Mathematics faculty and graduate students are invited to attend the virtual MS Defense of Aaron Lear.

Date:  Tuesday, June 28, 2022
Time:  11:00 a.m.

To Join visit: https://zoom.us/j/95631356964?pwd=WTY4TmJ1Z2k2Z0NHN0dkaTVHVVVHQT09
Meeting ID: 956 3135 6964
Passcode: 318551

Advisor:  Dr. Anton Betten

Committee:  Dr. Henry Adams, Dr. Aaron Nielsen

Title: IMPRIMITIVELY GENERATED DESIGNS

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 $b \subset V$. 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.