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Normal Structures and Automorphism Groups of t-Designs


Lane, Richard Neil (1968) Normal Structures and Automorphism Groups of t-Designs. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/9TC8-YA06.


Combinatorial configurations known as t-designs are studied. These are pairs ˂B, ∏˃, where each element of B is a k-subset of ∏, and each t-design occurs in exactly λ elements of B, for some fixed integers k and λ. A theory of internal structure of t-designs is developed, and it is shown that any t-design 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 t-designs 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 t-design 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 t-design to be simple, purely in terms of the automorphism group of the design. Some constructions are given.

Finally, 2-designs 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 and English)
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Minor Option:English
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Hall, Marshall
Thesis Committee:
  • Unknown, Unknown
Defense Date:30 March 1968
Record Number:CaltechTHESIS:12212015-113506968
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9339
Deposited On:21 Dec 2015 22:15
Last Modified:02 Apr 2024 23:04

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