Date of Award

Spring 1-1-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

First Advisor

Nathaniel Thiem

Second Advisor

Farid Aliniaeifard

Third Advisor

Richard Green

Fourth Advisor

Martin Walter

Abstract

The set of supercharacter theories of a finite group forms a lattice under a natural partial order. An active area of research in the study of supercharacter theories is the classification of this lattice for various families of groups. One other active area of research is the formation of Hopf structures from compatible supercharacter theories over indexed families of groups. This thesis therefore has two goals. First, we will classify the supercharacter theory lattice of the dihedral groups D2n in terms of their cyclic subgroups of rotations, as well as for some semidirect products of the form ℤn ⋊ ℤp. Second, we will construct a pair of combinatorial Hopf algebras from natural supercharacter theories on the alternating and finite special linear groups and relate them using the theory of combinatorial Hopf algebras, as developed by Aguiar, Bergeron, and Sottile in 2006.

Included in

Algebra Commons

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