Desargues Desargues Outline

Finite ... a finite geometry package Geometry

Examples of Problems John Bamberg Features of Desargues Anton Betten Jan De Beule Maska Law Another example Max Neunh¨offer Michael Pauley Sven Reichard Desargues Outline

Outline

Finite Geometry Examples of 1 Outline Problems

Features of Desargues

Another 2 Finite Geometry example

3 Examples of Problems

4 Features of Desargues

5 Another example • Polar Spaces • Generalised Polygons

Incidence Geometry Consists of: • Objects (points, lines, planes, etc) • Incidence relation (anti-reflexive and symmetric) • A maximal flag contains an object of each type. The rank is the number of types of object.

Desargues Finite Geometry

Outline

Finite Geometry Examples of • Projective Geometry/Affine Geometry Problems

Features of Desargues

Another example • Generalised Polygons

Incidence Geometry Consists of: • Objects (points, lines, planes, etc) • Incidence relation (anti-reflexive and symmetric) • A maximal flag contains an object of each type. The rank is the number of types of object.

Desargues Finite Geometry

Outline

Finite Geometry Examples of • Projective Geometry/Affine Geometry Problems

Features of • Polar Spaces Desargues

Another example Incidence Geometry Consists of: • Objects (points, lines, planes, etc) • Incidence relation (anti-reflexive and symmetric) • A maximal flag contains an object of each type. The rank is the number of types of object.

Desargues Finite Geometry

Outline

Finite Geometry Examples of • Projective Geometry/Affine Geometry Problems

Features of • Polar Spaces Desargues • Generalised Polygons Another example Desargues Finite Geometry

Outline

Finite Geometry Examples of • Projective Geometry/Affine Geometry Problems

Features of • Polar Spaces Desargues • Generalised Polygons Another example Incidence Geometry Consists of: • Objects (points, lines, planes, etc) • Incidence relation (anti-reflexive and symmetric) • A maximal flag contains an object of each type. The rank is the number of types of object. • Incidence Relation: A ⊂ B or B ⊂ A • Types 1, 2, 3,..., d − 1

Theorem A projective space or polar space of rank at least 3 is classical, that is,

it comes from a vector space.

Desargues

Projective Space Outline Start with a vector space V (d, (q)) Finite GF Geometry • Objects Examples of Problems Points: 1-dim subspaces Features of Lines: 2-dim subspaces Desargues Planes: 3-dim subspaces, etc... Another example • Types 1, 2, 3,..., d − 1

Theorem A projective space or polar space of rank at least 3 is classical, that is,

it comes from a vector space.

Desargues

Projective Space Outline Start with a vector space V (d, (q)) Finite GF Geometry • Objects Examples of Problems Points: 1-dim subspaces Features of Lines: 2-dim subspaces Desargues Planes: 3-dim subspaces, etc... Another example • Incidence Relation: A ⊂ B or B ⊂ A Theorem A projective space or polar space of rank at least 3 is classical, that is,

it comes from a vector space.

Desargues

Projective Space Outline Start with a vector space V (d, (q)) Finite GF Geometry • Objects Examples of Problems Points: 1-dim subspaces Features of Lines: 2-dim subspaces Desargues Planes: 3-dim subspaces, etc... Another example • Incidence Relation: A ⊂ B or B ⊂ A • Types 1, 2, 3,..., d − 1 Desargues

Projective Space Outline Start with a vector space V (d, (q)) Finite GF Geometry • Objects Examples of Problems Points: 1-dim subspaces Features of Lines: 2-dim subspaces Desargues Planes: 3-dim subspaces, etc... Another example • Incidence Relation: A ⊂ B or B ⊂ A • Types 1, 2, 3,..., d − 1

Theorem A projective space or polar space of rank at least 3 is classical, that is,

it comes from a vector space. Points: All one-dim subspaces. 2 q6−1 Lines: (q + 1) q−1 two-dim subspaces. 3 q4−1 Planes: (q + 1) q−1 three-dim subspaces.

Spreads A spread of W(5, q) is a set of q3 + 1 planes which form a partition of the set of points.

Desargues Spreads of W(5, q)

Outline

Finite Geometry W(5, q) Examples of Consider V (6, q) equipped with an alternating form: Problems

Features of Desargues hu, vi = u1v2 − u2v1 + u3v4 − u4v3 + u5v6 − u6v5. Another example We get a polar geometry consisting of Spreads A spread of W(5, q) is a set of q3 + 1 planes which form a partition of the set of points.

Desargues Spreads of W(5, q)

Outline

Finite Geometry W(5, q) Examples of Consider V (6, q) equipped with an alternating form: Problems

Features of Desargues hu, vi = u1v2 − u2v1 + u3v4 − u4v3 + u5v6 − u6v5. Another example We get a polar geometry consisting of Points: All one-dim subspaces. 2 q6−1 Lines: (q + 1) q−1 two-dim subspaces. 3 q4−1 Planes: (q + 1) q−1 three-dim subspaces. Desargues Spreads of W(5, q)

Outline

Finite Geometry W(5, q) Examples of Consider V (6, q) equipped with an alternating form: Problems

Features of Desargues hu, vi = u1v2 − u2v1 + u3v4 − u4v3 + u5v6 − u6v5. Another example We get a polar geometry consisting of Points: All one-dim subspaces. 2 q6−1 Lines: (q + 1) q−1 two-dim subspaces. 3 q4−1 Planes: (q + 1) q−1 three-dim subspaces.

Spreads A spread of W(5, q) is a set of q3 + 1 planes which form a partition of the set of points. What we need... 6 Points: Easy, all one-dim subspaces of GF(3) . Planes: Take orbit of one plane

gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);;

Group: Sylow 13-subgroup of Sp(6, 3)

gap> syl13 := SylowSubgroup(sp, 13);

Solution: Stitch together orbits on planes

Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline

Finite order divisible by 13. Geometry

Examples of Problems

Features of Desargues

Another example Planes: Take orbit of one plane

gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);;

Group: Sylow 13-subgroup of Sp(6, 3)

gap> syl13 := SylowSubgroup(sp, 13);

Solution: Stitch together orbits on planes

Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline

Finite order divisible by 13. Geometry Examples of What we need... Problems 6 Features of Points: Easy, all one-dim subspaces of GF(3) . Desargues

Another example gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);;

Group: Sylow 13-subgroup of Sp(6, 3)

gap> syl13 := SylowSubgroup(sp, 13);

Solution: Stitch together orbits on planes

Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline

Finite order divisible by 13. Geometry Examples of What we need... Problems 6 Features of Points: Easy, all one-dim subspaces of GF(3) . Desargues

Another Planes: Take orbit of one plane example Group: Sylow 13-subgroup of Sp(6, 3)

gap> syl13 := SylowSubgroup(sp, 13);

Solution: Stitch together orbits on planes

Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline

Finite order divisible by 13. Geometry Examples of What we need... Problems 6 Features of Points: Easy, all one-dim subspaces of GF(3) . Desargues

Another Planes: Take orbit of one plane example gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);; Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline

Finite order divisible by 13. Geometry Examples of What we need... Problems 6 Features of Points: Easy, all one-dim subspaces of GF(3) . Desargues

Another Planes: Take orbit of one plane example gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);;

Group: Sylow 13-subgroup of Sp(6, 3)

gap> syl13 := SylowSubgroup(sp, 13);

Solution: Stitch together orbits on planes gap> syl := SylowSubgroup(sp, 13); gap> planes := Planes( w ); gap> planes := AsList( planes );; gap> orbits := Orbits(syl, planes , OnLieVarieties);; gap> Collected( List( orbits, Size )); [ [ 1, 2 ], [ 13, 86 ] ] gap> IsPartialSpread := s -> ForAll( Combinations(s,2), c -> ProjectiveDimension( Meet(c[1], c[2]) ) = -1 );; gap> partialspreads := Filtered(orbits, IsPartialSpread);; gap> 13s := Filtered(partialspreads, i -> Size(i) = 13);; gap> 26s := List(Combinations(13s,2), Union);; gap> two := Union(Filtered(partialspreads, i -> Size(i) = 1));; gap> 28s := List(26s, x -> Union(x, two) );; gap> spreads := Filtered( 28s, IsPartialSpread);; gap> Size(spreads); 5

Desargues In Desargues...

Outline gap> w := SymplecticSpace(5, 3); Finite W(5, 3) Geometry gap> sp := IsometryGroup( w ); PSp(6,3) Examples of Problems

Features of Desargues

Another example gap> orbits := Orbits(syl, planes , OnLieVarieties);; gap> Collected( List( orbits, Size )); [ [ 1, 2 ], [ 13, 86 ] ] gap> IsPartialSpread := s -> ForAll( Combinations(s,2), c -> ProjectiveDimension( Meet(c[1], c[2]) ) = -1 );; gap> partialspreads := Filtered(orbits, IsPartialSpread);; gap> 13s := Filtered(partialspreads, i -> Size(i) = 13);; gap> 26s := List(Combinations(13s,2), Union);; gap> two := Union(Filtered(partialspreads, i -> Size(i) = 1));; gap> 28s := List(26s, x -> Union(x, two) );; gap> spreads := Filtered( 28s, IsPartialSpread);; gap> Size(spreads); 5

Desargues In Desargues...

Outline gap> w := SymplecticSpace(5, 3); Finite W(5, 3) Geometry gap> sp := IsometryGroup( w ); PSp(6,3) Examples of Problems gap> syl := SylowSubgroup(sp, 13); Features of Desargues gap> planes := Planes( w ); Another example gap> planes := AsList( planes );; gap> IsPartialSpread := s -> ForAll( Combinations(s,2), c -> ProjectiveDimension( Meet(c[1], c[2]) ) = -1 );; gap> partialspreads := Filtered(orbits, IsPartialSpread);; gap> 13s := Filtered(partialspreads, i -> Size(i) = 13);; gap> 26s := List(Combinations(13s,2), Union);; gap> two := Union(Filtered(partialspreads, i -> Size(i) = 1));; gap> 28s := List(26s, x -> Union(x, two) );; gap> spreads := Filtered( 28s, IsPartialSpread);; gap> Size(spreads); 5

Desargues In Desargues...

Outline gap> w := SymplecticSpace(5, 3); Finite W(5, 3) Geometry gap> sp := IsometryGroup( w ); PSp(6,3) Examples of Problems gap> syl := SylowSubgroup(sp, 13); Features of Desargues gap> planes := Planes( w ); Another example gap> planes := AsList( planes );; gap> orbits := Orbits(syl, planes , OnLieVarieties);; gap> Collected( List( orbits, Size )); [ [ 1, 2 ], [ 13, 86 ] ] gap> 13s := Filtered(partialspreads, i -> Size(i) = 13);; gap> 26s := List(Combinations(13s,2), Union);; gap> two := Union(Filtered(partialspreads, i -> Size(i) = 1));; gap> 28s := List(26s, x -> Union(x, two) );; gap> spreads := Filtered( 28s, IsPartialSpread);; gap> Size(spreads); 5

Desargues In Desargues...

Outline gap> w := SymplecticSpace(5, 3); Finite W(5, 3) Geometry gap> sp := IsometryGroup( w ); PSp(6,3) Examples of Problems gap> syl := SylowSubgroup(sp, 13); Features of Desargues gap> planes := Planes( w ); Another example gap> planes := AsList( planes );; gap> orbits := Orbits(syl, planes , OnLieVarieties);; gap> Collected( List( orbits, Size )); [ [ 1, 2 ], [ 13, 86 ] ] gap> IsPartialSpread := s -> ForAll( Combinations(s,2), c -> ProjectiveDimension( Meet(c[1], c[2]) ) = -1 );; gap> partialspreads := Filtered(orbits, IsPartialSpread);; gap> spreads := Filtered( 28s, IsPartialSpread);; gap> Size(spreads); 5

Desargues In Desargues...

Outline gap> w := SymplecticSpace(5, 3); Finite W(5, 3) Geometry gap> sp := IsometryGroup( w ); PSp(6,3) Examples of Problems gap> syl := SylowSubgroup(sp, 13); Features of Desargues gap> planes := Planes( w ); Another example gap> planes := AsList( planes );; gap> orbits := Orbits(syl, planes , OnLieVarieties);; gap> Collected( List( orbits, Size )); [ [ 1, 2 ], [ 13, 86 ] ] gap> IsPartialSpread := s -> ForAll( Combinations(s,2), c -> ProjectiveDimension( Meet(c[1], c[2]) ) = -1 );; gap> partialspreads := Filtered(orbits, IsPartialSpread);; gap> 13s := Filtered(partialspreads, i -> Size(i) = 13);; gap> 26s := List(Combinations(13s,2), Union);; gap> two := Union(Filtered(partialspreads, i -> Size(i) = 1));; gap> 28s := List(26s, x -> Union(x, two) );; Desargues In Desargues...

Outline gap> w := SymplecticSpace(5, 3); Finite W(5, 3) Geometry gap> sp := IsometryGroup( w ); PSp(6,3) Examples of Problems gap> syl := SylowSubgroup(sp, 13); Features of Desargues gap> planes := Planes( w ); Another example gap> planes := AsList( planes );; gap> orbits := Orbits(syl, planes , OnLieVarieties);; gap> Collected( List( orbits, Size )); [ [ 1, 2 ], [ 13, 86 ] ] gap> IsPartialSpread := s -> ForAll( Combinations(s,2), c -> ProjectiveDimension( Meet(c[1], c[2]) ) = -1 );; gap> partialspreads := Filtered(orbits, IsPartialSpread);; gap> 13s := Filtered(partialspreads, i -> Size(i) = 13);; gap> 26s := List(Combinations(13s,2), Union);; gap> two := Union(Filtered(partialspreads, i -> Size(i) = 1));; gap> 28s := List(26s, x -> Union(x, two) );; gap> spreads := Filtered( 28s, IsPartialSpread);; gap> Size(spreads); 5 Desargues

Outline

Finite Geometry

Examples of Problems These five spreads of W(5, 3) Features of Desargues 3 Another 1 2× Albert semifield-spread: 3 · 13 : 3 example 2 2× Hering spread: PSL(2, 13) 3 1× Regular spread: PSL(2, 27) • Basic functionality • ResidualOfVariety( geometry, object, type) • ResidualOfFlag( geometry, flag, type ) • Varieties( geometry, type ) • Join( object, object ) • Meet( object, object ) • VectorSpaceToVariety( geometry, vector or matrix ) • Flexibility with polar spaces: PolarSpace( form )

Desargues

Features of Desargues, at the moment... Outline Finite • Construction of geometries Geometry • ProjectiveSpace( d, q ) Examples of Problems • EllipticQuadric( d, q ) 2 Features of • HermitianVariety( d, q ) Desargues • AffineSpace( d, q ) Another • example SplitCayleyHexagon( q ) • Flexibility with polar spaces: PolarSpace( form )

Desargues

Features of Desargues, at the moment... Outline Finite • Construction of geometries Geometry • ProjectiveSpace( d, q ) Examples of Problems • EllipticQuadric( d, q ) 2 Features of • HermitianVariety( d, q ) Desargues • AffineSpace( d, q ) Another • example SplitCayleyHexagon( q ) • Basic functionality • ResidualOfVariety( geometry, object, type) • ResidualOfFlag( geometry, flag, type ) • Varieties( geometry, type ) • Join( object, object ) • Meet( object, object ) • VectorSpaceToVariety( geometry, vector or matrix ) Desargues

Features of Desargues, at the moment... Outline Finite • Construction of geometries Geometry • ProjectiveSpace( d, q ) Examples of Problems • EllipticQuadric( d, q ) 2 Features of • HermitianVariety( d, q ) Desargues • AffineSpace( d, q ) Another • example SplitCayleyHexagon( q ) • Basic functionality • ResidualOfVariety( geometry, object, type) • ResidualOfFlag( geometry, flag, type ) • Varieties( geometry, type ) • Join( object, object ) • Meet( object, object ) • VectorSpaceToVariety( geometry, vector or matrix ) • Flexibility with polar spaces: PolarSpace( form ) • Group actions • OnLieVarieties • OnAffineVarieties • OnKantorFamily • Enumerators • Morphisms • NaturalEmbeddingByVariety • NaturalProjectionByVariety • KleinCorrespondence • ProjectiveCompletion

Desargues

• Generalised polygons Outline • Finite TwistedTrialityHexagon( q ) Geometry • EGQByKantorFamily( group, list1, list2 ) Examples of • EGQByqClan( q-clan, field ) Problems • BLTSetByqClan( q-clan, field ) Features of Desargues

Another example • Enumerators • Morphisms • NaturalEmbeddingByVariety • NaturalProjectionByVariety • KleinCorrespondence • ProjectiveCompletion

Desargues

• Generalised polygons Outline • Finite TwistedTrialityHexagon( q ) Geometry • EGQByKantorFamily( group, list1, list2 ) Examples of • EGQByqClan( q-clan, field ) Problems • BLTSetByqClan( q-clan, field ) Features of Desargues • Group actions Another • OnLieVarieties example • OnAffineVarieties • OnKantorFamily • Morphisms • NaturalEmbeddingByVariety • NaturalProjectionByVariety • KleinCorrespondence • ProjectiveCompletion

Desargues

• Generalised polygons Outline • Finite TwistedTrialityHexagon( q ) Geometry • EGQByKantorFamily( group, list1, list2 ) Examples of • EGQByqClan( q-clan, field ) Problems • BLTSetByqClan( q-clan, field ) Features of Desargues • Group actions Another • OnLieVarieties example • OnAffineVarieties • OnKantorFamily • Enumerators Desargues

• Generalised polygons Outline • Finite TwistedTrialityHexagon( q ) Geometry • EGQByKantorFamily( group, list1, list2 ) Examples of • EGQByqClan( q-clan, field ) Problems • BLTSetByqClan( q-clan, field ) Features of Desargues • Group actions Another • OnLieVarieties example • OnAffineVarieties • OnKantorFamily • Enumerators • Morphisms • NaturalEmbeddingByVariety • NaturalProjectionByVariety • KleinCorrespondence • ProjectiveCompletion • This polar space also contains lines and planes. • Ovoid: set of 28 points of Q(6, 3) which partition the planes of Q(6, 3). • Patterson: Unique up to projectivity. • We will use E. E. Shult’s beautiful construction.

Desargues Another example

Outline

Finite Geometry

Examples of Problems The Patterson ovoid of Q(6, 3) Features of 7 Desargues • Q(6, 3): points of GF(3) which are solutions of Another example 2 2 2 2 2 2 2 x1 + x2 + x3 + x4 + x5 + x6 + x7 = 0. • Ovoid: set of 28 points of Q(6, 3) which partition the planes of Q(6, 3). • Patterson: Unique up to projectivity. • We will use E. E. Shult’s beautiful construction.

Desargues Another example

Outline

Finite Geometry

Examples of Problems The Patterson ovoid of Q(6, 3) Features of 7 Desargues • Q(6, 3): points of GF(3) which are solutions of Another example 2 2 2 2 2 2 2 x1 + x2 + x3 + x4 + x5 + x6 + x7 = 0.

• This polar space also contains lines and planes. • Patterson: Unique up to projectivity. • We will use E. E. Shult’s beautiful construction.

Desargues Another example

Outline

Finite Geometry

Examples of Problems The Patterson ovoid of Q(6, 3) Features of 7 Desargues • Q(6, 3): points of GF(3) which are solutions of Another example 2 2 2 2 2 2 2 x1 + x2 + x3 + x4 + x5 + x6 + x7 = 0.

• This polar space also contains lines and planes. • Ovoid: set of 28 points of Q(6, 3) which partition the planes of Q(6, 3). • We will use E. E. Shult’s beautiful construction.

Desargues Another example

Outline

Finite Geometry

Examples of Problems The Patterson ovoid of Q(6, 3) Features of 7 Desargues • Q(6, 3): points of GF(3) which are solutions of Another example 2 2 2 2 2 2 2 x1 + x2 + x3 + x4 + x5 + x6 + x7 = 0.

• This polar space also contains lines and planes. • Ovoid: set of 28 points of Q(6, 3) which partition the planes of Q(6, 3). • Patterson: Unique up to projectivity. Desargues Another example

Outline

Finite Geometry

Examples of Problems The Patterson ovoid of Q(6, 3) Features of 7 Desargues • Q(6, 3): points of GF(3) which are solutions of Another example 2 2 2 2 2 2 2 x1 + x2 + x3 + x4 + x5 + x6 + x7 = 0.

• This polar space also contains lines and planes. • Ovoid: set of 28 points of Q(6, 3) which partition the planes of Q(6, 3). • Patterson: Unique up to projectivity. • We will use E. E. Shult’s beautiful construction. Group([ (4,6)(5,7), (1,2,4)(3,6,5) ])

Construct ovoid

gap> psl32 := PSL(3,2);

gap> reps := [[1,1,1,0,0,0,0], [-1,1,1,0,0,0,0], [1,-1,1,0,0,0,0], [1,1,-1,0,0,0,0]]*Z(3)^0;; gap> ovoid := Union( List(reps, x-> Orbit(psl32, x, Permuted)) );; gap> ovoid := List(ovoid, x -> VectorSpaceToVariety(ps, x));;

Desargues

Outline Construct specific polar space

Finite Geometry gap> id := IdentityMat(7, GF(3));; gap> form := QuadraticFormByMatrix(id, GF(3)); Examples of Problems < quadratic form > gap> ps := PolarSpace( form ); Features of Desargues

Another example Desargues

Outline Construct specific polar space

Finite Geometry gap> id := IdentityMat(7, GF(3));; gap> form := QuadraticFormByMatrix(id, GF(3)); Examples of Problems < quadratic form > gap> ps := PolarSpace( form ); Features of Desargues

Another example Construct ovoid

gap> psl32 := PSL(3,2); Group([ (4,6)(5,7), (1,2,4)(3,6,5) ]) gap> reps := [[1,1,1,0,0,0,0], [-1,1,1,0,0,0,0], [1,-1,1,0,0,0,0], [1,1,-1,0,0,0,0]]*Z(3)^0;; gap> ovoid := Union( List(reps, x-> Orbit(psl32, x, Permuted)) );; gap> ovoid := List(ovoid, x -> VectorSpaceToVariety(ps, x));; Check that it is an ovoid

gap> planes := AsList( Planes(pq) );; gap> ForAll(planes, p -> Number(ovoid2, x -> x in p) = 1); true

Desargues Look at it in the canonical polar space

Outline gap> pq := ParabolicQuadric(6, 3); Q(6, 3) Finite gap> iso := IsomorphismPolarSpaces(ps, pq); Geometry Examples of to Q(6, 3)> Problems gap> ovoid2 := ImagesSet(iso, ovoid); Features of [ , , , Desargues , , , Another , , , example , , , , , , , , , , , , , , , , , , ] Desargues Look at it in the canonical polar space

Outline gap> pq := ParabolicQuadric(6, 3); Q(6, 3) Finite gap> iso := IsomorphismPolarSpaces(ps, pq); Geometry Examples of to Q(6, 3)> Problems gap> ovoid2 := ImagesSet(iso, ovoid); Features of [ , , , Desargues , , , Another , , , example , , , , , , , , , , , , , , , , , , ]

Check that it is an ovoid

gap> planes := AsList( Planes(pq) );; gap> ForAll(planes, p -> Number(ovoid2, x -> x in p) = 1); true [ 336, 28 ]

G (size 1451520) | B(3,2) = O(7,2) ~ C(3,2) = S(6,2) 1 (size 1)

Orbits and composition series

gap> OrbitLengths(stabovoid,points,OnLieVarieties);

gap> DisplayCompositionSeries(stabovoid);

Desargues Find the stabiliser

Outline gap> g := CollineationGroup( pq ); Finite PGammaO(7,3) Geometry gap> points := AsList( Points(pq) );; Examples of gap> hom := ActionHomomorphism(g, points, OnLieVarieties); Problems Features of gap> omega := HomeEnumerator( UnderlyingExternalSet(hom) );; Desargues gap> imgs := Filtered([1..Size(omega)], i -> omega[i] in ovoid2);; Another gap> stab := Stabilizer(Image(hom), imgs, OnSets); example gap> stabovoid := PreImage(hom, stab);; Desargues Find the stabiliser

Outline gap> g := CollineationGroup( pq ); Finite PGammaO(7,3) Geometry gap> points := AsList( Points(pq) );; Examples of gap> hom := ActionHomomorphism(g, points, OnLieVarieties); Problems Features of gap> omega := HomeEnumerator( UnderlyingExternalSet(hom) );; Desargues gap> imgs := Filtered([1..Size(omega)], i -> omega[i] in ovoid2);; Another gap> stab := Stabilizer(Image(hom), imgs, OnSets); example gap> stabovoid := PreImage(hom, stab);;

Orbits and composition series

gap> OrbitLengths(stabovoid,points,OnLieVarieties); [ 336, 28 ] gap> DisplayCompositionSeries(stabovoid); G (size 1451520) | B(3,2) = O(7,2) ~ C(3,2) = S(6,2) 1 (size 1) Desargues For more information and Outline updates... Finite Geometry

Examples of Problems

Features of Desargues

Another example Ghent University (Dept. Pure Math.) website http://cage.ugent.be/geometry/software.php