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postgraduate thesis: The automorphism groups of unitary block designs and the existence of O'Nan configurations
Title  The automorphism groups of unitary block designs and the existence of O'Nan configurations 

Authors  
Issue Date  2014 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Tai, Y. [戴怡嘉]. (2014). The automorphism groups of unitary block designs and the existence of O'Nan configurations. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5295508 
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 nonabsolute 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 (nonabsolute) lines intersecting in six (absolute) points. In 1981 Piper [Piper, 1981] conjectured that the nonexistence 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 nonclassical, 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 nonDesarguesian 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 
Degree  Doctor of Philosophy 
Subject  Automorphisms Projective planes 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/216098 
HKU Library Item ID  b5295508 
DC Field  Value  Language 

dc.contributor.author  Tai, Yeeka   
dc.contributor.author  戴怡嘉   
dc.date.accessioned  20150821T23:11:35Z   
dc.date.available  20150821T23:11:35Z   
dc.date.issued  2014   
dc.identifier.citation  Tai, Y. [戴怡嘉]. (2014). The automorphism groups of unitary block designs and the existence of O'Nan configurations. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5295508   
dc.identifier.uri  http://hdl.handle.net/10722/216098   
dc.description.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 nonabsolute 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 (nonabsolute) lines intersecting in six (absolute) points. In 1981 Piper [Piper, 1981] conjectured that the nonexistence 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 nonclassical, 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 nonDesarguesian 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   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: AttributionNonCommerical 3.0 Hong Kong License   
dc.subject.lcsh  Automorphisms   
dc.subject.lcsh  Projective planes   
dc.title  The automorphism groups of unitary block designs and the existence of O'Nan configurations   
dc.type  PG_Thesis   
dc.identifier.hkul  b5295508   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5295508   