Characterizations and embedding of unitary block designs

Abstract

A (finite) projective plane of order m, m an integer greater than 1, is a 2- (m^2 + m + 1, m + 1, 1) design. A unital (or a unitary block design) U of order n, n an integer greater than 2, is a 2-(n^3 + 1, n + 1, 1) design. An embedded unital is one that is a subdesign of a projective plane. If a unital of order n is embedded in a projective planeπ, then the order of π is at least n^2. A unital of order n embedded in a projective planeπ of order n^2 is called a polar unital if it consists of the absolute points and non-absolute lines of a unitary polarity of π. In particular, if π is the classical (Desarguesian) plane PG(2, q^2) coordinatized by the finite field Fq^2, then the polar unital is called a classical unital.

The main problem in the study of unitals is their characterization and classification. The classical unital does not contain a configuration of four lines meeting in six points (an O'Nan configuration) [O'Nan, 1972]. It is conjectured that this property characterizes the classical unital [Piper, 1979]. The classical unital is characterized by three conditions (I), (II) and (III): (I) is the absence of O'Nan configurations; (II) and (III) are further configurational requirements [Wilbrink, 1983]. The result depends on the classification of finite doubly transitive groups. Furthermore, when the order of a unital is even, (III) is a necessary condition of (I) and (II) [Wilbrink, 1983]. As for group theoretic characterizations, the only unitals that admit doubly transitive automorphism groups are the classical unitals and the Ree unitals [Kantor, 1985]. The classical unital is also characterized by the existence of sufficiently many translations [Grundhöfer, Stroppel, Van Maldeghem, 2013].

In this thesis, a necessary and sufficient condition is given for embedding a unital into a projective plane as a polar unital. A strengthened version of the condition is introduced and is shown to be necessary for a classical unital. Using the strengthened condition and results of [Wilbrink,1983] and [Grundhöfer, Stroppel and Van Maldeghem, 2013], a new intrinsic characterization of the classical unital is given without assuming the absence of O'Nan configurations. Finally, a unital of even order satisfying the first two intrinsic characterization conditions of Wilbrink is shown to be classical without invoking deep results from group theory.