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gap> pg1 := PG(2,3);
ProjectiveSpace(2, 3)
gap> pg2 := PG(2,9);
ProjectiveSpace(2, 9)
gap> em := NaturalEmbeddingBySubfield(pg1,pg2);
<geometry morphism from <All elements of ProjectiveSpace(2,
3)> to <All elements of ProjectiveSpace(2, 9)>>
gap> points := AsList(Points( pg1 ));
[ <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
<a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
<a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
<a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
<a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
<a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
<a point in ProjectiveSpace(2, 3)> ]
gap> image := ImagesSet(em, points);
[ <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
<a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
<a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
<a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
<a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
<a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
<a point in ProjectiveSpace(2, 9)> ]
gap> hom := Intertwiner(em);
MappingByFunction( The FinInG projectivity group PGL(3,3), <projective colline
ation group of size 5616 with
2 generators>, function( x ) ... end, function( y ) ... end )
gap> group1 := ProjectivityGroup(pg1);
The FinInG projectivity group PGL(3,3)
gap> gens := GeneratorsOfGroup(group1);
[ < a collineation: <cmat 3x3 over GF(3,1)>, F^0>,
< a collineation: <cmat 3x3 over GF(3,1)>, F^0> ]
gap> group1_image := Group(List(gens,x->x^hom));
<projective collineation group with 2 generators>
gap> Order(group1_image);
5616
gap> group2 := ProjectivityGroup(pg2);
The FinInG projectivity group PGL(3,9)
gap> Order(group2);
42456960
gap> g := Random(group2);
< a collineation: <cmat 3x3 over GF(3,2)>, F^0>
gap> PreImageElm(hom,g);
#I <el> is not in the range of the intertwiner
fail
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