, 3);
> Plane, Pts, Lns := ProjectivePlane(V);
> G := CollineationGroup(Plane);
>
> /*Construct a Baer subplane*/
>
> Subplane := SubfieldSubplane(Plane, GF(q));
>
> /*The Baer segment consists of those points of the Baer subplane that
> lie on the line at infinity. Take the line x = 0 as the line at
> infinity.*/
>
> LineInf := Lns![1, 0, 0];
>
> BaerSeg := Support(Subplane) meet LineInf;
>
> /*We now find the subgroup of the collineation group that fixes the Baer
> segment. The translates of the Baer subplane under this subgroup will
> give us those Baer subplanes that contain the set BaerSeg.*/
>
> StabSeg := Stabilizer(G, BaerSeg);
>
> /*Rather than computing the translates of the entire Baer subplane, we
> compute the translates of SubPlane - BaerSeg so that we get exactly
> those sets which become new affine lines.*/
>
> BaerLines := (Support(Subplane) diff BaerSeg)^StabSeg;
>
> /*We complete the new plane by taking those lines of PG(2, q^2) which
> intersect the line at infinity at points other than those in the Baer
> segment. Upon removing the intersection point with LineInf, each such
> line becomes a line of the new affine plane.*/
>
> AffLines := BaerLines join { Set(l) diff LineInf : l in Lns |
> (BaerSeg meet l) eq {} };
>
> return AffinePlane< SetToIndexedSet(&join(AffLines)) | Setseq(AffLines)
> : Check := false >;
>
> end function; /*BaerDerivation*/
>
> A := BaerDerivation(3);
> print A;
Affine plane of order 9
IsTransitive(P) : Plane -> BoolElt
True iff the collineation group of the plane P acts transitively on the points of P.
True iff the collineation group of the plane P acts transitively on the lines of P.[Next] [Prev] [Right] [Left] [Up] [Index] [Root]