Imprimitively generated designs
Date
2022
Authors
Lear, Aaron, author
Betten, Anton, advisor
Adams, Henry, committee member
Nielsen, Aaron, committee member
Journal Title
Journal ISSN
Volume Title
Abstract
Designs are a type of combinatorial object which uniformly cover all pairs in a base set V with subsets of V known as blocks. One important class of designs are those generated by a permutation group G acting on V and single initial block b subset of V. The most atomic examples of these designs would be generated by a primitive G. This thesis focuses on the less atomic case where G is imprimitive. Imprimitive permutation groups can be rearranged into a subset of easily understood groups which are derived from G and generate very symmetrical designs. This creates combinatorial restrictions on which group and block combinations can generate a design, turning a question about the existence of combinatorial objects into one more directly involving group theory. Specifically, the existence of imprimitively generated designs turns into a question about the existence of pair orbits of an appropriate size, for smaller permutation groups. This thesis introduces two restrictions on combinations of G and b which can generate designs, and discusses how they could be used to more efficiently enumerate imprimitively generated designs.