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gap> pg2 := PG(2,5);
ProjectiveSpace(2, 5)
gap> pg3 := PG(3,5);
ProjectiveSpace(3, 5)
gap> hyp := VectorSpaceToElement(pg3,[[1,2,4,0],[0,3,2,0],[1,1,0,1]]*Z(5)^0);
<a plane in ProjectiveSpace(3, 5)>
gap> em := NaturalEmbeddingBySubspace( pg2, pg3, hyp );
<geometry morphism from <All elements of ProjectiveSpace(2,
5)> to <All elements of ProjectiveSpace(3, 5)>>
gap> l := Random(Lines(pg2));
<a line in ProjectiveSpace(2, 5)>
gap> l^em;
<a line in ProjectiveSpace(3, 5)>
gap> p := Random(Points(hyp));
<a point in ProjectiveSpace(3, 5)>
gap> PreImageElm(em,p);
<a point in ProjectiveSpace(2, 5)>
gap> mat := [[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]*Z(3);
[ [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3) ], [ 0*Z(3), 0*Z(3), Z(3), 0*Z(3) ],
[ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ] ]
gap> form := BilinearFormByMatrix(mat,GF(3));
< bilinear form >
gap> w3 := PolarSpace(form);
<polar space in ProjectiveSpace(3,GF(3)): -x1*y4-x2*y3+x3*y2+x4*y1=0 >
gap> w5 := SymplecticSpace(5, 3);
W(5, 3)
gap> pg := AmbientSpace( w5 );
ProjectiveSpace(5, 3)
gap> solid := VectorSpaceToElement(pg,[[1,0,0,0,0,0],[0,1,0,0,0,0],
> [0,0,1,0,0,0],[0,0,0,1,0,0]]*Z(3)^0);
<a solid in ProjectiveSpace(5, 3)>
gap> TypeOfSubspace(w5,solid);
"symplectic"
gap> em := NaturalEmbeddingBySubspace( w3, w5, solid );
<geometry morphism from <Elements of <polar space in ProjectiveSpace(
3,GF(3)): -x1*y4-x2*y3+x3*y2+x4*y1=0 >> to <Elements of W(5, 3)>>
gap> points := Points( w3 );
<points of W(3, 3): -x1*y4-x2*y3+x3*y2+x4*y1=0>
gap> points2 := ImagesSet(em, AsSet(points));;
#I Computing collineation group of canonical polar space...
gap> ForAll(points2, x -> x in solid);
true
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