Symmetric designs with bruck subdesigns

IfP is a finite projective plane of ordern with a proper subplaneQ of orderm which is not a Baer subplane, then a theorem of Bruck [Trans. AMS 78(1955), 464–481] asserts thatn≧m ²+m. If the equalityn=m ²+m were to occur thenP would be of composite order andQ should be called a Bruck subplane. It can be shown that if a projective planeP contains a Bruck subplaneQ, then in factP contains a designQ′ which has the parameters of the lines in a three dimensional projective geometry of orderm. A well known scheme of Bruck suggests using such aQ′ to constructP. Bruck’s theorem readily extends to symmetric designs [Kantor, Trans. AMS 146 (1969), 1–28], hence the concept of a Bruck subdesign. This paper develops the analoque ofQ′ and shows (by example) that the analogous construction scheme can be used to find symmetric designs.     

Compact projective planes with homogeneous ovals

Closed ovals exist only in 2-or 4-dimensional compact projective planes. We show that a plane of dimension 2 or 4 contains ahomogeneous closed oval iff the automorphism group contains SO₂ or SO₃, respectively. In 4-dimensional planes, existence of a homogeneous oval and existence of a homogeneous Baer subplane are equivalent. We determine the possible full automorphism groups of planes containing homogeneous ovals except for two possibilities in dimension 4 whose existence remains uncertain.     

On Restrictions of Automorphism Groups of Compact Projective Planes to Subplanes

It is shown that the restriction of automorphisms of a compact projective plane to a closed Baer subplane or to an open subgeometry, respectively, is a quotient mapping (in contrast to restrictions to arbitrary subgeometries). In the proof, we investigate lineations in towers of Baer subplanes.     

Nonexistence of disjoint compact Baer subplanes

We show that a certain fibration associated with a Baer subplane of a compact connected projective plane does not admit a cross section. This implies that, in marked contrast to the finite case, two Baer subplanes cannot be disjoint.     

A lower bound for the minimum weight of the dual 7-ary code of a projective plane of order 49

Existing bounds on the minimum weight d ^⊥ of the dual 7-ary code of a projective plane of order 49 show that this must be in the range 76 ≤ d ^⊥ ≤ 98. We use combinatorial arguments to improve this range to 88 ≤ d ^⊥ ≤ 98, noting that the upper bound can be taken to be 91 if the plane has a Baer subplane, as in the desarguesian case. A brief survey of known results for the minimum weight of the dual codes of finite projective planes is also included. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

On small {;k; q};-arcs in planes of order q2

In this paper we examine some properties of complete {;k; q};-arcs in projective planes of order q². In particular, we derive a lower bound for k, and we exhibit a family of arcs having low values of k which exist in every such plane having a Baer subplane. In addition we resolve the existence problem for complete {;k; 3};-arcs in PG(2, 9).     

On the embedding of some linear spaces in finite projective planes

The problem of embedding of linear spaces in finite projective planes has been examined by several authors ([1], [2], [3], [4], [5], [6]). In particular, it has been proved in [1] that a linear space which is the complement of a projective or affine subplane of order m is embeddable in a unique way in a projective plane of order n. In this article, we give a generalization of this result by embedding linear spaces in a finite projective plane of order n, which are complements of certain regularA-affine linear spaces with respect to a finite projective plane.     

The André/Bruck and Bose representation of conics in Baer subplanes of PG(2,q 2)

The André/Bruck and Bose representation ([1], [5,6]) of PG(2,q ²) in PG(4,q) is a tool used by many authors in the proof of recent results. In this paper the André/Bruck and Bose representation of conics in Baer subplanes of PG(2,q ²) is determined. It is proved that a non-degenerate conic in a Baer subplane of PG(2,q ²) is a normal rational curve of order 2, 3, or 4 in the André/Bruck and Bose representation. Moreover the three possibilities (classes) are examined and we classify the conics in each class. Received 1 September 1999; revised 17 July 2000.     

Subplanes of a tactical decomposition and singer groups of a projective plane

Sufficient and necessary conditions have been obtained for the following: (1) the substructure formed by a member of the partition of points and a member of the partition of lines to be a subplane; (2) the centralizer of a multiplier to be a Baer subplane. We establish the cyclicity of a Sylow 3-subgroup of the multiplier group of an abelian Singer group of square planar order. Sufficient conditions for the existence of a Type II divisor of a Singer group are given. For a Singer group of orderpq, p<q, we prove that if the order of the multiplier group is divisible byp, then the plane will admit a cyclic Singer group.Partially supported by a NSA grant     

Perfect Baer Subplane Partitions and Three-Dimensional Flag-Transitive Planes

The classification of perfectBaer subplane partitions of PG(2, q²) is equivalentto the classification of 3-dimensional flag-transitive planeswhose translation complements contain a linear cyclic group actingregularly on the line at infinity. Since all known flag-transitiveplanes admit a translation complement containing a linear cyclicsubgroup which either acts regularly on the points of the lineat infinity or has two orbits of equal size on these points,such a classification would be a significant step towards theclassification of all 3-dimensional flag-transitive planes. Usinglinearized polynomials, a parametric enumeration of all perfectBaer subplane partitions for odd q is described.Moreover, a cyclotomic conjecture is given, verified by computerfor odd prime powers q < 200, whose truth would implythat all perfect Baer subplane partitions arise from a constructionof Kantor and hence the corresponding flag-transitive planesare all known.     

On Planes of Order p2 in Which Every Quadrangle Generates a Subplane of Order p

After Gleason's result, in the late fifties the following conjecture appeared: if in a finite projective plane every quadrangle is contained in a unique Desarguesian proper subplane of order p, then the plane is Desarguesian (and its order is p ^d for some d). In this paper we prove the conjecture in the case when the plane is of order p ² and p is a prime.     

On regular {v,n}-arcs in finite projective spaces

A regular {v, n}-arc of a projective space P of order q is a set S of v points such that each line of P has exactly 0,1 or n points in common with S and such that there exists a line of P intersecting S in exactly n points. Our main results are as follows: (1) If P is a projective plane of order q and if S is a regular {v, n}-arc with n ≥ √q + 1, then S is a set of n collinear points, a Baer subplane, a unital, or a maximal arc. (2) If P is a projective space of order q and if S is a regular {v, n}-arc with n ≥ √q + 1 spanning a subspace U of dimension at least 3, then S is a Baer subspace of U, an affine space of order q in U, or S equals the point set Of U. © 1993 John Wiley & Sons, Inc.     

Unitals which meet Baer subplanes in 1 moduloq points

We prove that a parabolic unitalU in a translation plane of orderq ² with kernel containing GF(q) is a Buekenhout-Metz unital if and only if certain Baer subplanes containing the translation line of meetU in 1 moduloq points. As a corollary we show that a unital 16-03 in PG(2,q ²) is classical if and only if it meets each Baer subplane of PG(2,q ²) in 1 moduloq points.     

Baer subplanes generated by collineations between pencils of lines

In this paper we define a degenerateC _F-set in PG (2,q ²) as the set of points of intersection of corresponding lines under a suitable collineation between two pencils of lines with vertices two distinct pointsA andB mapping the lineA ∨B onto itself. We prove that every such a set is the union of the lineA ∨B and a Baer subplane and vice versa every Baer subplane can be seen as a subset of a degenerateC _F-set.     

Sharply transitive subsets of PΓL(2,F) and spreads covered by derivable partial spreads

It is well known that one can associate a spread and hence a translation plane with every flock of a Miquelian Minkowski plane. If the Miquelian Minkowski plane is described via the sharply 3-fold transitive action of the projective group PGL(2,F) on the projective line over the commutative fieldF then the flocks correspond to sharply transitive subsets of PGL(2,F). We show that under certain conditions one can also associate a spread with a sharply transitive subset of PL(2,F) where PL(2,F) acts on the projective line over the not necessarily commutative fieldF in the natural way.Dedicated to Professor J. Joussen on his 60th birthday     

Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

On tactical decompositions of class number 2

In this article we consider tactical decompositions of class number 2 of symmetric designs. Our main result says that if the orders are prime, then the only decompositions are of affine type. Moreover, we study symmetric decompositions of finite projective planes and show that, except in some cases, they are related to Baer subplanes, unitals, or 2 - ((m ² - m + 1)m, m, 1)designs.     

On a problem of Neumann

A conjecture widely attributed to Neumann is that all finite non-desarguesian projective planes contain a Fano subplane. In this note, we show that any finite projective plane of even order which admits an orthogonal polarity contains many Fano subplanes. The number of planes of order less than $n$ previously known to contain a Fano subplane was $O(logn)$, whereas the number of planes of order less than $n$ that our theorem applies to is not bounded above by any polynomial in $n$.     

Grosse Baer-Untergruppen auf Translationsebenen gerader Ordnung

We determine all collineation groups of finite translation planes of even order, which are generated by sufficiently large elementary abelian 2-subgroups all of whose involutions centralize a Baer subplane.     

Fano subplanes in finite Figueroa planes

It has been conjectured that all non-desarguesian projective planes contain a Fano subplane. The Figueroa planes are a family of non-translation planes that are defined for both infinite orders and finite order q ³ for q > 2 a prime power. We will show that there is an embedded Fano subplane in the Figueroa plane of order q ³ for q any prime power.