Paige Mankey

Publication Date

2-1-2016

Abstract

During the past five years, a number of mathematicians have conducted research involving closure operations on the ideals of commutative rings. The most accessible paper on this is written by Neil Epstein, entitled "A Guide to Closure Operations in Commutative Algebra". This paper compiles much of the research done on the topic, and gives the reader an overview of closure operations on ideals, and includes examples, methods for constructions, and various special properties that arise from these operations as they pertain to ideals. However, very little research--if any--has been done on closures of subgroups. This thesis aims to give a comprehensive overview of closure operations on subgroups, much in the spirit of Epstein's paper. We begin with an introduction to the notion of closure operations on subgroups, as well as providing examples to familiarize the reader with the idea of a "closed" subgroup. We then deviate from following Epstein's work to investigate ways homomorphisms preserve this notion of closures across groups, ultimately arriving at a closed-subgroup equivalent of the First Isomorphism Theorem. We also provide constructions for these operations, as well as include a chapter involving the dual notion of interior operations and how they can interact with closure operations. In the interest of keeping the thesis self-contained, included is a preliminary chapter containing material from group theory that will resurface during the chapters on closure and interior operations. This section largely focuses on examples to prepare the reader for the material that lies ahead, and provides theory only when appropriate.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Janet Vassilev

Second Committee Member

Alexandru Buium

Third Committee Member

Charles Boyer

Language

English

Keywords

Closure operation, Subgroup, Group theory

Document Type

Thesis

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