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postgraduate thesis: Characterizations and embedding of unitary block designs

TitleCharacterizations and embedding of unitary block designs
Authors
Advisors
Advisor(s):Wong, PPW
Issue Date2014
PublisherThe 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
AbstractA (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.
DegreeDoctor of Philosophy
SubjectProjective planes
Dept/ProgramMathematics
Persistent Identifierhttp://hdl.handle.net/10722/206337

 

DC FieldValueLanguage
dc.contributor.advisorWong, PPW-
dc.contributor.authorHui, Man-wa-
dc.contributor.author許敏華-
dc.date.accessioned2014-10-23T23:14:27Z-
dc.date.available2014-10-23T23:14:27Z-
dc.date.issued2014-
dc.identifier.citationHui, 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.urihttp://hdl.handle.net/10722/206337-
dc.description.abstractA (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.-
dc.languageeng-
dc.publisherThe University of Hong Kong (Pokfulam, Hong Kong)-
dc.relation.ispartofHKU Theses Online (HKUTO)-
dc.rightsThe author retains all proprietary rights, (such as patent rights) and the right to use in future works.-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subject.lcshProjective planes-
dc.titleCharacterizations and embedding of unitary block designs-
dc.typePG_Thesis-
dc.identifier.hkulb5312346-
dc.description.thesisnameDoctor of Philosophy-
dc.description.thesislevelDoctoral-
dc.description.thesisdisciplineMathematics-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.5353/th_b5312346-

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