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postgraduate thesis: Characterizations and embedding of unitary block designs
Title  Characterizations and embedding of unitary block designs 

Authors  
Advisors  Advisor(s):Wong, PPW 
Issue Date  2014 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Hui, M. [許敏華]. (2014). Characterizations and embedding of unitary block designs. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5312346 
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 nonabsolute 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. 
Degree  Doctor of Philosophy 
Subject  Projective planes 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/206337 
HKU Library Item ID  b5312346 
DC Field  Value  Language 

dc.contributor.advisor  Wong, PPW   
dc.contributor.author  Hui, Manwa   
dc.contributor.author  許敏華   
dc.date.accessioned  20141023T23:14:27Z   
dc.date.available  20141023T23:14:27Z   
dc.date.issued  2014   
dc.identifier.citation  Hui, M. [許敏華]. (2014). Characterizations and embedding of unitary block designs. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5312346   
dc.identifier.uri  http://hdl.handle.net/10722/206337   
dc.description.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 nonabsolute 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.   
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  Projective planes   
dc.title  Characterizations and embedding of unitary block designs   
dc.type  PG_Thesis   
dc.identifier.hkul  b5312346   
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_b5312346   