Date of Award

Spring 8-31-2019

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

First Advisor

Nathaniel Thiem

Second Advisor

Richard M. Green

Third Advisor

Martin Walter

Fourth Advisor

Farid Aliniaeifard

Fifth Advisor

Amanda A. Schaeffer Fry

Abstract

Since its introduction, supercharacter theory has been used to study a wide variety of problems, often involving only very specific families of groups. For an arbitrary finite group G, however, the structure of the set of supercharacter theories of G remains mysterious, as does its connection to the group G itself. This thesis aims to provide the basics of a framework for studying a group via its set of supercharacter theories. We define analogs of the center and commutator subgroup, and show that these subgroups enjoy many properties analogous to those of their classical counterparts. The analog of the commutator subgroup additionally allows us to illustrate a relationship between Camina triples and Δ-products, and to ultimately prove a supercharacter characterization of Δ-products. We also associate to any supercharacter theory S of G and supernormal subgroup N a supercharacter theory of N that generalizes the difference between G-conjugacy and N-conjugacy, which we call refined restriction. Refined restriction allows us to develop a generalized version of subnormality, which has a similar relationship to supernormality as classical subnormality has to normality. We then introduce S-normal series, which are normal series defined using S-normal subgroups. Among these S-normal series are S-central series and S-abelian series, which are defined using the analogs of the center and commutator subgroup. Using S-central series, we develop the concept of S-nilpotence, a coarser version of nilpotence. Finally, we use refined restriction and S-abelian series to define S-solvability, an analog of solvability. We show that S-solvability is related to S-nilpotence in a way that recovers the usual relationship when S is the finest supercharacter theory of G.

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