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