Citation
Lane, Richard Neili (1968) Normal structures and automorphism groups of tdesigns. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/9TC8YA06. https://resolver.caltech.edu/CaltechTHESIS:12212015113506968
Abstract
Combinatorial configurations known as tdesigns are studied. These are pairs ˂B, ∏˃, where each element of B is a ksubset of ∏, and each tdesign occurs in exactly λ elements of B, for some fixed integers k and λ. A theory of internal structure of tdesigns is developed, and it is shown that any tdesign can be decomposed in a natural fashion into a sequence of “simple” subdesigns. The theory is quite similar to the analysis of a group with respect to its normal subgroups, quotient groups, and homomorphisms. The analogous concepts of normal subdesigns, quotient designs, and design homomorphisms are all defined and used.
This structure theory is then applied to the class of tdesigns whose automorphism groups are transitive on sets of t points. It is shown that if G is a permutation group transitive on sets of t letters and ф is any set of letters, then images of ф under G form a tdesign whose parameters may be calculated from the group G. Such groups are discussed, especially for the case t = 2, and the normal structure of such designs is considered. Theorem 2.2.12 gives necessary and sufficient conditions for a tdesign to be simple, purely in terms of the automorphism group of the design. Some constructions are given.
Finally, 2designs with k = 3 and λ = 2 are considered in detail. These designs are first considered in general, with examples illustrating some of the configurations which can arise. Then an attempt is made to classify all such designs with an automorphism group transitive on pairs of points. Many cases are eliminated of reduced to combinations of Steiner triple systems. In the remaining cases, the simple designs are determined to consist of one infinite class and one exceptional case.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  30 March 1968 
Record Number:  CaltechTHESIS:12212015113506968 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:12212015113506968 
DOI:  10.7907/9TC8YA06 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  9339 
Collection:  CaltechTHESIS 
Deposited By:  INVALID USER 
Deposited On:  21 Dec 2015 22:15 
Last Modified:  21 Dec 2019 04:42 
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