A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

Translation planes of order q 2 admitting a two-transitive orbit of length q + 1 on the line at infinity

A classification is given of all translation planes of order q ² that admit a collineation group G admitting a two-transitive orbit of q + 1 points on the line at infinity.     

On a generalization of Kantor's likeable planes

Generalizing an idea of Kantor [7], Johnson and Wilke [5] introduced elusive sets of functions over GF(q) to represent translation planes of order q ² that admit a collineation group of order q ² in the linear translation complement and whose kern contains GF(q). In this paper we determine explicitly all elusive sets for q even. We obtain another translation plane of order 8².Deicated to Professors Adriano Barlotti and Luigi Antonio Rosati on the occasion of their 60th birthdayThis research was supported in part by a grant from the M.P.I. (40% funds).     

A characterization of Buekenhout-Metz unitals in PG(2, q 2), q even

The known examples of embedded unitals (i.e. Hermitian arcs) in PG(2, q ²) are B-unitals, i.e. they can be obtained from ovoids of PG(3, q) by a method due to Buekenhout. B-unitals arising from elliptic quadrics are called BM-unitals. Recently, BM-unitals have been classified and their collineation groups have been investigated. A new characterization is given in this paper. We also compute the linear collineation group fixing the B-unital arising from the Segre-Tits ovoid of PG(3, 2 ^r ), r3 odd. It turns out that this group is an Abelian group of order q ².Research supported by MURST.     

Projective Planes of Order q whose Collineation Groups have Order q 2

Translation planes of order q are constructed whose full collineation groups have order q ².     

On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

The translation planes of order q2 that admit SL(2,q) as a collineation group. II. Odd order

Let be a translation plane of odd order q², where q=p^r and p is a prime. If admits SL(2,q) (or PSL(2,q)) as a collineation group then is a Desarguesian, Hall, or Hering plane, or one of two Walker planes of order 25.Partially supported by grants from the National Science Foundation.     

PSL(2,q) as a collineation group of projective planes of small order

Let Π be a projective plane of order n admitting a collineation group G≅PSL(2, q) for some prime power q. It is well known for n=q that Π must be Desarguesian. We show that if n<q then only finitely many cases may occur for П, all of which are Desarguesian. We obtain some information in case n=q ² with q odd, notably that G acts irreducibly on П for q≠3, 5, 9. The material herein was presented to the University of Toronto in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author is grateful to Professor Chat Y. Ho, presently at the University of Florida, for guidance in this research.     

A class of semifields of order q4

This paper gives a class of semifields of order q⁴ for any prime power q = p^r with p greater than 3. It is shown that this class has left nucleus GF(q²), and right and middle nucleus GF(q). Although it is not proved, it is believed that all the semifields in this class are new.Supported in part by NSF     

On PSL(2,q) as a totally irregular collineation group

Non-abelian simple totally irregular collineation groups containing an involutorial perspectivity have been classified by the authors in a recent paper. They are PSL(2,q), PSL(3,q), PSU(3,q), Sz(q), the alternating group on 7 letters, and the second Janko sporadic simple group. In this article, we study PSL(2,q),q congruent to 1 modulo 4, as a collineation group containing an involutory homology.C. Y. Ho was partially supported by a NSA grant.     

The Classification of Projective Planes of Order q3 with Cone-Representations in PG (6, q)

The classification of cone-representations of projective planes of orderq ³ of index 3 and rank 4 (and so in PG(6,^q)) is completed. Any projective plane with a non-spread representation (being a cone-representation of the second kind) is a dual generalised Desarguesian translation plane, as found by Jha and Johnson, and conversely. Indeed, given any collineation of PG(2,^q) with no fixed points, there exists such a projective plane of order q³ , where q is a prime power, that has the second kind of cone-representation of index 3 and rank 4 in PG(6,q). An associated semifield plane of order q ³ is also constructed at most points of the plane. Although Jha and Johnson found this plane before, here we can show directly the geometrical connection between these two kinds of planes.     

On translation planes of orderq 2 that admit a group of orderq 2 (q−1); bartolone's theorem

This article is concerned with translation planesP of orderq ² and kernelK isomorphic toG F(q). IfP admits a collineation groupG in the linear translation. complement and the order ofG K/K isq ²(q?1) then it is shown thatP is either a semifield plane or is a Lüneburg-Tits, Walker or Betten plane. This generalizes earlier work of Bartolone.     

Translation planes of large dimension admitting nonsolvable groups

In this article, the question is considered whether there exist finite translation planes with arbitrarily small kernels admitting nonsolvable collineation groups. For any integerN, it is shown that there exist translation planes of dimension >N and orderq ³ admittingGL(2,q) as a collineation group.     

A nearfield-free definition of regular Hughes planes and some embeddings of PG(3,q) in Hughes planes of order q 4

We give a nearfield-free definition of some finite and infinite incidence systems by means of half-points and half-lines and show that they are projective planes. We determine a planar ternary ring for these planes and use it to determine the full collineation group and to demonstrate some embeddings of these planes among themselves. We show that these planes include all finite regular Hughes planes and many infinite ones. We also show that PG(3, q) embeds in Hu(q ⁴) (and show infinite versions of this embedding). Dan Hughes 80th Birthday.     

Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q² that admit a collineation group G admitting a two-transitive orbit of q+1 points is applied to show that the only projective translation planes of order q² admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.     

Combinatorial geometries representable over GF(3) and GF(q). I. The number of points

Letq be a prime power not divisible by 3. We show that the number of points (or rank-1 flats) in a combinatorial geometry (or simple matroid) of rankn representable over GF(3) and GF(q) is at mostn ². Whenq is odd, this bound is sharp and is attained by the Dowling geometries over the cyclic group of order 2.This research was partially supported by National Science Foundation Grant DMS-8521826 and a North Texas State University Faculty Research Grant.     

A group theoretic characterization of Buekenhout-Metz unitals

It is shown that a unital U embedded in PG(2,q²) is a Buekenhout-Metz unital if and only if U admits a linear collineation group that is a semidirect product of a Sylow p-subgroup of order q³ by a subgroup of order q − 1. This is the full linear collineation group of U except for two equivalence classes of unitals: (i) the classical unitals, and (ii) the Buekenhout-Metz unitals which can be expressed as a union of a partial pencil of conics. The unitals in class (ii) only occur when q is odd, and any two of them are projectively equivalent. © 1996 John Wiley & Sons, Inc.     

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.     

Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries

Letq be an odd prime power not divisible by 3. In Part I of this series, it was shown that the number of points in a rank-n combinatorial geometry (or simple matroid) representable over GF(3) and GF(q) is at mostn ². In this paper, we show that, with the exception ofn = 3, a rank-n geometry that is representable over GF(3) and GF(q) and contains exactlyn ² points is isomorphic to the rank-n Dowling geometry based on the multiplicative group of GF(3).This research was partially supported by the National Science Foundation under Grants DMS-8521826 and DMS-8500494.