Projective Planes, Finite and Infinite

G. Eric Moorhouse

Department of Mathematics University of Wyoming

Waterloo C&O Seminar 7 September 2020

G. Eric Moorhouse Projective Planes, Finite and Infinite Projective Planes

A projective plane is a point-line incidence structure for which • every pair of distinct points lies on a unique line; • every pair of distinct lines meets in a unique point; and • there exist four points with no three collinear.

Equivalently, a projective plane is a bipartite graph of diameter 3 and girth 6, containing an 8-cycle. Finite case: order n N = n2+n+1 points and lines n+1 points/line n+1 lines/point

G. Eric Moorhouse Projective Planes, Finite and Infinite Projective Planes

A projective plane is a point-line incidence structure for which • every pair of distinct points lies on a unique line; • every pair of distinct lines meets in a unique point; and • there exist four points with no three collinear.

Equivalently, a projective plane is a bipartite graph of diameter 3 and girth 6, containing an 8-cycle. Finite case: order n N = n2+n+1 points and lines n+1 points/line n+1 lines/point

G. Eric Moorhouse Projective Planes, Finite and Infinite Projective Planes

A projective plane is a point-line incidence structure for which • every pair of distinct points lies on a unique line; • every pair of distinct lines meets in a unique point; and • there exist four points with no three collinear.

Equivalently, a projective plane is a bipartite graph of diameter 3 and girth 6, containing an 8-cycle. Finite case: order n N = n2+n+1 points and lines n+1 points/line n+1 lines/point

G. Eric Moorhouse Projective Planes, Finite and Infinite Why work on infinite structures?

I don’t always work on infinite stuff. . .

But when I do, I consider arbitrary cardinalities.

shed light on the finite case, and vice versa source of tractable problems source of interesting problems expand the current interest in finite geometry

G. Eric Moorhouse Projective Planes, Finite and Infinite Why work on infinite structures?

I don’t always work on infinite stuff. . .

But when I do, I consider arbitrary cardinalities.

shed light on the finite case, and vice versa source of tractable problems source of interesting problems expand the current interest in finite geometry

G. Eric Moorhouse Projective Planes, Finite and Infinite Why work on infinite structures?

I don’t always work on infinite stuff. . .

But when I do, I consider arbitrary cardinalities.

shed light on the finite case, and vice versa source of tractable problems source of interesting problems expand the current interest in finite geometry

G. Eric Moorhouse Projective Planes, Finite and Infinite Why work on infinite structures?

I don’t always work on infinite stuff. . .

But when I do, I consider arbitrary cardinalities.

shed light on the finite case, and vice versa source of tractable problems source of interesting problems expand the current interest in finite geometry

G. Eric Moorhouse Projective Planes, Finite and Infinite Why work on infinite structures?

I don’t always work on infinite stuff. . .

But when I do, I consider arbitrary cardinalities.

shed light on the finite case, and vice versa source of tractable problems source of interesting problems expand the current interest in finite geometry

G. Eric Moorhouse Projective Planes, Finite and Infinite Classical Planes

Given a field F, the 1- and 2-dimensional subspaces of F 3 form the Pappian plane over F, of order |F|.

A similar construction over a skewfield F gives the Desarguesian plane over F, of order |F|.

Every finite skewfield is a field (so Desarguesian is equivalent to Pappian).

G. Eric Moorhouse Projective Planes, Finite and Infinite Classical Planes

Given a field F, the 1- and 2-dimensional subspaces of F 3 form the Pappian plane over F, of order |F|.

A similar construction over a skewfield F gives the Desarguesian plane over F, of order |F|.

Every finite skewfield is a field (so Desarguesian is equivalent to Pappian).

G. Eric Moorhouse Projective Planes, Finite and Infinite Classical Planes

Given a field F, the 1- and 2-dimensional subspaces of F 3 form the Pappian plane over F, of order |F|.

A similar construction over a skewfield F gives the Desarguesian plane over F, of order |F|.

Every finite skewfield is a field (so Desarguesian is equivalent to Pappian).

G. Eric Moorhouse Projective Planes, Finite and Infinite Known finite planes of small order Number of planes up to isomorphism (i.e. collineations): number of number of n planes of n planes of order n order n 2 1 16 > 22 3 1 17 > 1 4 1 19 > 1 5 1 23 > 1 7 1 25 > 193 8 1 27 > 13 9 4 29 > 1 11 > 1 · ·· · ·· 13 > 1 49 > 500,000 Problem: Are there any finite planes of non-prime-power order? Problem: Are there any nonclassical planes of prime order?

G. Eric Moorhouse Projective Planes, Finite and Infinite Known finite planes of small order Number of planes up to isomorphism (i.e. collineations): number of number of n planes of n planes of order n order n 2 1 16 > 22 3 1 17 > 1 4 1 19 > 1 5 1 23 > 1 7 1 25 > 193 8 1 27 > 13 9 4 29 > 1 11 > 1 · ·· · ·· 13 > 1 49 > 500,000 Problem: Are there any finite planes of non-prime-power order? Problem: Are there any nonclassical planes of prime order?

G. Eric Moorhouse Projective Planes, Finite and Infinite Known finite planes of small order Number of planes up to isomorphism (i.e. collineations): number of number of n planes of n planes of order n order n 2 1 16 > 22 3 1 17 > 1 4 1 19 > 1 5 1 23 > 1 7 1 25 > 193 8 1 27 > 13 9 4 29 > 1 11 > 1 · ·· · ·· 13 > 1 49 > 500,000 Problem: Are there any finite planes of non-prime-power order? Problem: Are there any nonclassical planes of prime order?

G. Eric Moorhouse Projective Planes, Finite and Infinite Subplanes Generated by Quadrangles

In a projective plane Π, every quadrangle (four points, no three collinear) generates a subplane. Every field F is a field has a prime field K ⊆ F (its unique ∼ minimal subfield). Here K = Fp or Q. In the Pappian plane P2(F), every quadrangle generates a subplane isomorphic to P2(K ). Problem: Are there any nonclassical planes in which every quadrangle generates a proper subplane?

Theorem (Gleason, 1956) Let Π be a projective plane. Then Π is Desarguesian of characteristic 2, iff every quadrangle generates a subplane of order 2. No analogue is known for subplanes of order 3 or larger.

G. Eric Moorhouse Projective Planes, Finite and Infinite Subplanes Generated by Quadrangles

In a projective plane Π, every quadrangle (four points, no three collinear) generates a subplane. Every field F is a field has a prime field K ⊆ F (its unique ∼ minimal subfield). Here K = Fp or Q. In the Pappian plane P2(F), every quadrangle generates a subplane isomorphic to P2(K ). Problem: Are there any nonclassical planes in which every quadrangle generates a proper subplane?

Theorem (Gleason, 1956) Let Π be a projective plane. Then Π is Desarguesian of characteristic 2, iff every quadrangle generates a subplane of order 2. No analogue is known for subplanes of order 3 or larger.

G. Eric Moorhouse Projective Planes, Finite and Infinite Subplanes Generated by Quadrangles

In a projective plane Π, every quadrangle (four points, no three collinear) generates a subplane. Every field F is a field has a prime field K ⊆ F (its unique ∼ minimal subfield). Here K = Fp or Q. In the Pappian plane P2(F), every quadrangle generates a subplane isomorphic to P2(K ). Problem: Are there any nonclassical planes in which every quadrangle generates a proper subplane?

Theorem (Gleason, 1956) Let Π be a projective plane. Then Π is Desarguesian of characteristic 2, iff every quadrangle generates a subplane of order 2. No analogue is known for subplanes of order 3 or larger.

G. Eric Moorhouse Projective Planes, Finite and Infinite Subplanes Generated by Quadrangles

In a projective plane Π, every quadrangle (four points, no three collinear) generates a subplane. Every field F is a field has a prime field K ⊆ F (its unique ∼ minimal subfield). Here K = Fp or Q. In the Pappian plane P2(F), every quadrangle generates a subplane isomorphic to P2(K ). Problem: Are there any nonclassical planes in which every quadrangle generates a proper subplane?

Theorem (Gleason, 1956) Let Π be a projective plane. Then Π is Desarguesian of characteristic 2, iff every quadrangle generates a subplane of order 2. No analogue is known for subplanes of order 3 or larger.

G. Eric Moorhouse Projective Planes, Finite and Infinite Subplanes Generated by Quadrangles

In a projective plane Π, every quadrangle (four points, no three collinear) generates a subplane. Every field F is a field has a prime field K ⊆ F (its unique ∼ minimal subfield). Here K = Fp or Q. In the Pappian plane P2(F), every quadrangle generates a subplane isomorphic to P2(K ). Problem: Are there any nonclassical planes in which every quadrangle generates a proper subplane?

Theorem (Gleason, 1956) Let Π be a projective plane. Then Π is Desarguesian of characteristic 2, iff every quadrangle generates a subplane of order 2. No analogue is known for subplanes of order 3 or larger.

G. Eric Moorhouse Projective Planes, Finite and Infinite Subplanes Generated by Quadrangles

In a projective plane Π, every quadrangle (four points, no three collinear) generates a subplane. Every field F is a field has a prime field K ⊆ F (its unique ∼ minimal subfield). Here K = Fp or Q. In the Pappian plane P2(F), every quadrangle generates a subplane isomorphic to P2(K ). Problem: Are there any nonclassical planes in which every quadrangle generates a proper subplane?

Theorem (Gleason, 1956) Let Π be a projective plane. Then Π is Desarguesian of characteristic 2, iff every quadrangle generates a subplane of order 2. No analogue is known for subplanes of order 3 or larger.

G. Eric Moorhouse Projective Planes, Finite and Infinite Subplanes Generated by Quadrangles

Theorem (Blokhuis, Sziklai, 2000) Let Π be a finite plane of order p2, p prime. Then Π is classical (Pappian) iff every quadrangle generates a subplane of order p.

This was subsequently generalized using CFSG: Theorem (Kantor, Penttila, 2012) Let Π be a finite plane of order m2 in which every quadrangle lies in a unique subplane of order m. Then m is prime and Π is classical (Pappian).

G. Eric Moorhouse Projective Planes, Finite and Infinite Subplanes Generated by Quadrangles

Theorem (Blokhuis, Sziklai, 2000) Let Π be a finite plane of order p2, p prime. Then Π is classical (Pappian) iff every quadrangle generates a subplane of order p.

This was subsequently generalized using CFSG: Theorem (Kantor, Penttila, 2012) Let Π be a finite plane of order m2 in which every quadrangle lies in a unique subplane of order m. Then m is prime and Π is classical (Pappian).

G. Eric Moorhouse Projective Planes, Finite and Infinite Neumann’s Conjecture

Does every finite nonclassical plane have a subplane of order 2?

This does not hold in the infinite case.

Hanna Neumann (1914–1971)

G. Eric Moorhouse Projective Planes, Finite and Infinite Neumann’s Conjecture

Does every finite nonclassical plane have a subplane of order 2?

This does not hold in the infinite case.

Hanna Neumann (1914–1971)

G. Eric Moorhouse Projective Planes, Finite and Infinite The Embedding Problem

A partial linear space is a point-line incidence structure in which any two distinct points are joined by at most one line. (Its incidence graph has no 4-cycles.)

Easy Fact: Every partial linear space embeds in a projective plane.

Problem: Does every finite partial linear space embed in a finite projective plane?

G. Eric Moorhouse Projective Planes, Finite and Infinite The Embedding Problem

A partial linear space is a point-line incidence structure in which any two distinct points are joined by at most one line. (Its incidence graph has no 4-cycles.)

Easy Fact: Every partial linear space embeds in a projective plane.

Problem: Does every finite partial linear space embed in a finite projective plane?

G. Eric Moorhouse Projective Planes, Finite and Infinite The Embedding Problem

A partial linear space is a point-line incidence structure in which any two distinct points are joined by at most one line. (Its incidence graph has no 4-cycles.)

Easy Fact: Every partial linear space embeds in a projective plane.

Problem: Does every finite partial linear space embed in a finite projective plane?

G. Eric Moorhouse Projective Planes, Finite and Infinite ‘Rigid’ Planes

Infinite planes having no nontrivial automorphisms are easily constructed.

Problem: Construct finite planes with no nontrivial automorphisms.

G. Eric Moorhouse Projective Planes, Finite and Infinite ‘Rigid’ Planes

Infinite planes having no nontrivial automorphisms are easily constructed.

Problem: Construct finite planes with no nontrivial automorphisms.

G. Eric Moorhouse Projective Planes, Finite and Infinite Doubly Transitive Planes

Theorem (Ostrom, Dembowski, Wagner, 1959, 1965) Let Π be a finite projective plane Π. Then Π has an automorphism group which is doubly transitive on the lines, iff Π is Pappian.

N.B. This result precedes CFSG.

In the infinite case, there exist non-classical planes having automorphism groups which are doubly transitive on points (even transitive on ordered quadrangles).

G. Eric Moorhouse Projective Planes, Finite and Infinite Doubly Transitive Planes

Theorem (Ostrom, Dembowski, Wagner, 1959, 1965) Let Π be a finite projective plane Π. Then Π has an automorphism group which is doubly transitive on the lines, iff Π is Pappian.

N.B. This result precedes CFSG.

In the infinite case, there exist non-classical planes having automorphism groups which are doubly transitive on points (even transitive on ordered quadrangles).

G. Eric Moorhouse Projective Planes, Finite and Infinite Dembowski-Hughes-Parker Theorem a.k.a. Block’s Lemma

Theorem (Dembowski, Hughes, Parker, c. 1950’s)) Let G be an automorphism group of a finite projective plane Π. Then G has equally many point and line orbits.

Idea of Proof (Brauer, 1941): G has two permutation representations of degree N = n2+n+1, one on points and one on lines. In general they are not permutation-equivalent. But they are linearly equivalent (they are intertwined by the adjacency matrix of Π.

What about in the infinite case?

G. Eric Moorhouse Projective Planes, Finite and Infinite Dembowski-Hughes-Parker Theorem a.k.a. Block’s Lemma

Theorem (Dembowski, Hughes, Parker, c. 1950’s)) Let G be an automorphism group of a finite projective plane Π. Then G has equally many point and line orbits.

Idea of Proof (Brauer, 1941): G has two permutation representations of degree N = n2+n+1, one on points and one on lines. In general they are not permutation-equivalent. But they are linearly equivalent (they are intertwined by the adjacency matrix of Π.

What about in the infinite case?

G. Eric Moorhouse Projective Planes, Finite and Infinite Dembowski-Hughes-Parker Theorem a.k.a. Block’s Lemma

Theorem (Dembowski, Hughes, Parker, c. 1950’s)) Let G be an automorphism group of a finite projective plane Π. Then G has equally many point and line orbits.

Idea of Proof (Brauer, 1941): G has two permutation representations of degree N = n2+n+1, one on points and one on lines. In general they are not permutation-equivalent. But they are linearly equivalent (they are intertwined by the adjacency matrix of Π.

What about in the infinite case?

G. Eric Moorhouse Projective Planes, Finite and Infinite Dembowski-Hughes-Parker Theorem a.k.a. Block’s Lemma

Theorem (Dembowski, Hughes, Parker, c. 1950’s)) Let G be an automorphism group of a finite projective plane Π. Then G has equally many point and line orbits.

Idea of Proof (Brauer, 1941): G has two permutation representations of degree N = n2+n+1, one on points and one on lines. In general they are not permutation-equivalent. But they are linearly equivalent (they are intertwined by the adjacency matrix of Π.

What about in the infinite case?

G. Eric Moorhouse Projective Planes, Finite and Infinite Dembowski-Hughes-Parker Theorem a.k.a. Block’s Lemma

Theorem (Dembowski, Hughes, Parker, c. 1950’s)) Let G be an automorphism group of a finite projective plane Π. Then G has equally many point and line orbits.

Idea of Proof (Brauer, 1941): G has two permutation representations of degree N = n2+n+1, one on points and one on lines. In general they are not permutation-equivalent. But they are linearly equivalent (they are intertwined by the adjacency matrix of Π.

What about in the infinite case?

G. Eric Moorhouse Projective Planes, Finite and Infinite Arbitrarily Many Orbits on Points and Lines

Theorem (M. and Penttila, 2012) There exists a Desarguesian plane Π admitting a group G < Aut Π having two orbits on points, and more than two orbits on lines. Given any two nonempty sets A and B, there exists a projective plane Π admitting a group G 6 Aut Π having exactly |A| orbits on points and |B| orbits on lines.

Problem: In the Desarguesian case, how many orbits are possible on points and lines? Problem: Prove a stronger result by realizing G as the full automorphism (collineation) group of Π. Problem: For which classes of groups must the number of point and line orbits be the same? How about for amenable groups?

G. Eric Moorhouse Projective Planes, Finite and Infinite Arbitrarily Many Orbits on Points and Lines

Theorem (M. and Penttila, 2012) There exists a Desarguesian plane Π admitting a group G < Aut Π having two orbits on points, and more than two orbits on lines. Given any two nonempty sets A and B, there exists a projective plane Π admitting a group G 6 Aut Π having exactly |A| orbits on points and |B| orbits on lines.

Problem: In the Desarguesian case, how many orbits are possible on points and lines? Problem: Prove a stronger result by realizing G as the full automorphism (collineation) group of Π. Problem: For which classes of groups must the number of point and line orbits be the same? How about for amenable groups?

G. Eric Moorhouse Projective Planes, Finite and Infinite Arbitrarily Many Orbits on Points and Lines

Theorem (M. and Penttila, 2012) There exists a Desarguesian plane Π admitting a group G < Aut Π having two orbits on points, and more than two orbits on lines. Given any two nonempty sets A and B, there exists a projective plane Π admitting a group G 6 Aut Π having exactly |A| orbits on points and |B| orbits on lines.

Problem: In the Desarguesian case, how many orbits are possible on points and lines? Problem: Prove a stronger result by realizing G as the full automorphism (collineation) group of Π. Problem: For which classes of groups must the number of point and line orbits be the same? How about for amenable groups?

G. Eric Moorhouse Projective Planes, Finite and Infinite Arbitrarily Many Orbits on Points and Lines

Theorem (M. and Penttila, 2012) There exists a Desarguesian plane Π admitting a group G < Aut Π having two orbits on points, and more than two orbits on lines. Given any two nonempty sets A and B, there exists a projective plane Π admitting a group G 6 Aut Π having exactly |A| orbits on points and |B| orbits on lines.

Problem: In the Desarguesian case, how many orbits are possible on points and lines? Problem: Prove a stronger result by realizing G as the full automorphism (collineation) group of Π. Problem: For which classes of groups must the number of point and line orbits be the same? How about for amenable groups?

G. Eric Moorhouse Projective Planes, Finite and Infinite Arbitrarily Many Orbits on Points and Lines

Theorem (M. and Penttila, 2012) There exists a Desarguesian plane Π admitting a group G < Aut Π having two orbits on points, and more than two orbits on lines. Given any two nonempty sets A and B, there exists a projective plane Π admitting a group G 6 Aut Π having exactly |A| orbits on points and |B| orbits on lines.

Problem: In the Desarguesian case, how many orbits are possible on points and lines? Problem: Prove a stronger result by realizing G as the full automorphism (collineation) group of Π. Problem: For which classes of groups must the number of point and line orbits be the same? How about for amenable groups?

G. Eric Moorhouse Projective Planes, Finite and Infinite Counting Orbits on n-tuples of Points

Let F be an infinite field. The Pappian plane Π = P2(F) has ∼ automorphism group G = PΓL3(F).

G is transitive on points of Π.

G has one orbit on pairs (P, Q) of distinct points.

G has two orbits on triples (P, Q, R) of distinct points (collinear triples; noncollinear triples).

G has infinitely many orbits on 4-tuples (P, Q, R, S) of distinct points: quadrangles (one orbit), three collinear (one orbit), four collinear (infinitely many orbits).

Problem: Does there exist an infinite projective plane whose automorphism group has only finitely many orbits on 4-tuples of points?

G. Eric Moorhouse Projective Planes, Finite and Infinite Counting Orbits on n-tuples of Points

Let F be an infinite field. The Pappian plane Π = P2(F) has ∼ automorphism group G = PΓL3(F).

G is transitive on points of Π.

G has one orbit on pairs (P, Q) of distinct points.

G has two orbits on triples (P, Q, R) of distinct points (collinear triples; noncollinear triples).

G has infinitely many orbits on 4-tuples (P, Q, R, S) of distinct points: quadrangles (one orbit), three collinear (one orbit), four collinear (infinitely many orbits).

Problem: Does there exist an infinite projective plane whose automorphism group has only finitely many orbits on 4-tuples of points?

G. Eric Moorhouse Projective Planes, Finite and Infinite Counting Orbits on n-tuples of Points

Let F be an infinite field. The Pappian plane Π = P2(F) has ∼ automorphism group G = PΓL3(F).

G is transitive on points of Π.

G has one orbit on pairs (P, Q) of distinct points.

G has two orbits on triples (P, Q, R) of distinct points (collinear triples; noncollinear triples).

G has infinitely many orbits on 4-tuples (P, Q, R, S) of distinct points: quadrangles (one orbit), three collinear (one orbit), four collinear (infinitely many orbits).

Problem: Does there exist an infinite projective plane whose automorphism group has only finitely many orbits on 4-tuples of points?

G. Eric Moorhouse Projective Planes, Finite and Infinite Counting Orbits on n-tuples of Points

Let F be an infinite field. The Pappian plane Π = P2(F) has ∼ automorphism group G = PΓL3(F).

G is transitive on points of Π.

G has one orbit on pairs (P, Q) of distinct points.

G has two orbits on triples (P, Q, R) of distinct points (collinear triples; noncollinear triples).

G has infinitely many orbits on 4-tuples (P, Q, R, S) of distinct points: quadrangles (one orbit), three collinear (one orbit), four collinear (infinitely many orbits).

Problem: Does there exist an infinite projective plane whose automorphism group has only finitely many orbits on 4-tuples of points?

G. Eric Moorhouse Projective Planes, Finite and Infinite Counting Orbits on n-tuples of Points

Let F be an infinite field. The Pappian plane Π = P2(F) has ∼ automorphism group G = PΓL3(F).

G is transitive on points of Π.

G has one orbit on pairs (P, Q) of distinct points.

G has two orbits on triples (P, Q, R) of distinct points (collinear triples; noncollinear triples).

G has infinitely many orbits on 4-tuples (P, Q, R, S) of distinct points: quadrangles (one orbit), three collinear (one orbit), four collinear (infinitely many orbits).

Problem: Does there exist an infinite projective plane whose automorphism group has only finitely many orbits on 4-tuples of points?

G. Eric Moorhouse Projective Planes, Finite and Infinite Counting Orbits on n-tuples of Points

Let F be an infinite field. The Pappian plane Π = P2(F) has ∼ automorphism group G = PΓL3(F).

G is transitive on points of Π.

G has one orbit on pairs (P, Q) of distinct points.

G has two orbits on triples (P, Q, R) of distinct points (collinear triples; noncollinear triples).

G has infinitely many orbits on 4-tuples (P, Q, R, S) of distinct points: quadrangles (one orbit), three collinear (one orbit), four collinear (infinitely many orbits).

Problem: Does there exist an infinite projective plane whose automorphism group has only finitely many orbits on 4-tuples of points?

G. Eric Moorhouse Projective Planes, Finite and Infinite Finitely Many Orbits on n-tuples of Points

Problem: Does there exist an infinite plane whose automorphism group has only finitely many orbits on n-tuples of points? (Or n-tuples of distinct points. Or n-sets of points. Or n-sets of points and lines. Same problem in each case.) An affirmative answer is unlikely as it would imply: Without loss of generality, Π is countably infinite; and G fixes pointwise a finite subplane of Π. There would exist infinitely many finite nonclassical planes in which every quadrangle generates a proper subplane. Same thing with n-tuples of points instead of 4-tuples. G is uncountable. But the group of projectivities is countable. Surely this cannot happen?!

G. Eric Moorhouse Projective Planes, Finite and Infinite Finitely Many Orbits on n-tuples of Points

Problem: Does there exist an infinite plane whose automorphism group has only finitely many orbits on n-tuples of points? (Or n-tuples of distinct points. Or n-sets of points. Or n-sets of points and lines. Same problem in each case.) An affirmative answer is unlikely as it would imply: Without loss of generality, Π is countably infinite; and G fixes pointwise a finite subplane of Π. There would exist infinitely many finite nonclassical planes in which every quadrangle generates a proper subplane. Same thing with n-tuples of points instead of 4-tuples. G is uncountable. But the group of projectivities is countable. Surely this cannot happen?!

G. Eric Moorhouse Projective Planes, Finite and Infinite Finitely Many Orbits on n-tuples of Points

Problem: Does there exist an infinite plane whose automorphism group has only finitely many orbits on n-tuples of points? (Or n-tuples of distinct points. Or n-sets of points. Or n-sets of points and lines. Same problem in each case.) An affirmative answer is unlikely as it would imply: Without loss of generality, Π is countably infinite; and G fixes pointwise a finite subplane of Π. There would exist infinitely many finite nonclassical planes in which every quadrangle generates a proper subplane. Same thing with n-tuples of points instead of 4-tuples. G is uncountable. But the group of projectivities is countable. Surely this cannot happen?!

G. Eric Moorhouse Projective Planes, Finite and Infinite Finitely Many Orbits on n-tuples of Points

Problem: Does there exist an infinite plane whose automorphism group has only finitely many orbits on n-tuples of points? (Or n-tuples of distinct points. Or n-sets of points. Or n-sets of points and lines. Same problem in each case.) An affirmative answer is unlikely as it would imply: Without loss of generality, Π is countably infinite; and G fixes pointwise a finite subplane of Π. There would exist infinitely many finite nonclassical planes in which every quadrangle generates a proper subplane. Same thing with n-tuples of points instead of 4-tuples. G is uncountable. But the group of projectivities is countable. Surely this cannot happen?!

G. Eric Moorhouse Projective Planes, Finite and Infinite Finitely Many Orbits on n-tuples of Points

Problem: Does there exist an infinite plane whose automorphism group has only finitely many orbits on n-tuples of points? (Or n-tuples of distinct points. Or n-sets of points. Or n-sets of points and lines. Same problem in each case.) An affirmative answer is unlikely as it would imply: Without loss of generality, Π is countably infinite; and G fixes pointwise a finite subplane of Π. There would exist infinitely many finite nonclassical planes in which every quadrangle generates a proper subplane. Same thing with n-tuples of points instead of 4-tuples. G is uncountable. But the group of projectivities is countable. Surely this cannot happen?!

G. Eric Moorhouse Projective Planes, Finite and Infinite Finitely Many Orbits on n-tuples of Points

Problem: Does there exist an infinite plane whose automorphism group has only finitely many orbits on n-tuples of points? (Or n-tuples of distinct points. Or n-sets of points. Or n-sets of points and lines. Same problem in each case.) An affirmative answer is unlikely as it would imply: Without loss of generality, Π is countably infinite; and G fixes pointwise a finite subplane of Π. There would exist infinitely many finite nonclassical planes in which every quadrangle generates a proper subplane. Same thing with n-tuples of points instead of 4-tuples. G is uncountable. But the group of projectivities is countable. Surely this cannot happen?!

G. Eric Moorhouse Projective Planes, Finite and Infinite Thank You!

Questions?

G. Eric Moorhouse Projective Planes, Finite and Infinite