Characterizations And Embedding Of Unitary Block Designs

Author : Man-Wa Hui
Publisher : Open Dissertation Press
ISBN 13 : 9781361346860
Total Pages : pages
Book Rating : 4.3/5 (468 download)

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Download or read book Characterizations and Embedding of Unitary Block Designs written by Man-Wa Hui and published by Open Dissertation Press. This book was released on 2017-01-27 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation, "Characterizations and Embedding of Unitary Block Designs" by Man-wa, Hui, 許敏華, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: A (finite) projective plane of order m, m an integer greater than 1, is a 2- (m DEGREES2 + 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 DEGREES3 + 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 DEGREES2. A unital of order n embedded in a projective planeπ of order n DEGREES2 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 DEGREES2) coordinatized by the finite field Fq DEGREES2, 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 [Grundhofer, 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 [Grundhofer, 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. DOI: 10.5353/th_b5312346 Subjects: Projective plane