The automorphism groups of unitary block designs and the existence of O'Nan configurations

Abstract

A unital is a 2-(n^3 + 1, n + 1, 1) design. An important invariant of a unital is its automorphism group. A projective plane is a 2-(m^2 +1, m+1, 1) design. A polar unital is a unital that consists of the absolute points and non-absolute lines of a unitary polarity in a projective plane. The collineation group stabilizing a polar unital in a projective plane is always a subgroup of the automorphism group of the unital. The classical example of a unital is a classical unital H of order q. It is an embedded unital in PG(2, q^2). In 1972 O'Nan [O'Nan, 1972] proved that Aut(H), the design automorphism group, is isomorphic to Col(H), the collineation subgroup of PG(2, q^2) stabilizing H. It was also observed in [O'Nan, 1972] that in the classical unital there is no O'Nan configuration: four (non-absolute) lines intersecting in six (absolute) points. In 1981 Piper [Piper, 1981] conjectured that the non-existence of O'Nan configurations characterizes the classical unital.

In this thesis, we study two classes of unitary block designs: the Ganley unital in the Dickson semifield plane and the Figueroa unital in the Figueroa plane. We study the existence of O'Nan configurations in these unitals and also investigate their automorphism groups.

A Ganley unital is defined by a unitary polarity in a Dickson semifield plane [Ganley, 1972]. For a Dickson semifield plane II(K(σ)), let U(σ) be the Ganley unital defined. We prove that every Ganley unital U(σ), parametrized by a field automorphism σ, is non-classical, extending a result of Ganley's [Ganley, 1972]; we prove that U(σ_1) is isomorphic to U(σ_2) if and only if σ_1 = σ_2 or σ_1 = σ_2^(-1) ; and we determine the automorphism group of U(σ). The finite Figueroa plane is another class of non-Desarguesian projective planes [Figueroa, 1982; Hering and Schaeffer, 1982]. A synthetic construction of the finite Figueroa plane is known [Grundhofer, 1986]. A Figueroa planes of finite square order possess a unitary polarity [de Resmini and Hamilton, 1998]. The unital defined is called the Figueroa unital. We introduce an alternative synthetic description of the Figueroa plane leading to an alternative synthetic description of the Figueroa unital. We demonstrate the existence of O'Nan configurations in the Figueroa unital, thus providing support to Piper's conjecture. We extend and complete some of the partial structural results obtained in [Hui and Wong, 2012], and also demonstrate the existence of many classical unitals of order q in a Figueroa unital of order q^3