On Projective Planes by CAFER C¸ALIS¸KAN
A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
Florida Atlantic University Boca Raton, Florida May 2010 Copyright by CAFER C¸ALIS¸KAN 2010
ii
Acknowledgements
First of all, I would like to express my gratitude to my advisor Spyros S. Magliveras. Without his expertise, never-ending support and guidance I could not have written this thesis. I also thank the members of my committee – Professors Ron C. Mullin, Lee Klingler and Rainer Steinwandt – who were more than generous with their expertise and precious time. A very special thanks goes to Heinrich Niederhausen, who has supported me in many ways. I would also like to use this opportunity to thank the faculty members and my colleagues at the Department of Mathematical Sciences at Florida Atlantic University. I would like to thank my family for all the support they provided and in particular, I must acknowledge my dearest, Burcu. Without her love and understanding, I would not have finished this manuscript.
Boca Raton, Florida Cafer C¸alıs¸kan March 30th, 2010
iv Abstract
Author: CAFER C¸ALIS¸KAN Title: On Projective Planes Institution: Florida Atlantic University Dissertation Advisor: Dr. Spyros S. Magliveras Degree: Doctor of Philosophy Year: 2010
This work was motivated by the well-known question: “Does there exist a non- desarguesian projective plane of prime order?” For a prime p < 11, there is only the pap- pian plane of order p. Hence, such planes are indeed desarguesian. Thus, it is of interest to examine whether there are non-desarguesian planes of order 11. A suggestion by As- cher Wagner in 1985 was made to Spyros S. Magliveras: “Begin with a non-desarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations, and check whether one of these is non-desarguesian.” In this manuscript we use a group-theoretic methodology to determine the subplane structures of some non-desarguesian planes. In particular, we determine orbit representatives of all proper Q-subplanes both of a Veblen-Wedderburn (VW) plane Π of order 121 and of the Hughes plane Σ of order 121, under their full collineation groups. In Π, there are 13 orbits of Baer subplanes, all of which are desarguesian, and approximately 3000 orbits of Fano subplanes. In Σ, there are 8 orbits of Baer subplanes, all of which are desarguesian, 2 orbits of subplanes of order 3, and at most 408, 075 distinct Fano subplanes. In addition to the above results, we also study the subplane structures of some non-desarguesian planes, such as the Hall plane of
v order 25, the Hughes planes of order 25 and 49, and the Figueora planes of order 27 and 125. A surprising discovery by L. Puccio and M. J. de Resmini was the existence of a plane of order 3 in the Hughes plane of order 25. We generalize this result, showing that there are subplanes of order 3 in the Hughes planes of order q2, where q is a prime power and q ≡ 5 (mod 6). Furthermore, we analyze the structure of the full collineation groups of certain Veblen- Wedderburn (VW) planes of orders 25, 49 and 121, and discuss how to recover the planes from their collineation groups.
vi Dedication
To my love Burcu Sayım C¸alıs¸kan.
“Science is the most genuine guide in life.” – Mustafa Kemal (1881 – 193∞) On Projective Planes
List of Tables...... ix
List of Figures ...... x
1 Introduction ...... 1
2 Preliminaries ...... 4
3 Some non-desarguesian planes ...... 21
3.1 A VW plane π ...... 21 3.2 The Hall plane ς ...... 22 3.3 The Hughes and Figueroa planes ...... 23 3.3.1 The Hughes plane σ ...... 23 3.3.2 The Figueroa plane ξ ...... 24
4 Subplane structures of planes ...... 26
4.1 Closures of quadrangles and their orbits ...... 26 4.2 Subplanes of some non-desarguesian planes of small order ...... 29 4.3 Some non-desarguesian planes of order 121 ...... 31 4.3.1 π of order 121 (Π)...... 32 4.3.2 σ of order 121 (Σ) ...... 35 4.4 Subplanes of order 3 in the Hughes Planes ...... 37 vii 4.4.1 Case: q ≡ 5 (mod 12) ...... 39 4.4.2 Case: q ≡ 11 (mod 12)...... 40 4.5 Further substructures of the Hughes Planes...... 42
5 Reconstructing planes from their groups...... 45
5.1 The structure of Gπ ...... 45
5.2 Reconstruction from Gπ ...... 47 5.2.1 Counting Principle...... 47 5.2.2 Reconstruction ...... 48
List of Notations ...... 51
Appendices ...... 52
A VW plane of order 25 (α)...... 52
B VW plane of order 49 (β)...... 53
C VW plane of order 121 (Π)...... 55
Bibliography ...... 57
viii List of Tables
2.1 Projective planes of small order ...... 16
ix List of Figures
2.1 Desargues’ configuration ...... 8 2.2 Pappus’ configuration ...... 9 2.3 Coordinatization of a projective plane ...... 16
4.1 Finding the orbit representatives of the point-sets of planes in Dt...... 28
4.2 The full collineation group GΠ ...... 32
4.3 The full collineation group GΣ ...... 36
5.1 The full collineation group Gπ ...... 46 5.2 Counting principle ...... 47 5.3 Representing certain points and lines by involutions ...... 48 5.4 Determining lines of Type I by certain group elements of order p ...... 49 5.5 Determining lines of Type I by certain group elements of order p ...... 50
x Chapter 1
Introduction
A question that is still outstanding in finite geometry is whether all finite projective planes of prime order are desarguesian. For a prime p < 11, there is only the pappian plane of order p [3]. Hence, such planes are indeed desarguesian. Thus, it is still of interest to examine whether there are non-desarguesian planes of order 11. A suggestion by Ascher Wagner in 1985 was made to Spyros S. Magliveras. It is to begin with a non-desarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations and check whether one of these is non-desarguesian. In 1985, the computational complexity of the problem was certainly high enough to make the problem rather intractable, even for k = 2. Since there are planes which are not generated by quadrangles, we reserve the term Q-plane (Q-subplane) for a plane (sub- plane) generated by a quadrangle. In this manuscript, we determine all proper subplanes of some well-known non-desarguesian planes of small order and all proper Q-subplanes both of a Veblen-Wedderburn (VW) plane Π of order 121 and the Hughes plane Σ of order 121 up to collineations, paying special attention to the Baer subplanes of Π and Σ. In particular, there are 19 orbits of Fano subplanes and 6 orbits of subplanes of order 5 in the Hall plane of order 25. The Hughes plane of order 25 has 56 orbits of Fano subplanes, only 1 orbit of subplanes of order 3 and 5 orbits of subplanes of order 5. There are 3524 orbits of Fano subplanes and 59 orbits of subplanes of order 3 in the Figueroa plane of 1 order 27. None of these planes, namely the Hall plane of order 25, the Hughes plane of order 25 or the Figueroa plane of order 27, have a subplane of order 4. We also show that the Hughes plane of order 49 does not have subplanes of order 3. Moreover, the VW plane Π of order 121 contains only Fano subplanes and desargue- sian Baer subplanes as proper Q-subplanes. In Π, there are 13 orbits of subplanes of order 11 all of which are desarguesian and approximately 3000 orbits of Fano subplanes. Subplanes of order 4 or 8 may still exist, since these projective planes are not Q-planes. In a desarguesian plane of order 9, every quadrangle generates a subplane of order 3. However, Π does not have a subplane of order 3. This implies that Π does not have a desarguesian plane of order 9 either. In the Hughes plane Σ of order 121, the proper Q-subplanes are Fano subplanes, subplanes of order 3 and desarguesian Baer subplanes. There are 8 orbits of subplanes of order 11 all of which are desarguesian, 2 orbits of sub- planes of order 3 and at most 408, 075 distinct Fano subplanes in Σ. The existence of subplanes of order 3 is interesting, although not totally surprising in view of L. Puccio and M. J. de Resmini’s discovery of subplanes of order 3 in the Hughes plane of order 25 [30]. We presently show that every Hughes plane of order q2, where q is a prime power and q ≡ 5 (mod 6), has subplanes of order 3. In Σ, there may exist subplanes of order 4 or 8, and desarguesian planes of order 9 as proper subplanes. Examining the closures of all possible representatives of orbits of 5-arcs under the full collineation group GΠ (GΣ) will answer whether subplanes of order 4 or 8 (4, 8 or 9) can occur in Π (Σ). Although we now feel it is very unlikely, a non-desarguesian subplane of order 11 may still exist in Π or Σ as such planes are not necessarily Q-planes. Furthermore, we determine the closures of some randomly chosen quadrangles in the Figueroa plane Ξ of order 125. It results in that Ξ has Fano subplanes, subplanes of order 3 and subplanes of order 5. “Is it always possible to construct projective planes from their collineation groups?” In [7], Brown shows how to construct both the Hughes plane of order q2 and the Figueroa
2 plane of order q3, q is an odd prime power, from the linear group GL(3, q). In our study, we discuss a reconstruction method for a particular non-desarguesian VW plane of order p2, p = 5, 7 or 11, from its collineation group. It has been long recognized that incidence structures such as projective planes can be used to produce codes. Most of these applications in Coding Theory are using either the incidence matrices of the desarguesian planes or some substructures of general projective planes. See [2], [8] for some examples. Incidence structures are also used to construct au- thentication codes in the area of information security. For instance, it was first introduced in [14] how to use generalized quadrangles to construct authentication codes. See [28] for a comprehensive introduction on authentication codes arising from incidence struc- tures. Desarguesian planes are all known as finite fields are all known. However, there is still little known about the non-desarguesian planes and their substructures. Understand- ing the substructures of these planes may result in some improvements in the existing applications or new application areas. In Chapter 2, we introduce some notation, definitions and preliminaries. In Chapter 3, we show how to construct some classes of non-desarguesian projective planes, namely a particular VW plane, the Hall planes, Hughes planes and Figueroa planes. In Chapter 4, we present a methodology for determining a complete set of orbit representatives of Q-subplanes under the action of the full collineation groups. We computationally analyze the subplane structures of some non-desarguesian planes, and then show the existence of subplanes of order 3 in the Hughes planes of order q2, where q is a prime power and q ≡ 5 (mod 6). We further show that there exist finite partial linear spaces which cannot embed in any Hughes plane. In Chapter 5, we discuss a reconstruction method for a VW plane of order p2, p = 5, 7 or 11, using its collineation group.
3 Chapter 2
Preliminaries
In what follows we deal with simple incidence structures involving three sets: a set of points P, a set of lines L and a relation I ⊆ P × L called an incidence relation. If (A, a) ∈ I, then we write A I a and say that point A is on line a, or that a contains A.
An incidence structure (P, L , I) is called a partial linear space (linear space) if it satisfies the following: (i) any line contains at least two points, and (ii) any two points are on at most (exactly) one line.
A set S ⊆ P is said to be collinear if and only if there exists a line a ∈ L such that A I a for all A ∈ S. By a quadrangle we mean a set of four points no three of which are collinear. We say that two lines a and b intersect, if there exists C ∈ P such that C I a and C I b. An affine plane A is a linear space in which: (i) any point A not on a line b is on exactly one line a not intersecting b, and (ii) there is a set of three points which are not collinear. A projective plane α is a linear space in which: (i) any two lines intersect, and (ii) there exists a quadrangle.
Let α = (Pα, Lα, Iα) be a projective plane with the set of points Pα, set of lines
Lα and incidence relation Iα ⊆ Pα × Lα, where |Pα| < ∞. Then we say α is finite. We consider finite projective planes throughout this dissertation. One can easily show that given a finite projective plane α there is a natural number n ≥ 2 such that any line 4 a ∈ Lα has exactly n + 1 points on it and any point A ∈ Pα lies on exactly n + 1 lines.
2 An elementary argument shows that |Pα| = |Lα| = n + n + 1. We call n the order of a projective plane α. The Fano plane is the finite projective plane with the smallest order n = 2.
0 0 0 0 0 Let us assume that P = Lα, L = Pα, and I ⊆ P × L is an incidence relation such that a I0 A if and only if A I a. Then α0 = (P0, L 0, I0) is a projective plane called the dual plane of α.
Theorem 2.1. (The principle of duality, [1, p.13]) Let R be any theorem about projective planes. If R0 is the statement obtained by interchanging the words “points” and “lines”, then R0 is a theorem about the corresponding dual planes. Hence, R0 is also a theorem about projective planes.
We denote by (a) the set of points incident with a ∈ Lα and by (A) the set of lines incident with A ∈ Pα. If A, B ∈ Pα and A 6= B, we denote by AB the line in Lα incident with A and B. Symmetrically, if a, b ∈ Lα, a 6= b, ab denotes the point in Pα incident with a and b. We denote the collection of all quadrangles of a plane α by Qα.A subplane β = (Pβ, Lβ, Iβ) of α = (Pα, Lα, Iα) is a projective plane where Pβ ⊆ Pα,
Lβ ⊆ Lα, and Iβ = Iα | Pβ × Lβ. If Pβ ⊂ Pα, then β is said to be proper. Even though there is no known example of a subplane of order m in a projective plane of order n, when m2 + m = n, the following lemma is important in determining the order of subplanes in a given finite projective plane.
Lemma 2.1. (Bruck, [19, p.81]) Let α be a plane of order n and β a proper subplane of α of order m. Then m2 = n or m2 + m ≤ n.
Let α be a plane and S ⊆ Pα. The closure S of S is defined recursively as follows: (i) determine the set L of lines through all pairs of points of S, (ii) determine the set Sˆ of intersection points of the lines in L, if S 6= Sˆ replace S by Sˆ and go to (i) otherwise 5 set S = (S,Lˆ ). If S contains a quadrangle, S is the smallest subplane β of α such that
S ⊆ Pβ. We denote by PS the point set of the closure S. We say that a plane α is said to be a Q-plane if there is a quadrangle Q whose closure is α. If α is a Q-plane and Q ∈ Qα a quadrangle whose closure gives rise to α, we say that α is generated by Q. A Baer subset B is a subset of points and lines of α such that every point in α lies on a line of B and every line of α meets B in at least one point. If B is a subplane, it is said to be a Baer subplane of α.
Lemma 2.2. [3, p.46] If α is a plane of order n and β a subplane of α of order m, then β is a proper Baer subplane of α if and only if n = m2.
A k-arc Ω of α is a subset of Pα of size k with no three points collinear. Let a, b and c be three lines of α with |(a) ∩ Ω| = 0, |(b) ∩ Ω| = 1 and |(c) ∩ Ω| = 2; Then we call a, b and c an external line, a tangent, and a chord of Ω, respectively. A k-arc Ω is said to be complete if it is not contained in a (k + 1)-arc. In a plane of order n, the size of a k-arc Ω is at most n + 1 if n is odd, and n + 2 if n is even. We call Ω an oval, or a hyperoval, if |Ω| = n + 1, or |Ω| = n + 2, respectively. If Ω is a hyperoval, then there is a unique tangent line to each point of Ω, and these tangents are all passing through a point called the nucleus of Ω.
Let α = (Pα, Lα, Iα) and β = (Pβ, Lβ, Iβ). An isomorphism from α to β is an ordered pair Φ = (Φ1, Φ2) where Φ1 : Pα → Pβ and Φ2 : Lα → Lβ are bijections such that A Iα a if and only if Φ1(A) Iβ Φ2(a). We say α and β are isomorphic if there is such an isomorphism. A collineation ρ of α is an isomorphism of α onto itself. If ρ = (ρ1, ρ2) is a collineation of α, then ρ(A), A ∈ Pα, and ρ(a), a ∈ Lα, denote ρ1(A) and ρ2(a), respectively. The set of all collineations of α forms a group under composition, called the full collineation group Gα of α. A collineation ρ is called a planar collineation (Baer collineation) if F(ρ) results in a subplane (Baer subplane) of α, where F(ρ) is the set of all points and lines fixed by ρ.A correlation % of α is a one-to-one mapping of the points 6 of α onto the lines of α and the lines of α onto the points of α such that A I a if and only if %(a) I %(A). Note that the product of two correlations of α is a collineation of α. Moreover, a correlation % is said to be a polarity if % is of order 2.
Proposition 2.1. [19, p.94] Let ρ be a non-identity collineation of a projective plane α.
If ρ fixes a line a ∈ Lα pointwise, then there is a point A ∈ Pα fixed linewise by ρ. Furthermore, ρ fixes no other point or line.
A collineation ρ is called an (A, a)-central collineation, or an (A, a)-perspectivity, if A is fixed linewise and a pointwise by ρ. The point A and line a are called the center and the axis of ρ, respectively. An (A, a)-central collineation ρ of α is said to be an elation if AIαa, and a homology otherwise. The set of all (A, a)-central collineations forms a subgroup in Gα.
A projective plane α is said to be (A, a)-transitive if, for any distinct points B0 and
B1 such that B0A = B1A, B0,B1 ∈/ (a) and B0 6= A 6= B1, there is a (A, a)-central collineation ρ ∈ Gα such that ρ(B0) = B1. Moreover, if α is (A, a)-transitive for all points on a line a0, then α is said to be (a0, a)-transitive. Symmetrically, if α is (A, a)- transitive for all lines incident with a point A0, then α is said to be (A, A0)-transitive.
If a ∈ Lα (A ∈ Pα) is a line (point) such that α is (a, a)-transitive ((A, A)-transitive ), then a (A) is called a translation line (translation point) of α and the plane α is called a translation plane (dual translation plane) with respect to a (A). See [21] for details.
Proposition 2.2. [19, p.100] If a plane α is (A, a)-transitive and (B, a)-transitive where
A, B ∈ Pα, A 6= B, A Iα a and B Iα a, then a is a translation line of α.
Let T = T (Gα) = {(A, a) | Gα is (A, a)-transitive}. The Lenz-Barlotti classification of projective planes is based on the possibilities for T . See [13] for more details about the Lenz-Barlotti classification.
Two non-degenerate triangles (3-arcs) A1B1C1 and A2B2C2 are said to be perspec- tive with respect to a point D if corresponding vertices are on lines through D, i.e. that 7 DA1A2, DB1B2, DC1C2 are lines, and perspective with respect to a line d if correspond- ing sides intersect in points of d. In plane Euclidean geometry the following theorem holds:
Theorem 2.2. (Desargues’ theorem, [9, p.15]) Two triangles are perspective with respect to a point if and only if they are perspective with respect to a line.
Figure 2.1: Desargues’ configuration
If D is incident with d (See Figure 2.1), the configuration is called the little Desargues’ configuration. A projective plane is said to be (D, d)-desarguesian if, for each pair of non- degenerate triangles ∆i (i = 1, 2) with vertices Ai,Bi,Ci and opposite sides ai, bi, ci such that (i) ∆1 and ∆2 are perspective with respect to D and (ii) a1a2, b1b2 are on d, it follows that c1c2 is on d [19].
Proposition 2.3. (Ostrom, [19, p.107]) A finite projective plane contains a Desargues’ configuration.
It is known that the existence of an (A, a)-central collineation guarantees the exis- tence of a Desargues’ configuration. However, it is still of interest whether the exis- tence of a Desargues’ configuration implies the existence of a central collineation. More- over, it is not known whether there is a finite projective plane admitting no non-identity collineations. 8 Proposition 2.4. (Baer, [19, p.108]) A projective plane α is (A, a)-transitive if and only if α is (A, a)-desarguesian.
In a general finite projective plane, there may exist two triangles which are perspective with respect to a point, but which fail to be perspective with respect to a line (or the dual).
A plane α is said to be desarguesian if it is (A, a)-desarguesian for all A ∈ Pα and a ∈ Lα. Otherwise, it is called non-desarguesian.
Figure 2.2: Pappus’ configuration
Theorem 2.3. (Pappus’ theorem, [9, p.16]) Let A1,A2,A3 and B1,B2,B3 be distinct points on distinct lines a and b, respectively, not intersecting each other at any of these points. Then the intersection points C1,C2 and C3, as described in Figure 2.2, are collinear.
Although Pappus’ theorem holds in plane Euclidean geometry, it does not necessarily hold for projective planes in general. A plane α is said to be pappian if Pappus’ theorem holds. The following theorem is true for any pappian plane. However, the converse is not true for infinite projective planes in general.
Theorem 2.4. [3, p.58] A pappian projective plane α is desarguesian.
Among various methods for constructing projective planes, the following is for con- structing the projective planes called the field planes denoted by PG(2, Fq), or PG(2, q). 9 Let Fq be a finite field of order q, where q ≥ 2 is a prime power, and V = {(x, y, z) | x, y, z ∈
Fq} a 3-dimensional (left) vector space over Fq. Then we define the set of points (set of lines) of PG(2, Fq) to be the set of all equivalence classes of elements of V \{(0, 0, 0)}, under the equivalence (x, y, z) ∼ k(x, y, z) ([a, b, c] ∼ k[a, b, c]) for k ∈ Fq \{0}. The incidence relation for PG(2, Fq) is defined as follows: Point (x, y, z) is incident with line
[a, b, c] if and only if ax + by + cz = 0. It is known that α = PG(2, Fq) is a desargue- sian projective plane of order q and satisfies the little Desargues’ axiom with any pair of triangles perspective from any point D ∈ Pα, and any axis d ∈ Lα. Moreover, if
r s q = p , where p is a prime, then α = PG(2, Fq) has a subplane of order p if and only if s | r [19]. Furthermore, the full collineation group Gα is PΓL(3, q), which is the pro- jective linear group PGL(3, q) extended by the field automorphism. It is also known that PSL(3, q) < PΓL(3, q). We also construct projective planes by extending affine planes. Every affine plane can be extended to a projective plane by adding a line called the “line at infinity” or “line at ∞” as follows: Let A be an affine plane. We say two lines a0 and a1 to be parallel if a0 = a1 or (a0) ∩ (a1) = ∅. It can be shown easily that parallelism of lines in A is an equivalence relation. We denote by [a] the equivalence class of a ∈ LA. For each equivalence class [a] we assign a point A[a], which is also called a point at ∞, such that the lines in [a] intersect at A[a]. If S = {A[a] | [a] is an equivalence class in A}, then S is collinear. We adjoin LA a new line a∞ called the “line at infinity” where (a∞) = S. Then the incidence structure with the “line at ∞” and “points at ∞” as described above is a projective plane. Hence, we have the following theorem:
Theorem 2.5. Any affine plane A is embeddable in a projective plane α so that the points of the affine plane are in Pα \ (a∞).
An affine plane A is said to be desarguesian if and only if the corresponding projective plane is desarguesian. The so called Moulton planes are examples of non-desarguesian 10 affine planes. See [25] for details of Moulton planes. A latin square L of order n is an n × n array of distinct symbols from an alphabet
Ψ = {ψ1, ψ2, ..., ψn} in which each symbol in Ψ appears exactly once in each row and
th th column of L. We denote by Li,j the symbol located at i row and j column of the array. Let L and L0 be two latin squares of order n over an alphabet Ψ of size n. Then L and L0 are said to be orthogonal if each ordered pair in Ψ × Ψ appears exactly once in the set
0 {(Li,j, L i,j) | 1 ≤ i, j ≤ n}.A graeco-latin square of order n is an n×n array of ordered pairs from an alphabet of n symbols, say Ψ, such that in each row and each column of the array, each symbol in Ψ appears exactly once in each coordinate, and that each of the n2 possible ordered pairs appears exactly once in the entire array. Therefore, finding a graeco-latin square of order n is equivalent to finding two orthogonal latin squares of order n. See Examples 2.1 and 2.2 for constructing two orthogonal latin squares and a graeco-latin square of order 3, respectively.
Example 2.1. (Latin squares of order 3) Let F = {0, 1, 2} be a finite field of order 3.
(k) ∗ Define f (i,j) = ki + j, where k ∈ F . Then
0 1 2 0 1 2
(1) (2) f (i,j) = 1 2 0 f (i,j) = 2 0 1 2 0 1 1 2 0
Example 2.2. (Graeco-latin square of order 3)
(0,0) (1,1) (2,2) (1,2) (2,0) (0,1) (2,1) (0,2) (1,0)
Lemma 2.3. [37, p.252] Let N(n), n > 1, be the largest integer v for which there exist v latin squares of order n that are mutually orthogonal. Then 1 ≤ N(n) ≤ n − 1. 11 A set of n − 1 mutually orthogonal latin squares of order n is said to be complete. Quasigroups are some algebraic structures whose multiplication tables are latin squares. Alternatively, a quasigroup (Q, ∗) is a set Q with a binary operation (∗) such that for each g and h in Q, there exist unique elements x and y in Q such that (i) g ∗ x = h and (ii) y ∗ g = h. Moreover, a loop is a quasigroup with an identity element which is not neces- sarily associative.
We define an operation called a ternary operation as follows. If R is any nonempty set, then a ternary operation T on R is a rule which assigns to any three ordered elements x, y, z ∈ R a unique element T (x, y, z) ∈ R [19, p.113]. An algebraic system (R, T ) satisfying the following axioms is called a (Planar) Ternary Ring (PTR):
(i) T (x1, 0, x2) = T (0, x1, x2) = x2, ∀x1, x2 ∈ R.
(ii) T (1, x, 0) = T (x, 1, 0) = x, ∀x ∈ R.
(iii) For every x1, x2, x3 ∈ R, T (x1, x2, x) = x3 has a unique solution x ∈ R.
(iv) For every x1, x2, x3, x4 ∈ R, where x1 6= x3, T (x, x1, x2) = T (x, x3, x4) has a unique solution x ∈ R.
(v) For every x1, x2, x3, x4 ∈ R, where x1 6= x3, each of T (x1, x, y) = x2 and
2 T (x3, x, y) = x4 has a unique solution (x, y) ∈ R .
We note that axiom (v) is redundant if R is finite. A PTR (R, T ) is said to be linear if T (x1, x2, x3) = x1 ·x2 +x3 for all x1, x2, x3 ∈ R. For further information about PTR’s see [19]. Orthogonal latin squares and PTR’s are closely related structures in the sense that a PTR can be used to construct a complete set of mutually orthogonal latin squares, or vice versa. Moreover, it is known that a PTR uniquely determines a projective plane, or there is a PTR for a given plane.
12 Theorem 2.6. (Bose, [4]) N(n) = n − 1 if and only if there exists a projective plane of order n.
Euler first introduced graeco-latin squares by stating the “36-officer problem”. It is to arrange 36 officers from six different regiments and of six different ranks in a 6×6 array so that each row and each column contained one officer of each rank and one officer of each regiment. The solution requires a graeco-latin square of order 6. Then he conjectured that a graeco-latin square of order n does not exist for n ≡ 2 (mod 4). In 1900, Tarry [36] showed that Euler’s conjecture holds for n = 6. In 1984, Stinson [34] gave a shorter proof for the nonexistence of a pair of orthogonal latin squares of order 6. Hence, Theorem 2.6 implies that there does not exist a projective plane of order 6. The following well- known theorem, namely the Bruck-Ryser theorem, also implies the non-existence of the projective plane of order 6.
Theorem 2.7. (Bruck-Ryser, [19, p.80]) If n ≡ 1, or 2 (mod 4), there cannot be a pro- jective plane of order n unless n can be expressed as a sum of two integral squares.
In 1960, Bose and Shrikhande [5] constructed a pair of orthogonal latin squares of order 22. Then Bose, Shrikhande and Parker [6] show that there exist at least two orthog- onal latin squares for all n ≡ 2 (mod 4), n ≥ 10. These were the first examples showing that Euler’s conjecture is false. Also, Stinson [35, p.152] gave a shorter and elegant proof that Euler’s conjecture is false. The existence of at least two orthogonal latin squares of order 10 gave a hope for the existence of the projective plane of order 10. Moreover, the Bruck-Ryser theorem is not sufficient to establish the non-existence of a projective plane of order 10. However, Lam [23] in 1991 showed the non-existence of a projective plane of order 10 by a very complex procedure, which included heavy computer use. Of course this result implies that there cannot exist 9 mutually orthogonal latin squares of order 10, however it is of interest that to this day no one has been able to construct 3 mutually orthogonal latin squares of order 10, inspite considerable effort. 13 Let us assume that there are n − 1 mutually orthogonal latin squares of order n over the alphabet Ψ = {ψ1, ψ2, ..., ψn}. We name the rows and columns of each latin square by the symbols in Ψ. For instance, we name the first row and column of a latin square by
2 ψ1, the second row and column by ψ2, and so on. We now consider an n × (n + 1) array
T such that the first two columns of T consist of all pairs of symbols in Ψ and there is a one-to-one correspondence between the other n − 1 columns and the latin squares. Let us take the kth column, 3 ≤ k ≤ (n + 1) and assume that the corresponding latin square is L. We choose a row of T and assume that the symbols ψi and ψj appear at the first and second columns on this particular row, respectively. Then we look up the symbol at the row ψi and column ψj of the latin square L and put this symbol at this row and kth column of T . The array T is called an orthogonal array, or a Tableau. Once the orthogonal array T is constructed, the corresponding projective plane α of order n is constructed as follows: Let T 0 be an 1 × (n + 1) array where the only row of T 0 consists of all integers 1 through (n + 1). Let us take the kth column of T , 1 ≤ k ≤ (n + 1) and
0 consider a symbol ψi, 1 ≤ i ≤ n, in Ψ. Then we adjoin to T the new row containing the integer k as the first entry which is followed by all indices of rows containing ψi at the kth column of T . We continue this process by considering every symbol in Ψ for the kth column. If the entire process is done for every column of T , the rows of the resulting array T 0 give rise to the lines of a projective plane α. See Example 2.3. If n = 2, 3, 4, 5, 7 or 8, then there is exactly one projective plane α which is isomor- phic with PG(2, n) [3, p.43]. Thus, α is desarguesian. There are exactly 4 planes of order 9, namely PG(2, 9) and the first examples of non-desarguesian planes: the Hughes plane, Hall plane and dual Hall plane of order 9. There is no projective plane of order 6 or 10, and at least one projective plane of order 11, namely PG(2, 11). It is still not known whether there is a non-desarguesian plane of order 11 or not. See table 2.1 for the list of projective planes of small order. The general conjecture is to prove or disprove that there
14 is no non-desarguesian projective plane of prime order. Unfortunately, it seems that the existing knowledge about projective planes is not promising to settle this question soon. Another well-known conjecture, the prime power conjecture for projective planes, is to prove or disprove that there is no projective plane of composite order. Although there have been attempts, this conjecture is not resolved either.
Example 2.3. (Constructing a plane of order 3 from a Tableau)
Orthogonal latin squares
ψ1 ψ2 ψ3 ψ1 ψ2 ψ3
ψ1 ψ1 ψ2 ψ3 ψ1 ψ1 ψ2 ψ3
ψ2 ψ2 ψ3 ψ1 ψ2 ψ3 ψ1 ψ2
ψ3 ψ3 ψ1 ψ2 ψ3 ψ2 ψ3 ψ1
1 2 3 4 1 2 3 4
5 ψ1 ψ1 ψ1 ψ1 1 5 6 7
6 ψ1 ψ2 ψ2 ψ2 1 8 9 10
7 ψ1 ψ3 ψ3 ψ3 1 11 12 13
0 Tableau T : 8 ψ2 ψ1 ψ2 ψ3 → T : 2 5 8 11
9 ψ2 ψ2 ψ3 ψ1 2 6 9 12
10 ψ2 ψ3 ψ1 ψ2 2 7 10 13
11 ψ3 ψ1 ψ3 ψ2 3 5 10 12
12 ψ3 ψ2 ψ1 ψ3 3 6 8 13
13 ψ3 ψ3 ψ2 ψ1 3 7 9 11 4 5 9 13 4 6 10 11 4 7 8 12
15 n 2 all isomorphic with PG(2,2) 3 all isomorphic with PG(2,3) 4 all isomorphic with PG(2,4) 5 all isomorphic with PG(2,5) 6 impossible 7 all isomorphic with PG(2,7) 8 all isomorphic with PG(2,8) 9 PG(2,9) and three ND planes 10 impossible 11 PG(2,11), ?
Table 2.1: Projective planes of small order
Let β be a field plane PG(2, Fq), q prime power, and A = (x, y, z), where x, y, z ∈ Fq, a point in Pβ. We call (x, y, z) the homogeneous coordinates of A. If z 6= 0, A can alternatively be represented by (x/z, y/z). If z = 0, we assume that A is incident with the line at ∞. Thus, any point not incident with the line at ∞ can be represented in the form (x0, y0), where x0 = x/z and y0 = y/z. Let α be an arbitrary plane of order q. In order to determine whether α is isomorphic to β, we can use a concept called coordinatization which adopts the same representation for the points of α as in β.
Figure 2.3: Coordinatization of a projective plane
16 In the following, we present a method for coordinatizing an arbitrary projective plane α with a PTR. There is a quadrangle Q = {O,X,Y,Z} in α by the definition of a projec- tive plane. Let XY be the line at ∞. Without loss of generality we also assume that X, Y , O and Z have the homogeneous coordinates (1, 0, 0), (0, 1, 0), (0, 0, 1) and (1, 1, 1), respectively. See Figure 2.3. As discussed above, we assign (0, 0) and (1, 1) to O and Z, respectively, since O and Z are not incident with XY . In a similar manner, each point on line OZ, except the intersection point A of XY and OZ, is represented by (x, x), where x is different for different points on OZ. We label the line OZ as y = x. Moreover, we rep- resent A by (1) since the slope of y = x is 1. We also define R = {x | (x, x) is on OA} which we will use later. Let B be any point which is not incident with XY . If XB and
YB intersect OZ in (y0, y0) and (x0, x0), then we say that XB and YB intersect OY and OX in (0, y0) and (x0, 0), respectively. Moreover, (x0, y0) is assigned to B. If we let the line through O and (1, m) intersect XY at the point C, then we assign (m) to C. Intuitively, the slopes of OY and OX are ∞ and 0, hence we assign to Y and X the coordinates (∞) and (0), respectively. We call XY the line at ∞ and denote it by [∞]. We also denote by ((∞)) and ([∞]) the set of lines through (∞) and set of points on [∞], respectively.
Consider the line through Y and (x0, x0). Any point on this line has the same first coordinate (x-coordinate) x0, thus the line has the equation x = x0. Similarly any line through Y except XY has the equation x = x0, for some x0. We take the line through (1) and (0, y1). If (x, y) is a point on this line, then we define a binary addition (+) by the equation y = x + y1 and this equation represents this particular line. OZ has the equation y = x since O has coordinates (0, 0). Similarly, if (x, y) is a point on OC, then we define a binary multiplication (·) by the equation y = m · x and this equation represents OC.
If R is a set (as described above) with operations (+) and (·), we have the following proposition:
17 Proposition 2.5. [9, p.101,103] (R, +) and (R \{0}, ·) are loops.
Having defined two binary operations (+), (·) on R, we can now define a ternary
3 operation T : R → R geometrically as follows : Let CD intersect OY at (0, y1) in Figure 2.3. If (x, y) is any point on CD, we define a ternary operation T such that y = T (x, m, y1). Thus, y = x + y1 = T (x, 1, y1) and y = m · x = T (m, x, 0). We call T a Hall ternary operation.
Proposition 2.6. [17, p.355] Let α be a projective plane coordinatized by a set R, containing distinct elements 0 and 1, and T the Hall ternary operation defined above.
Then (R, T ) is a PTR.
There are different methods for coordinatizing a projective plane i.e. a projective plane can be coordinatized by different PTR’s. However, a PTR uniquely determines a projective plane [17, p.355], [19], [29].
Theorem 2.8. [17, p.355] Every choice of a quadrangle Q determines a PTR.
Conversely, for given a certain PTR, the corresponding projective plane α, with points
Pα, lines Lα and incidence I ⊂ Pα × Lα, can be constructed as follows [19, p.114]:
(i) Pα = {(x1, x2): x1, x2 ∈ R} ∪ {(x1): x1 ∈ R} ∪ {(∞)},
(ii) Lα = {[x1, x2]: x1, x2 ∈ R} ∪ {[x1]: x1 ∈ R} ∪ {[∞]},
(iii) For all x1, x2, x3, x4 ∈ R, (x1, x2) I [x3, x4] if and only if T (x3, x1, x2) = x4,
(iv) (x1, x2) I [x3], (x1) I [x3, x4] if and only if x1 = x3,
(v) (x1) I [∞], (∞) I [x3], (∞) I [∞],
(vi) (x1, x2) 6 I [∞], (x1) 6 I [x3], (∞) 6 I [x3, x4].
18 Proposition 2.7. [17, p.360] Let α be a projective plane coordinatized by a Hall ternary ring (R, T ) such that any two triangles perspective with respect to the point at ∞ are also perspective with respect to the line at ∞. Then R is a group under (+) and the ternary ring (R,T ) is linear.
A Veblen-Wedderburn (VW) system (quasifield) is an algebraic system used to coordi- natize projective planes. We say that a plane is a VW plane if it is coordinatized by VW systems. A VW system (quasifield) (R, +, ·) is a set R of elements with operations (+) and (·) satisfying the following axioms [17, p.362]:
(i) (R, +) is an Abelian group.
(ii) (R \{0}, ·) is a loop.
(iii) (x1 + x2) · x3 = x1 · x3 + x2 · x3, ∀x1, x2, x3 ∈ R.
(iv) If x1 6= x2, x · x1 = x · x2 + x3 has a unique solution x ∈ R.
A linear ternary ring is called a cartesian group if its additive loop is associative thus a group. Then a VW system (quasifield) is basically a cartesian group such that exactly one of the right or left distributive law holds and it is commutative under (+).A semifield is a quasifield satisfying both of the left and right distributive laws. A planar nearfield
(nearfield) (N , +, ·) is a quasifield where the multiplication is associative. Thus (N , ·) is a group.
Example 2.4. Let (K, +, ·) be a finite field of order q2, and (F, +, ·) its subfield of order q, where q is an odd prime power. We define a nearfield (N , +, ◦) of order q2, where N has the same elements as K and the same addition (+). However, the multiplication ◦ is defined as follows: a ◦ b = a · b if a is a square in K, and a ◦ b = a · bq otherwise.
An alternative division ring is a semifield satisfying x · (x · y) = (x · x) · y and (x · y) · y = x · (y · y) for all x, y. 19 Proposition 2.8. (Artin-Zorn Theorem, [19, p.152]) A finite alternative division ring is a field.
A skew-field (division ring) has all the properties of a field, except that its multiplica- tion is not required to be commutative. Alternatively, a skew-field is a near-field which has both distributive laws, or a semifield in which the associative law holds.
Any projective plane α can be coordinatized by means of a set R with operations (+) and (·) as described above. Let α be a plane of order n > 2 and R a set which is used to coordinatize α, then α is desarguesian if and only if R is a skew-field, and pappian if and only if R is a field. See [19, p.154] for details.
Theorem 2.9. (Wedderburn, [19, p.7]) A finite skew-field is a field.
Corollary 2.1. A finite desarguesian projective plane α is pappian.
20 Chapter 3
Some non-desarguesian planes
In this chapter, we show how to construct some classes of non-desarguesian projective planes. A non-desarguesian plane has at least one pair of triangles which are perspective with respect to a point, but which fail to be perspective with respect to a line (or the converse). Therefore, it suffices to find such a pair of triangles in a plane α to show that α is non-desarguesian. Although there is no quick way for this determination, it has long been recognized as the Hanna Neumann’s conjecture that any non-desarguesian plane of odd order contains a Fano plane. No one has ever found a counterexample for this statement. Assuming that the statement is true, the existence of a Fano subplane in a plane α of odd order provides a strong hint (test) that α is non-desarguesian.
3.1 A VW plane π
O. Veblen and J. Wedderburn have defined a particular planar ternary ring (R, T ), which we will call the VW ternary ring. This ternary ring (of order p2) is defined as follows: Let
K be a finite field of order p2, p an odd prime, and R the set of elements of K. We define T : R3 → R as follows: T (a, b, c) = ab+c if b is a square in K, and T (a, b, c) = apb+c if b is not a square in K.
Proposition 3.1. Let R and T be as described above. Then (R, T ) is a PTR. 21 Proof. Let a, b, c ∈ R and a be a square in R. Then T (a, 0, b) = a0 + b = b = 0a + b = T (0, a, b), T (a, 1, 0) = a1 + 0 = a = 1a + 0 = T (1, a, 0), and T (b, a, x) = ba + x = c has a unique solution x ∈ R. If a is not a square in R, then T (a, 0, b) = a0 + b = b = 0pa + b = T (0, a, b), T (a, 1, 0) = a1 + 0 = a = 1pa + 0 = T (1, a, 0), and
T (b, a, x) = bpa + x = c has also a unique solution x ∈ R. Now, let a, b, c, d ∈ R, where a 6= c and a 6= 0 6= c. We have the following cases:
(i) If a and c are both squares in R, then T (x, a, b) = T (x, c, d) ⇔ xa + b = xc + d and xa + b = xc + d has a unique solution x ∈ R.
(ii) If a is not a square and c is a square, then T (x, a, b) = T (x, c, d) ⇔ xpa + b =
xc + d. This equation has a unique solution x ∈ R. See [9] for a proof.
(iii) If neither a nor c is a square in R, then T (x, a, b) = T (x, c, d) ⇔ xpa + b = xpc + d ⇔ xp = (v/u), where u = a − c 6= 0 and v = d − b. But there exists
t0 ∈ R such that (t0)p = (v/u). Therefore, xp = (t0)p. Hence there is a unique solution for T (x, a, b) = T (x, c, d).
Hence, (R, T ) is a PTR.
The VW ternary ring (of order p2) gives rise to a non-desarguesian plane of order p2. We denote by π the plane coordinatized by the VW ternary ring. For further details about VW planes, see [17].
3.2 The Hall plane ς
A Hall quasifield (of order q2) is constructed as follows: Let F be a finite field of order q and f(x) = x2 − rx − s be an irreducible quadratic over F. Let θ be a zero of f in the extension field K of order q2. Let R = {a + bθ | a, b ∈ K}. We define the addition ⊕ in
22 R as follows: (a + bθ) ⊕ (c + dθ) = (a + c) + (b + d)θ, and the multiplication in R as follows:
(a + bθ) c = ac + (bc)θ, and
(a + bθ) (c + dθ) = (ac − bd−1f(c)) + (ad − bc + br)θ, for d 6= 0.
A Hall plane ς (of order q2) is a non-desarguesian translation plane coordinatized by a Hall quasifield (of order q2). See [17] for more details about Hall quasifields and planes.
3.3 The Hughes and Figueroa planes
Some interesting constructions of finite non-desarguesian planes of prime power order n involve modifying a desarguesian plane α of order n by group theoretic means. These constructions involve a particular subgroup H of the full collineation group of α. More precisely, the points of the new, non-desarguesian plane β are taken to be the points of α and the lines of β are certain H-orbits of (n + 1)-subsets which are not lines in α.
3.3.1 The Hughes plane σ
Recall that for q a prime power the full collineation group of PG(2, q2) is G = PΓL(3, q2), and that H = PSL(3, q) < G. In [20], Daniel R. Hughes constructed a new family of non-desarguesian planes of order q2 by means of switching the lines of the desarguesian plane of order q2 with a new set of lines, that is a collection of H-orbits of (q2 + 1)-sets which were not collinear in the desarguesian plane. A synthetic construction for the Hughes plane σ (of order q2) was given by Rosati [31] and Zappa [38]. In our work, we used this synthetic technique to construct the Hughes 23 planes. The Rosati-Zappa method is as follows:
Let (N , +, ◦) be a nearfield of order q2 as described in Example 2.4 and V = {(x, y, z) | x, y, z ∈ N} the 3-dimensional left vector space over N . We define the set of points (set of lines) of σ to be the set of all equivalence classes of elements of V \{(0, 0, 0)}, under the equivalence (x, y, z) ∼ (k ◦ x, k ◦ y, k ◦ z) ([a, b, c] ∼ [k ◦ a, k ◦ b, k ◦ c]) for k ∈ N ∗. Then we fix a basis {1, t} for N as a vector space over its subfield F. The incidence relation for σ is defined as follows : Point (x, y, z) is incident with line [a, b, c], where a = a1 + ta2, b = b1 + tb2, and c = c1 + tc2, if and only if xa1 + yb1 + zc1 + (xa2 + yb2 + zc2) ◦ t = 0. The resulting plane σ is a non-desarguesian plane. σ is not (A, a)-transitive for any choice of A and a. Therefore, it is neither a translation plane nor a dual translation plane.
Moreover, Gσ does not fix a point or line of σ [19, p.196]. For further information about Hughes planes, see [19].
3.3.2 The Figueroa plane ξ
In [15], Figueroa constructed a new class of finite non-desargueisan projective planes of order q3 for each prime power q such that q 6≡ 1 (mod 3) and q > 2. This construction was generalized by Hering and Schaeffer in [18] for all prime powers. Similar to the construction of the Hughes planes, the Figueroa plane of order q3, q prime power, is constructed from α = PG(2, q3) by means of switching the lines of α with a new set of lines. In [16], Grundhofer¨ obtains a synthetic construction for the Figueroa planes of order q3, q a prime power, as follows:
3 Let α be the desarguesian plane PG(2, q ) over Fq3 . Gα has a planar collineation ρ of order 3. Then the points and lines of α partition into three classes. A point A ∈ Pα is said to be of Type I if ρ(A) = A, Type II if A, ρ(A) and ρ2(A) are collinear and distinct, and Type III if A, ρ(A) and ρ2(A) are not collinear. We dually define the types
24 of the lines. For a point A and a line a both of Type III, we define a bijection τ such that τ(A) = ρ(A)ρ2(A) and τ(a) = ρ(a) ∩ ρ2(a). Then we define the incidence structure of ξ as follows: A point A ∈ Pξ and a line a ∈ Lξ both of Type III are incident if and only if
τ(A) Iα τ(a). For other cases, A Iξ a if and only if A Iα a. Then ξ is a non-desarguesian plane of order q3 which is not a translation plane. For further information on the construction of Figueroa planes, see [16].
25 Chapter 4
Subplane structures of planes
If X is an ordered set and k ≤ |X|, we denote by Xk the collection of all ordered k- subsets of X, i.e. k-tuples (x1, x2, . . . , xk) where xi ∈ X and xi 6= xj when i 6= j. We X denote by k the collection of all k-subsets of X. A group action G | X induces actions X G | Xk and G | k respectively in the natural way. Let ∆1, ∆2,..., ∆r be the orbits of
G | X. We denote by δi the lexicographically smallest element of ∆i, 1 ≤ i ≤ r. If
x = (x1, x2, . . . , xk) ∈ Xk, we denote by G(x1,x2,...,xk) the pointwise stabilizer in G of x, i.e. G(x1,x2,...,xk) = Gx1 ∩ ... ∩ Gxk . Further, G{x1,...,xk} denotes the setwise stabilizer of {x1, . . . , xk} in G, i.e. the collection of all elements of G fixing {x1, . . . , xk} as a
whole, and O{x1,...,xk} the orbit of {x1, . . . , xk} under G. Clearly, G(x1,...,xk) ≤ G{x1,...,xk}.
(k) For each k, 1 ≤ k ≤ |X|, we denote by A = (ai,j) the matrix whose rows are the lexicographically minimal, distinct representatives of the orbits of G | Xk.
4.1 Closures of quadrangles and their orbits
Let α be a finite projective plane of order n and G = Gα. We assume Pα = X = {1, 2, . . . , n2 + n + 1} throughout this section. For k ≥ 1, the following proposition describes a method of finding all distinct G-orbit representatives of ordered (k + 1)- subsets of Pα. 26 Proposition 4.1. The matrix A(k+1) is constructed from A(k) as follows: (i) Initially set A(k+1) to be the empty matrix with 0 rows and k + 1 columns; (ii) For each row
(k) (ai,1, . . . , ai,k) of A , let ∆i,1, ∆i,2,..., ∆i,r be the orbits of G(ai,1,...,ai,k) on X\{ai,1,..., ai,k}; (iii) For each j ∈ {1, . . . , r}, select the lexicographically smallest representative
(k+1) δi,j ∈ ∆i,j and adjoin the (k + 1)-tuple (ai,1, . . . , ai,k, δi,j) as a new row to A . The
(k+1) final A will contain all distinct representatives of the orbits of G on Xk+1.
(k) Proof. Suppose that A = (ai,j), as defined above, is an m × k matrix and (b1, . . . , bk,
(k) bk+1) is any ordered (k + 1)-subset of X. Since the matrix A contains all distinct G- orbit representatives of ordered k-subsets of X, there exists g ∈ G such that for some
g (k) i ∈ {1, . . . , m}, (b1, . . . , bk) = (ai,1, . . . , ai,k), a particular row of A . Therefore
g (b1, . . . , bk, bk+1) = (ai,1, . . . , ai,k, c), c ∈ X, and since bk+1 6= bs for s < k + 1,
g c = bk+1 6= ai,s, for s < k + 1. Then c ∈ ∆i,j, for some j, where ∆i,j is one of the orbits of G(ai,1,...,ai,k) on X \{ai,1, . . . , ai,k}. Let δi,j be the lexicographically smallest
g h element of ∆i,j. Then there exists h ∈ G(ai,1,...,ai,k) such that ((b1, . . . , bk, bk+1) ) =
h (ai,1, . . . , ai,k, c) = (ai,1, . . . , ai,k, δi,j).
Now suppose that (ai,1, . . . , ai,k, ai,k+1) and (aj,1, . . . , aj,k, aj,k+1) are in the same G-
g orbit of Xk+1. Then there is g ∈ G such that (ai,1, . . . , ai,k+1) = (aj,1, . . . , aj,k+1).
g This implies that (ai,1, . . . , ai,k) = (aj,1, . . . , aj,k), that is, i = j and therefore g ∈
g G(ai,1,...,ai,k). Moreover, ai,k+1 = aj,k+1. Therefore ai,k+1 and aj,k+1 are in the same
(k+1) orbit of X \{ai,1, . . . , ai,k} under G(ai,1,...,ai,k). But in constructing A we selected a single (minimal) element from each G(ai,1,...,ai,k)-orbit on X \{ai,1, . . . , ai,k}, therefore ai,k+1 = aj,k+1.
Since we are interested in G-orbits of k-subsets, rather than G-orbits on ordered k- subsets, we form a new matrix A(k)0 by putting the rows of A(k) in ascending lexicographic order and deleting any duplicate rows.
Definition 4.1. Let B(k+1) be a matrix defined from A(k)0 as follows: Initially set B(k+1) to 27 (k)0 be an empty matrix with 0 rows and k + 1 columns. For each row {ai,1, . . . , ai,k} of A ,
(k+1) adjoin exactly |X| − ai,k rows to B , namely the (k + 1)-subsets {ai,1, . . . , ai,k, j} for all j such that ai,k < j ≤ |X|.
The following corollary follows easily, since the representatives of G{ai,1,...,ai,k} on
Y = X \{ai,1, . . . , ai,k} are among the elements of Y .
(k+1) X Corollary 4.1. B contains all representatives of the orbits of G on k+1 .
Computation of the A(k) for 1 ≤ k ≤ 3 is both time and space efficient. However, if the order n of the plane is large, computing A(4) is already inefficient in terms of time
(3) complexity, because for each row (ai,1, ai,2, ai,3) of A , we need to compute the stabilizer
Hi = G(ai,1,ai,2,ai,3) as well as the orbits of Hi on Yi = X \{ai,1, ai,2, ai,3}. To reduce the time complexity of our computation, we decided to use B(4) rather than A(4) (or A(4)0). Interestingly, this approach not only reduced the time complexity, but also the space complexity as the rows of B(4) are generated and used without storing B(4). We pay a price for this strategy, because many more closures of quadrangles are computed than if we had computed and stored A(4) (or A(4)0).
repst ← ∅ repeat V ←Random member of Dt Dt ← Dt \ OV repst ← repst ∪ {V } until Dt = ∅
Figure 4.1: Finding the orbit representatives of the point-sets of planes in Dt.
(4) Define Dt = {PS | S is a row of B , S ∈ Qα, |PS| = t}. Note that closures of two rows of B(4), which are quadrangles, may result in the same point-set of a subplane
(3) of α. Since we started with A rather than all triangles of α, Dt will not contain all distinct t-point Q-subplanes. However, by Corollary 4.1, Dt will contain representatives 28 of all G-orbits of t-point Q-subplanes. From Dt we select one representative from each G-orbit (See Figure 4.1).
4.2 Subplanes of some non-desarguesian planes of small
order
If a plane α is desarguesian of prime order, any quadrangle generates the whole plane. However, if α is non-desarguesian, there are many possibilities: If the order is p2, a quadrangle can generate the whole plane, a Fano subplane, a Baer subplane or a subplane of order m where 2 < m < p. For instance, L. Puccio and M. J. de Resmini [30] showed that subplanes of order 3 exist in the Hughes plane of order 25.
Proposition 4.2. In a desarguesian plane of order pn, p prime, every quadrangle gener- ates a subplane of order p.
Proof. Let α be a desarguesian plane of order pn, where p is a prime. By the Fundamental
Theorem of Projective Plane Geometry [13, p.160], Gα is transitive on quadrangles. It is also known that there is a subplane β of order p in α [19]. Moreover, any quadrangle
S ∈ Qβ generates β, since it is of order p. Hence, any quadrangle in α generates a subplane of order p.
It follows easily from Proposition 4.2 that projective planes α and β of order 4 and 8, respectively, are not Q-planes, since any quadrangle in α or β generates a Fano subplane. By Bruck’s theorem, α or β has only Fano subplanes as proper subplanes. Let Ω be a 5-arc in α (β). Since a Fano subplane does not contain a 5-arc, α (β) is generated by Ω.
Furthermore, if S ⊂ Pα (S ⊂ Pβ) contains the point-set of a Fano subplane of α (β) as a proper subset, then S = α (S = β). We say that this particular Fano subplane extends to α (β). 29 Moreover, a desarguesian projective plane γ of order 9 is not a Q-plane, since any quadrangle in γ generates a subplane of order 3. It is known that there are exactly 3 non- isomorphic non-desarguesian planes of order 9 and each contains a quadrangle generating the whole plane. This leads us to the following.
Proposition 4.3. A non-desarguesian projective plane of order 9 is a Q-plane.
We remark in passing that a plane of order 4 can be partitioned into 3 mutually disjoint Fano planes, i.e. the set of points (lines) of a plane of order 4 partition into three subsets such that each subset is the point-set (line-set) of a Fano plane. Similarly, a desarguesian plane of order 9 can be partitioned into 7 mutually disjoint planes of order 3. We analyze the subplane structures of some well-known non-desarguesian planes of small order, namely the Hall plane of order 25, the Hughes plane of order 25 and the Figueroa plane of order 27. Firstly, we compute the full collineation groups and A(k)0, 1 ≤ k ≤ 4, for each of these planes. After we determine the closures of all the rows of the matrices A(4)0 and delete duplicate planes, we compute orbit representatives of all proper Q-subplanes under their full collineation groups. This process results in the following propositions.
Proposition 4.4. There are 19 orbits of Fano subplanes and 6 orbits of subplanes of order 5 in the Hall plane of order 25.
Proposition 4.5. The Hughes plane of order 25 has 56 orbits of Fano subplanes, only 1 orbit of subplanes of order 3 and 5 orbits of subplanes of order 5.
Proposition 4.6. There are 3524 orbits of Fano subplanes and 59 orbits of subplanes of order 3 in the Figueroa plane of order 27.
Let α be one these planes, namely the Hall plane of order 25, the Hughes plane of order 25 or the Figueroa plane of order 27. The possible proper subplanes of α are the 30 planes of order 2, 3, 4, or 5. However, examining the closures of quadrangles in α is not enough to determine the existence of the planes of order 4, since a plane of order 4 is not a Q-plane. We firstly determine the orbit representatives of Fano subplanes of α, and then compute that none of these representatives extend to a plane of order 4.
Proposition 4.7. There are no subplanes of order 4 in the Hall plane of order 25, the Hughes plane of order 25 or the Figueroa plane of order 27.
We also compute A(4)0 for the Hughes plane of order 49. However, there is no row of A(4)0 whose closure is a subplane of order 3.
Proposition 4.8. The Hughes plane of order 49 does not have subplanes of order 3.
We use the same methodology to determine some subplanes of the Figueroa plane Ξ of order 125. Closures of some randomly chosen quadrangles give rise to Fano subplanes, subplanes of order 3 or subplanes of order 5. Although the numbers of orbits on these subplanes are not determined yet, it is interesting to have subplanes of order 3 since 3 does not divide the order of the plane Ξ.
Proposition 4.9. The Figueroa plane of order 125 has Fano subplanes, subplanes of order 3 and subplanes of order 5.
4.3 Some non-desarguesian planes of order 121
A question that is still outstanding in finite geometry is whether all finite projective planes of prime order are desarguesian. The first open case is for p = 11. There is at least one plane of order 11, namely the pappian plane of order 11. It is still of interest to examine whether there are non-desarguesian planes of order 11. In 1985, a suggestion by Ascher Wagner was made to Spyros S. Magliveras. It is to begin with a non-desarguesian plane of
31 order pk, k > 1, determine all subplanes of order p up to collineations and check whether one of these is non-desarguesian. In this section we determine all proper Q-subplanes of a Veblen-Wedderburn (VW) plane π of order 121 (Π) and the Hughes plane σ of order 121 (Σ) up to collineations, paying special attention to the Baer subplanes of Π and Σ.
Let α be a plane of order 121. If S is a quadrangle in α and PS = Pβ, where β is a proper subplane of α, then by Bruck’s theorem |PS| ∈ {7, 13, 21, 31, 57, 73, 91, 133}.
However, since projective planes of order 4 and 8 are not Q-planes, |PS|= 6 21, 73. In the following sections 4.3.1 and 4.3.2, we determine all proper Q-subplanes of a Veblen- Wedderburn (VW) plane π of order 121 and the Hughes plane σ of order 121 up to collineations. We denote these planes by Π and Σ, respectively.
4.3.1 π of order 121 (Π)
In this study, we use the VW ternary ring of order 121 to construct the non-desarguesian projective plane Π of order 121. The plane Π has 14763 points and 14763 lines. We assume PΠ = {1, 2,..., 14763} throughout this section.
GΠ
H H ′ K .m K′ U U ′ 11 .b 11 .b′ .a ′ .y .a ′ .y′ C60 C .x 60 . x′ .u
Figure 4.2: The full collineation group GΠ
8 2 2 4 We have that |GΠ| = 843, 321, 600 = 2 · 3 · 5 · 11 , and GΠ is not transitive on
32 points or lines. Our analysis of the structure of GΠ shows the following : There are three orbits on points, namely Θ1, Θ2 and Θ3, of lengths 1, 242 and 14520 and three orbits on lines, namely Γ1, Γ2 and Γ3, of lengths 2, 120 and 14641, respectively. In our representa- tion, Γ1 = {l1, l2}, Γ2 = ((∞))\Γ1, Γ3 = LΠ\((∞)), where (l1) = {1,..., 121, 14763},
(l2) = {14642,..., 14762, 14763}, and (∞) = l1l2 = 14763. Moreover, Θ1 = {(∞)},
Θ2 = ((l1)∪(l2))\{(∞)} and Θ3 = PΠ \((l1)∪(l2)). The actions GΠ | Θ2 and GΠ | Θ3 are faithful and Π can be completely recovered from GΠ | Θ2. We present a reconstruction method for Π from its collineation group in Chapter 5.
There is a subgroup K ≤ GΠ, of order 14520, with the following presentation: K = hx, y, a, b | x60, a11, b11, aba−1b−1, y2x30, y−1xyx49, x−1axb−1a3, y−1ayb2a−1, x−1bxba6, y−1byba−1i
Moreover, K is normal in a subgroup H of GΠ, with [H : K] = 2, and there exists an involution m ∈ GΠ such that: H = hx, y, a, b, m | m2, x60, a11, b11, aba−1b−1, y2x30, y−1xyx49, x−1axb9, y−1ayb9a2, x−1bxba2, y−1byb9a8, mamba6, mbmb5a9, mxmx49, mymy−1x45i. In Appendix C, we display the generators of H, namely x, y, a, b and m, in their faithful permutation representation on the subset {1,..., 121} of points. Finally, GΠ =
0 0 −1 0 hH, ui = hH,H , ui for an involution u ∈ GΠ \ H such that H = u Hu, and H ∩ H = hmi. For Π, our computation shows that A(1)0 is a 3 × 1 matrix, A(2)0 a 29 × 2 matrix, A(3)0 a 4895 × 3 matrix, and B(4) a 37, 405, 758 × 4 matrix. In the computation of A(k),
1 ≤ k ≤ 3, we use the permutation group GΠ represented on PΠ = {1,..., 14763},
(k)0 hence for 1 ≤ k ≤ 3, the below matrices A = (ai,j), have ai,j ∈ PΠ.
Proposition 4.10. There are no subplanes of order 3, 5, 7 or 9 in Π. Determination of all proper Q-subplanes yields 974 distinct Baer Q-subplanes and 362, 380 distinct Fano subplanes. 33 1 2 1 122 1 14642 1 14763 122 123 122 243 122 364 122 606 122 848 122 1211 122 1332 122 1453 122 1574 1 122 1695 (1)0 A = 122 A(2)0 = 122 1816 14763 122 1937 3×1 122 2058 122 2179 122 2663 122 2784 122 3026 122 3631 122 3873 122 4115 122 4478 122 4841 122 5204 122 5325 122 14763 29×2
1 2 3 4 .... .... 1 2 3 .... 1 2 4 1 2 3 14763 ... .... (3)0 A = ... B(4) = .... ... .... 122 5325 14038 122 5325 14038 14039 122 5325 14763 .... 4895×3 .... .... 122 5325 14038 14763 37,405,758×4
Proof. Planes of order 3, 5, 7 and non-desarguesian planes of order 9 are Q-planes. By the discussion in Section 4.1, any quadrangle generates a subplane of order 3 in a desar- 34 guesian plane of order 9. The statement of the proposition follows by considering the planes which are closures of all the rows of B(4) and deleting duplicate planes.
Proposition 4.11. In Π, there are exactly 13 orbits of Baer Q-subplanes and approxi- mately 3000 orbits of Fano subplanes under the full collineation group GΠ of Π.
Proof. The result follows by computing the orbits under GΠ on the collection of planes obtained in Proposition 4.10.
Note that there may exist subplanes of order 4 or 8 as proper subplanes in Π, since these subplanes are not Q-subplanes.
4.3.2 σ of order 121 (Σ)
In this section we analyze the subplane structure of the Hughes plane of order 121. We assume PΣ = {1, 2,..., 14763} throughout this section.
5 2 3 We have that |GΣ| = 424, 855, 200 = 2 · 3 · 5 · 11 · 19, and GΣ is not transitive on points or lines. Σ is self-dual and GΣ has two orbits on points (lines) of lengths 133 and 14630. GΣ has a faithful representation on the point orbit of size 14630 and the point orbit of size 133 gives rise to a Baer subplane of order 11 in the configuration of the plane Σ.
In Figure 4.3, H is a subgroup of GΣ, where |H| = 29040, and has the following presentation: H = hx, y, a, b | y2, x120, a11, b11, y−1xyx109, aba−1b−1, x−1axb−1a−1, y−1aya−1, x−1bxb2a9, y−1byba3i.
0 −1 0 Furthermore, there is an involution u ∈ GΣ \ H such that for H = u Hu, H ∩ H =
0 hmi and GΣ = hH, ui = hH,H , ui. For Σ, our computation determines that A(1)0 is a 2 × 1 matrix, A(2)0 a 16 × 2 matrix, A(3)0 a 8277 × 3 matrix and B(4) a 79, 762, 036 × 4 matrix. In the computation of A(k), 35 GΣ
H H ′ .m U U ′ 11 .b 11 .b′ .a ′ .y .a ′ .y′ C120 C .x 120 . x′ .u
Figure 4.3: The full collineation group GΣ
1 ≤ k ≤ 3, we use the permutation group GΣ represented on the set {1,..., 14763},
(k)0 hence for 1 ≤ k ≤ 3, the below matrices A = (ai,j), have ai,j ∈ PΣ.
Proposition 4.12. There are no subplanes of order 5 or 7 in Σ, and no non-desarguesian planes of order 9. Determination of all proper Q-subplanes yields 349 distinct Baer subplanes, 104 distinct subplanes of order 3 and 408, 075 distinct Fano subplanes.
1 2 1 12 1 133 12 13 12 14 12 15 12 16 (1)0 1 12 23 A = A(2)0 = 12 12 25 2×1 12 26 12 28 12 37 12 80 12 121 12 122 12 133 16×2
36 1 2 3 4 .... .... 1 2 3 .... 1 2 12 1 2 3 14763 ... .... (3)0 A = ... B(4) = .... ... .... 122 133 14757 122 133 14757 14758 12 133 14763 .... 8277×3 .... .... 122 133 14757 14763 79,762,036×4
Proof. Planes of order 5, 7 and non-desarguesian planes of order 9 are Q-planes. The statement of the proposition follows by considering the planes which are closures of all the rows of B(4) and deleting duplicate planes.
Proposition 4.13. In Σ, there are exactly 8 orbits of Baer Q-subplanes, and exactly 2 orbits of subplanes of order 3.
Proof. The result follows by computing the orbits under GΣ on the collection of planes obtained in Proposition 4.12.
The existence of subplanes of order 3 in Σ is interesting, although not totally surpris- ing in view of L. Puccio and M. J. de Resmini’s discovery of subplanes of order 3 in the Hughes plane of order 25 [30]. We also note that there may exist subplanes of order 4 or 8, and desarguesian planes of order 9 as proper subplanes in Σ, since these subplanes are not Q-subplanes.
4.4 Subplanes of order 3 in the Hughes Planes
In this section, we show that every Hughes plane of order q2, where q is a prime power and q ≡ 5 (mod 6), has subplanes of order 3. No subplanes of order 3 have ever been 37 found in the Hughes planes of order q2 for q ≡ 1 (mod 6); and computational evidence for small values of q suggests that subplanes of order 3 do not occur in this case. It is also an open problem whether there exists a Hughes plane with a subplane of order 4. It has long been recognized by M. J. de Resmini and others that every Hughes plane has subplanes of order 2. On the other hand, this is not totally surprising since for a quadrilateral to generate a subplane of order 2 it is only required that a single algebraic condition holds.
Throughout this section, let K and F be the fields of orders q2 and q, respectively. If d is a nonsquare in F and δ ∈ K an element such that δ2 = d, then the seven points
(1, 0, 0), (0, 1, 0), (0, 0, 1) (0, d, 1), (1, δ, 0), (−δ, 0, 1), (−δ, d, 1) and the seven lines
[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 0, δ], [−δ, 1, 0], [0, 1, −d], [−δ, 1, d] give a subplane of order 2. In order for a quadrilateral to generate a subplane of order 3, several inequivalent conditions must hold.
Let θ ∈ K such that θ2 = −3. Note that K contains two primitive cube roots of 1, which we denote by ω = (−1 + θ)/2 and ω2 = (−1 − θ)/2 throughout. Note that {1, θ} is a basis for K over F.
Lemma 4.1. x = a + bθ ∈ K (with a, b ∈ F) is a square, if and only if its norm xq+1 = a2 + 3b2 is a square in F.
2 Proof. The element x = a+bθ is a square in K if and only if 1 = x(q −1)/2 = (xq+1)(q−1)/2 = (a2 + 3b2)(q−1)/2, if and only if a2 + 3b2 is a square in F.
38 4.4.1 Case: q ≡ 5 (mod 12)
Let q ≡ 5 (mod 12). Then there is an element i ∈ F of order 4, since q ≡ 1 (mod 4). Thus ζ = iω is of order 12. We have that ζ = iω = (−i + iθ)/2 and i2 = −1. Then we compute that ζ2 = (1 + θ)/2, ζ4 = ω = (−1 + θ)/2, and ζ5 = iω2 = (−i − iθ)/2. Moreover, ζ + ζ7 = ζ2 + ζ8 = ζ4 + ζ10 = ζ5 + ζ11 = 0, since ζ6 = −1. Hence, ζ7 = (i − iθ)/2, ζ8 = (−1 − θ)/2, ζ10 = (1 − θ)/2, and ζ11 = (i + iθ)/2. The following Lemma follows easily from Lemma 4.1.
Lemma 4.2. 1 ± θ are squares and θ, 3 ± θ not squares in K.
We now define α, a set of 13 points, and β, a set of 13 lines, as follows :
p1 (0, 0, 1) `1 [0, 0, 1] p2 (0, 1, 0) `2 [0, 1, 0] 5 p3 (0, 1, ζ) `3 [0, 1, ζ ] 7 11 p4 (0, 1, ζ ) `4 [0, 1, ζ ] p5 (1, 0, 0) `5 [1, 0, 0] 2 4 α : p6 (1, 0, ζ ) β : `6 [1, 0, ζ ] 8 10 p7 (1, 0, ζ ) `7 [1, 0, ζ ] 5 p8 (1, ζ, 0) `8 [1, ζ , 0] 2 5 4 p9 (1, ζ, ζ ) `9 [1, ζ , ζ ] 8 5 10 p10 (1, ζ, ζ ) `10 [1, ζ , ζ ] 7 11 p11 (1, ζ , 0) `11 [1, ζ , 0] 7 2 11 4 p12 (1, ζ , ζ ) `12 [1, ζ , ζ ] 7 8 11 10 p13 (1, ζ , ζ ) `13 [1, ζ , ζ ]
Theorem 4.1. Let q be a prime power, q ≡ 5 (mod 12), and σ a Hughes plane of order q2. Then α is the set of points, and β the set of lines, of a subplane of order 3 in σ. This subplane is invariant under the polarity (x, y, z) ↔ [xq, yq, zq] of σ.
Proof. It is known that all elements of F are squares in K. We use Lemma 4.2 and the incidence relation described by Rosati [31] to determine whether pi and `j are incident for each pair of a point pi, 1 ≤ i ≤ 13, in α and a line `j, 1 ≤ j ≤ 13, in β. This gives rise to the following incidence matrix M: 39 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 M = 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0
T An easy computation shows that MM = J13 + 3I13, where J13 denotes the 13 × 13 matrix in which every entry is a “1” and I13 the 13 × 13 identity matrix. By Rosati [32], the map (x, y, z) ↔ [xq, yq, zq] is a polarity of σ. One easily checks that this map interchanges α and β. This completes the proof of Theorem 4.1.
4.4.2 Case: q ≡ 11 (mod 12)
Let us now assume that q ≡ 11 (mod 12), so that both −1 and −3 are nonsquares, and in particular 3 is a square in F.
Lemma 4.3. There exists c ∈ F such that c2 − c + 1 is a nonsquare in F.
Proof. By the Chevalley-Warning Theorem [33, p.5], there exist a, b, c ∈ F, not all zero, such that c2 − bc + b2 + a2 = 0. Clearly b 6= 0, so (c/b)2 − (c/b) + 1 = −(a/b)2, a nonsquare in F.
Fixing c ∈ F as in Lemma 4.3, we readily obtain the following from Lemma 4.1.
Lemma 4.4. The elements θ, 1 ± θ and 3 ± θ are squares in K. The elements c − 2 ± cθ, c + 1 ± (c − 1)θ and 2c − 1 ± θ are nonsquares in K.
40 We shall use Lemma 4.4 along with the fact that c∈ / {0, 1}. Now we define α0, a set of 13 points, and β0, a set of 13 lines, as follows :
2 2 p1 (1, ω, ω ) `1 [1, ω, ω ] p2 (1, 0, −ω) `2 [0, −ω, 1] p3 (−ω, 1, 0) `3 [1, 0, −ω] p4 (0, −ω, 1) `4 [−ω, 1, 0] 2 2 p5 (1/(c − 1), ω, ω ) `5 [ω , c/(1 − c), ω] 0 2 0 2 α : p6 (−c, ω, ω ) β : `6 [ω , c − 1, ω] 2 2 p7 ((1 − c)/c, ω, ω ) `7 [ω , −1/c, ω] 2 2 p8 (ω , (1 − c)/c, ω) `8 [ω, ω , c/(1 − c)] 2 2 p9 (ω , 1/(c − 1), ω) `9 [ω, ω , c − 1] 2 2 p10 (ω , −c, ω) `10 [ω, ω , −1/c] 2 2 p11 (ω, ω , 1/(c − 1)) `11 [c − 1, ω, ω ] 2 2 p12 (ω, ω , −c) `12 [−1/c, ω, ω ] 2 2 p13 (ω, ω , (1 − c)/c) `13 [c/(1 − c), ω, ω ]
Theorem 4.2. Let q be a prime power, q ≡ 11 (mod 12), and σ a Hughes plane of order q2. Then α0 is the set of points, and β0 the set of lines, of a subplane of order 3 in σ.
Proof. By Lemma 4.3 and 4.4, our computation gives rise to the following incidence
0 0 0T matrix M , where M M = J13 + 3I13. This proves Theorem 4.2.
1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 M = 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0
41 4.5 Further substructures of the Hughes Planes
In the following, we show that there exist finite partial linear spaces which cannot embed in any Hughes plane. This result is due to Moorhouse [12]. Let L be a finite partial linear space. As before, we denote by σ the Hughes plane of order q2. We say that f : L → σ is an embedding if f injectively maps points of L to points of σ, and f injectively maps lines of L to lines of σ, such that f(P ) lies on f(`) (in σ) if and only if the point P lies on the line ` (in L). (Replacing “if and only if” by “if” in the latter definition, does not change the essential difficulty of the embedding problem, or the validity of Theorem 4.3 below; see [26, Lemma 1].) In this language, our main result (above) is that the projective plane of order 3 embeds in σ whenever q ≡ 5 (mod 6).
Theorem 4.3. There exists a finite partial linear space which does not embed in any Hughes plane.
Proof. Let L0 be a finite partial linear space L which does not embed in any desarguesian plane of odd order. (We may take L0 to be a projective plane of order 2, or a configuration violating Desargues’ Theorem.) Let Γ0 be the incidence graph of L0, i.e. the graph whose vertices correspond to points and lines of L0; and whose edges correspond to incident point-line pairs of L0. Thus, Γ0 is a bipartite graph with no 4-cycles. By [27, Theorem 6.3] (see also [26, Lemma 2]), there exists a bipartite graph Γ having no 4-cycles, such that for every 2-coloring of the edges of Γ, there exists a subgraph isomorphic to Γ0, all of whose edges have the same color. We may regard Γ as the point-line incidence graph of a partial linear space L.
Suppose that f : L → σ is an embedding. For each point Pi and line `j of L, denote f(Pi) = (xi, yi, zi) and f(`j) = [aj, bj, cj]. (We have chosen arbitrary but fixed nonzero
42 3 2 vectors in N (N , nearfield of order q ) representing f(Pi) and f(`j).) Now write
3 (aj, bj, cj) = (aj1 + taj2, bj1 + tbj2, cj1 + tcj2), (ajk, bjk, cjk) ∈ F for all j, k, where {1, t} is a fixed basis for K over F.
Assuming Pi ∈ `j, we color the incident point-line pair (Pi, `j) red or blue according as aj2xi + bj2yi + cj2zi ∈ K is a square or a nonsquare.
Case 1: Γ has a subgraph isomorphic to Γ0, all of whose edges are red. In this case the map
Pi 7→ (xi, yi, zi), `j 7→ (aj, bj, cj)
2 restricts to an embedding of Γ0 in a desarguesian plane of order q , since
ajxi + bjyi + cjzi = (aj1xi + bj1yi + cj1zi) + (aj2xi + bj2yi + cj2zi) ◦ t = 0
for every red incident point-line pair Pi ∈ `j. This contradicts the choice of Γ0.
Case 2: Γ has a subgraph isomorphic to Γ0, all of whose edges are blue. In this case the map
q q q Pi 7→ (xi, yi, zi), `j 7→ (aj , bj , cj )
2 restricts to an embedding of Γ0 in a desarguesian plane of order q , since
q q q aj xi + bj yi + cj zi = (aj1xi + bj1yi + cj1zi) + (aj2xi + bj2yi + cj2zi) ◦ t = 0
for every blue incident point-line pair Pi ∈ `j . Again, this contradicts the choice of
Γ0.
The proof of Theorem 4.3 reveals a straightforward strategy for trying to embed a given finite partial linear space L (such as a finite projective plane) in the Hughes plane σ
43 of order q2: Choose an appropriate 2-coloring of the incident point-line pairs of L (i.e. the edges of its incidence graph Γ), such that both of the resulting monochromatic subgraphs of Γ correspond to partial linear spaces embeddable in a desarguesian plane of order q2. Unfortunately there are exponentially many 2-colorings of the edges of Γ to consider; and even for a projective plane of order 4, with 105 incident point-line pairs, this seems a daunting task. On the other hand, it is easy to 2-color these 105 incident point-line pairs without rendering any monochromatic subplane of order 2; so the argument of Theorem 4.3 seems ineffective in ruling out subplanes of order 4 in the Hughes planes.
44 Chapter 5
Reconstructing planes from their groups
In [22] and [24], it is shown how to construct a desarguesian plane of order pr from involutions and certains subgroups of PGL(2, pr). Points and lines are represented by some group elements and subgroups. The construction is using the subgroup structure to determine incidence relation. In [7], some group-theoretic methods for constructing both the Hughes plane of order q2 and the Figueroa plane of order q3, q an odd prime power, are discussed. The method is using the well-known linear group GL(3, q).
In this chapter, we use the VW ternary ring (R, T ) of order p2, p = 5, 7 or 11, to construct the non-desarguesian projective planes α, β, and Π of orders 25, 49 and 121, respectively. It follows easily from the definition that α, β, and Π are VW planes. We present how to construct these planes from their collineation groups.
5.1 The structure of Gπ
We compute the full collineation groups Gα, Gβ and GΠ of the planes α, β, and Π, respectively. Then we ask the following question: “Is it possible to reconstruct the planes α, β, and Π by only using their collineation groups?”
45 Let π be one of the planes α, β, or Π. Since π is of order p2, p = 5, 7 or 11, we assume that Pπ = {A0,A1, ..., Ap4+p2 } and Lπ = {a0, a1, ..., ap4+p2 } throughout the section. We observe that Gπ is not transitive on points and lines. Furthermore, there are three orbits
2 4 2 on points, namely Θ1, Θ2 and Θ3, of lengths 1, 2p and p − p , and three orbits on
2 4 lines, namely Γ1, Γ2 and Γ3, of lengths 2, p − 1 and p , respectively. Let Γ1 = {a0, a1}, where (a0) = {A0,A1, ..., Ap2 } and (a1) = {A0,Ap2+1, ..., A2p2 }. Then we have that
Γ2 = (A0) \ Γ1 and Γ3 = Lπ \ (A0). Moreover, Θ1 = {A0}, Θ2 = ((a0) ∪ (a1)) \{A0} and Θ3 = Pπ \ ((a0) ∪ (a1)). Furthermore, the actions Gπ | Θ2 and Gπ | Θ3 are faithful.
GΠ
H H ′ K .m K′ U U ′ p .b p .a .b′ .y .a′ .y′ C ′ C .x . x′ .u
Figure 5.1: The full collineation group Gπ
2 2 There is a subgroup K ≤ Gπ, of order p (p −1), and K is normal in a subgroup H <
Gπ, where [H : K] = 2. See Figure 5.1. Furthermore, there is a cyclic subgroup C < K of order (p2 − 1)/2. If C = hxi, then there is an element y ∈ K such that y2x(p2−1)/4 =
2 (p2−1)/8 1Gπ if p ≡ 3 (mod 4), and y x = 1Gπ if p ≡ 1 (mod 4). Moreover, the Sylow
2 p-Subgroup Up < K is of order p and K is the split extension of Up by the subgroup hx, yi generated by x and y. See the Appendices A, B and C for the presentations of K in Gα, Gβ and GΠ, respectively. In addition, there is an involution m such that H = hK, mi. The generators of the subgroup H, namely x, y, a, b and m, are represented as
2 permutations on the subset {1, ..., p }. Further, there is an involution u ∈ Gπ \ H such
46 0 −1 0 0 that for H = u Hu, H ∩ H = hmi and Gπ = hH, ui = hH,H , ui. See Appendices for the size and generators of the full collineation groups Gα, Gβ and GΠ.
5.2 Reconstruction from Gπ
In this section, we discuss a reconstruction method for π from its collineation group Gπ.
5.2.1 Counting Principle
Let a0 and a1 (as described above) intersect at A0. A point A is said to be of Type I if
A ∈ (a0) ∪ (a1), and of Type II, otherwise. Similarly, a line a is of Type I if a = AA0, where A 6= A0, and of Type II, otherwise. Let Ai 6= Aj, Ar 6= As be points of Type I, where Ai,Aj ∈ (a0) \{A0} and Ar,As ∈ (a1) \{A0}. Then it easily follows that Q = p2 p2 {Ai,Aj,Ar,As} is a quadrangle in π and there are 2 2 such quadrangles constructed by the points of a0 and a1.
Figure 5.2: Counting principle
The set of intersection points of lines passing through all pairs of the points of Q is
{Ai,Aj,Ar,As, A, B, A0}, where A and B are distinct points of Type II. See Figure 5.2. 47 p2 p2 Therefore, there are 2 2 2 points of Type II determined by the quadrangles which are p2 constructed as above. However, let A be any point of Type II, then there are 2 different pairs of lines of Type II intersecting at A. Hence, there are p2(p2 − 1) distinct points of Type II determined by such quadrangles. We also have that there are 2p2 + 1 distinct points of Type I. This leads us to the following lemma.
Lemma 5.1. All points of π are determined by the quadrangles as described above.
5.2.2 Reconstruction
−1 We define Sg,H = {h gh | h ∈ H} for any subgroup H ≤ Gπ and g ∈ Gπ. There is a cyclic subgroup C0 ≤ K0 of order (p2 − 1)/2 such that C0 = u−1Cu, where C ≤ K is cyclic and u is the involution described above. See Figure 5.1. Since p is odd and C0 is cyclic, C0 contains exactly one involution which we call τ 0.
Figure 5.3: Representing certain points and lines by involutions
0 0 0 0 0 Recall that K is the split extension of Up by the subgroup hx , y i generated by x and
0 0 −1 0 −1 0 0 y , where x = u xu and y = u yu. Now, consider the set Sτ ,Up . It easily follows that
0 2 2 0 0 0 0 |Sτ ,Up | = |Up| = p , i.e. Sτ ,Up contains exactly p involutions. Our analysis of the
0 0 elements in Sτ ,Up shows the following : 48 2 2 0 0 (i) {Ap +1,...,A2p ,A0} ⊂ Fix(s) for each s ∈ Sτ ,Up .
2 0 0 (ii) (a0) ∩ Fix(s) = {Ai,A0} for some i, 1 ≤ i ≤ p , and s ∈ Sτ ,Up .
0 0 (iii) (a0) ∩ Fix(s1) 6= (a0) ∩ Fix(s2) for distinct elements s1, s2 ∈ Sτ ,Up .
Therefore, there is a one-to-one correspondence between the points in (a0)\{A0} and
0 0 the involutions in Sτ ,Up . Moreover, we can represent the points on a0, except A0, by the involutions in S 0 0 . Hence, we write S 0 0 = {g , . . . , g }. Symmetrically, there τ ,Up τ ,Up A1 Ap2 is a single involution τ ∈ C and the points on a1, except A0, can be represented by the involutions in S . Similarly, we write S = {g , . . . , g }. See Figure 5.3. τ,Up τ,Up Ap2+1 A2p2
Let gAi,Aj = gAi gAj for some Ai ∈ (a0) \{A0} and Aj ∈ (a1) \{A0}. Then gAi,Aj ∈
Gπ is an involution such that Fix(gAi,Aj ) ∩ ((a0) ∪ (a1)) = {Ai,Aj,A0}. Therefore, the line through Ai and Aj can be represented by the involution gAi,Aj . See Figure 5.3. Hence, we can similarly represent the lines of Type II by some certain involutions.
Figure 5.4: Determining lines of Type I by certain group elements of order p
Let A be the intersection point of the lines represented by the involutions gAi,Ar and
2 2 2 gAj ,As , where i 6= j, 1 ≤ i, j ≤ p , and r 6= s, p + 1 ≤ r, s ≤ 2p , respectively. We also let a be the line of Type I passing through A0 and A. Our computation shows that 49 Fix(gAi,Ar ) ∩ Fix(gAj ,As ) = {A, A0} and (a) = Fix(gAi,Ar gAj ,As ), where gAi,Ar gAj ,As ∈
Gπ is of order p. See Figure 5.4.
Figure 5.5: Determining lines of Type I by certain group elements of order p
Proposition 5.1. Let π be one of the planes α, β, or Π. Then π can be reconstructed from Gπ.
Proof. Let a be a line of Type II passing through Ai and Aj. Then (a) = Fix(gAi,Aj ) \
{A0}, where gAi,Aj is the involution representing a. Let A0 and A be the intersection points of the line g with lines g and Ai,Ap2+1 A1,Ar 2 2 2 gA1,As , where r 6= s, p + 2 ≤ r, s ≤ 2p , and 2 ≤ i ≤ p . It easily follows from the definition of a projective plane that A0 and A are distinct points of Type II. See Figure 5.5. If i = 2, then we have that {(a) | a ∈ (A )} = {Fix(g g ) | p2 + 2 ≤ r ≤ 0 A2,Ap2+1 A1,Ar 2 2p } ∪ {(a0), (a1)}.
0 0 The lines of π can be determined by the sets Sτ,Up and Sτ ,Up as described above.
Hence, π can be reconstructed from Gπ.
50 List of Notations
α, β: A projective plane α0: Dual plane
Pα: Set of points of α Lα: Set of lines of α A I a: Point-line incidence A: An affine plane (A): Set of lines through A (a): Set of points on a AB: Line through A and B ab: Intersection point of a and b
Qα: Set of all quadrangles of α S: Closure of a set S B: Baer subset Ω: k-arc
ρ, τ: A collineation Gα: Full collineation group of α F(ρ): Configuration fixed by ρ %: A correlation
L, Lij: A latin square Ψ: An alphabet
T : A tableau Q: A quasigroup R: A quasifield (R, T ): A ternary ring N : A nearfield π: A VW plane σ: Hughes plane ς: Hall plane ξ: Figueroa plane Q-plane: A plane generated by a quadrangle
X G | X: A group action of G on X k : The collection of all k-subsets of X
Xk: The collection of all ordered k-subsets of X
(k) A = (ai,j): the matrix whose rows are the lexicographically minimal, distinct representatives of the orbits of G | Xk
51 Appendix A
VW plane of order 25 (α)
8 2 4 (i) |Gα| = 1, 440, 000 = 2 · 3 · 5
(ii) K has the following presentation: K = hx, y, a, b | x12, a5, b5, aba−1b−1, y2x3, y−1xy3x10, x−1axb2a3, y−1ayb4a2, x−1bxa3, y−1byb3ai
(iii) Generators of the full collineation group Gα: x : (27, 47, 44, 28, 43, 32, 30, 35, 38, 29, 39, 50)(31, 34, 42, 36, 37, 33, 46, 48, 40, 41, 45, 49) y : (27, 40, 29, 33, 30, 42, 28, 49)(31, 32, 41, 44, 46, 50, 36, 38)(34, 39, 45, 35, 48, 43, 37, 47) a : (26, 41, 31, 46, 36)(27, 42, 32, 47, 37)(28, 43, 33, 48, 38)(29, 44, 34, 49, 39) (30, 45, 35, 50, 40) b : (26, 49, 42, 40, 33)(27, 50, 43, 36, 34)(28, 46, 44, 37, 35)(29, 47, 45, 38, 31) (30, 48, 41, 39, 32) m : (6, 21)(7, 22)(8, 23)(9, 24)(10, 25)(11, 16)(12, 17)(13, 18)(14, 19)(15, 20) (31, 46)(32, 47)(33, 48)(34, 49)(35, 50)(36, 41)(37, 42)(38, 43)(39, 44)(40, 45)
Q25 u : v=1 (v, v + 25) 52 Appendix B
VW plane of order 49 (β)
10 2 4 (i) |Gβ| = 22, 127, 616 = 2 · 3 · 7
(ii) K has the following presentation: K = hx, y, a, b | x24, a7, b7, aba−1b−1, y2x12, y−1xyx17, x−1axb3a2, y−1ayb6a6, x−1bxa4, y−1byba2i
(iii) Generators of the full collineation group Gβ: x : (51, 77, 57, 94, 53, 68, 71, 84, 52, 97, 64, 89, 56, 79, 92, 62, 54, 88, 78, 72, 55, 59, 85, 67)(58, 65, 60, 63, 74, 95, 73, 75, 66, 80, 70, 69, 98, 91, 96, 93, 82, 61, 83, 81, 90, 76, 86, 87)
y : (51, 69, 56, 87)(52, 81, 55, 75)(53, 93, 54, 63)(57, 65, 92, 91)(58, 84, 98, 72) (59, 96, 97, 60)(61, 71, 95, 78)(62, 90, 94, 66)(64, 80, 85, 76)(67, 74, 89, 82) (68, 86, 88, 70)(73, 77, 83, 79)
a : (50, 84, 62, 89, 67, 94, 72)(51, 78, 63, 90, 68, 95, 73)(52, 79, 57, 91, 69, 96, 74) (53, 80, 58, 85, 70, 97, 75)(54, 81, 59, 86, 64, 98, 76)(55, 82, 60, 87, 65, 92, 77) (56, 83, 61, 88, 66, 93, 71)
53 b : (50, 64, 78, 92, 57, 71, 85)(51, 65, 79, 93, 58, 72, 86)(52, 66, 80, 94, 59, 73, 87) (53, 67, 81, 95, 60, 74, 88)(54, 68, 82, 96, 61, 75, 89)(55, 69, 83, 97, 62, 76, 90) (56, 70, 84, 98, 63, 77, 91)
m : (8, 43)(9, 44)(10, 45)(11, 46)(12, 47)(13, 48)(14, 49)(15, 36)(16, 37)(17, 38) (18, 39)(19, 40)(20, 41)(21, 42)(22, 29)(23, 30)(24, 31)(25, 32)(26, 33)(27, 34) (28, 35)(57, 92)(58, 93)(59, 94)(60, 95)(61, 96)(62, 97)(63, 98)(64, 85)(65, 86) (66, 87)(67, 88)(68, 89)(69, 90)(70, 91)(71, 78)(72, 79)(73, 80)(74, 81)(75, 82) (76, 83)(77, 84)
Q49 u : v=1 (v, v + 49)
54 Appendix C
VW plane of order 121 (Π)
8 2 2 4 (i) |GΠ| = 843, 321, 600 = 2 · 3 · 5 · 11
(ii) K has the following presentation: K = hx, y, a, b | x60, a11, b11, aba−1b−1, y2x30, y−1xyx49, x−1axb−1a3, y−1ayb2a−1,
x−1bxba6, y−1byba−1i.
(iii) Generators of the full collineation group GΠ: x : (2, 84, 80, 12, 70, 26, 3, 35, 38, 23, 18, 51, 5, 69, 75, 45, 24, 90, 9, 16, 17, 89, 47, 58, 6, 31, 33, 56, 93, 115, 11, 50, 54, 111, 64, 108, 10, 99, 96, 100, 116, 83, 8, 65, 59, 78, 110, 44, 4, 118, 117, 34, 87, 76, 7, 103, 101, 67, 41, 19)(13, 32, 105, 14, 104, 63, 25, 52, 88, 27, 86, 114, 49, 92, 43, 53, 39, 106, 97, 62, 74, 94, 77, 79, 61, 112, 15, 66, 21, 36, 121, 102, 29, 120, 30, 71, 109, 82, 46, 107, 48, 20, 85, 42, 91, 81, 95, 28, 37, 72, 60, 40, 57, 55, 73, 22, 119, 68, 113, 98)
y : (2, 22, 11, 112)(3, 32, 10, 102)(4, 42, 9, 92)(5, 52, 8, 82)(6, 62, 7, 72)(12, 21, 111, 113)(13, 31, 121, 103)(14, 41, 120, 93)(15, 51, 119, 83)(16, 61, 118, 73)(17, 71, 117, 63)(18, 81, 116, 53)(19, 91, 115, 43)(20, 101, 114, 33)(23, 30, 100, 104)(24, 40, 110, 94)(25, 50, 109, 84)(26, 60, 108, 74)(27, 70, 107, 64)(28, 80, 106, 54)(29, 90, 105, 44)(34, 39, 89, 95)(35, 49, 99, 85)(36, 59, 98, 75)(37, 69, 97, 65)(38, 79, 96, 55) 55 (45, 48, 78, 86)(46, 58, 88, 76)(47, 68, 87, 66)(56, 57, 67, 77)
a : (1, 21, 30, 39, 48, 57, 77, 86, 95, 104, 113)(2, 22, 31, 40, 49, 58, 67, 87, 96, 105, 114) (3, 12, 32, 41, 50, 59, 68, 88, 97, 106, 115)(4, 13, 33, 42, 51, 60, 69, 78, 98, 107, 116) (5, 14, 23, 43, 52, 61, 70, 79, 99, 108, 117)(6, 15, 24, 44, 53, 62, 71, 80, 89, 109, 118) (7, 16, 25, 34, 54, 63, 72, 81, 90, 110, 119)(8, 17, 26, 35, 55, 64, 73, 82, 91, 100, 120) (9, 18, 27, 36, 45, 65, 74, 83, 92, 101, 121)(10, 19, 28, 37, 46, 66, 75, 84, 93, 102, 111) (11, 20, 29, 38, 47, 56, 76, 85, 94, 103, 112)
b : (1, 5, 9, 2, 6, 10, 3, 7, 11, 4, 8)(12, 16, 20, 13, 17, 21, 14, 18, 22, 15, 19)(23, 27, 31, 24, 28, 32, 25, 29, 33, 26, 30)(34, 38, 42, 35, 39, 43, 36, 40, 44, 37, 41)(45, 49, 53, 46, 50, 54, 47, 51, 55, 48, 52)(56, 60, 64, 57, 61, 65, 58, 62, 66, 59, 63)(67, 71, 75, 68, 72, 76, 69, 73, 77, 70, 74)(78, 82, 86, 79, 83, 87, 80, 84, 88, 81, 85)(89, 93, 97, 90, 94, 98, 91, 95, 99, 92, 96)(100, 104, 108, 101, 105, 109, 102, 106, 110, 103, 107)(111, 115, 119, 112, 116, 120, 113, 117, 121, 114, 118)
m : (12, 111)(13, 112)(14, 113)(15, 114)(16, 115)(17, 116)(18, 117)(19, 118) (20, 119)(21, 120)(22, 121)(23, 100)(24, 101)(25, 102)(26, 103)(27, 104)(28, 105) (29, 106)(30, 107)(31, 108)(32, 109)(33, 110)(34, 89)(35, 90)(36, 91)(37, 92) (38, 93)(39, 94)(40, 95)(41, 96)(42, 97)(43, 98)(44, 99)(45, 78)(46, 79)(47, 80) (48, 81)(49, 82)(50, 83)(51, 84)(52, 85)(53, 86)(54, 87)(55, 88)(56, 67)(57, 68) (58, 69)(59, 70)(60, 71)(61, 72)(62, 73)(63, 74)(64, 75)(65, 76)(66, 77)
Q121 u : v=1 (v, v + 121)
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