Algebra Commons^™

Counting The Moduli Space Of Pentagons On Finite Projective Planes, Maxwell Hosler

Senior Independent Study Theses

Finite projective planes are finite incidence structures which generalize the concept of the real projective plane. In this paper, we consider structures of points embedded in these planes. In particular, we investigate pentagons in general position, meaning no three vertices are colinear. We are interested in properties of these pentagons that are preserved by collineation of the plane, and so can be conceived as properties of the equivalence class of polygons up to collineation as a whole. Amongst these are the symmetries of a pentagon and the periodicity of the pentagon under the pentagram map, and a generalization of …

On The Local Theory Of Profinite Groups, Mohammad Shatnawi

Dissertations

Let G be a finite group, and H be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of H. The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group G into H/H^' an abelian group where H is a subgroup of G and H' is the derived group of H. One important first application is Burnside’s normal p-complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it. …

New Theorems For The Digraphs Of Commutative Rings, Morgan Bounds

Rose-Hulman Undergraduate Mathematics Journal

The digraphs of commutative rings under modular arithmetic reveal intriguing cycle patterns, many of which have yet to be explained. To help illuminate these patterns, we establish a set of new theorems. Rings with relatively prime moduli a and b are used to predict cycles in the digraph of the ring with modulus ab. Rings that use Pythagorean primes as their modulus are shown to always have a cycle in common. Rings with perfect square moduli have cycles that relate to their square root.

Abstract Algebra: Theory And Applications, Thomas W. Judson

eBooks

Tom Judson's Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. Rob Beezer has contributed complementary material using the open source system, Sage.

An HTML version on the PreText platform is available here.

The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and …

Groups Of Divisibility, Seth J. Gerberding

Honors Thesis

In this thesis, we examine a part of abstract algebra known as Groups of Divisibility. We construct these special groups from basic concepts. We begin with partially-ordered sets, then build our way into groups, rings, and even structures akin to rings of polynomials. In particular, we explore how elementary algebra evolves when an ordering is included with the operations. Our results follow the work done by Anderson and Feil, however we include more explicit proofs and constructions. Our primary results include proving that a group of divisibility can be provided with an order to make it a partially-ordered group; that …

Galois Theory And The Quintic Equation, Yunye Jiang

Honors Theses

Most students know the quadratic formula for the solution of the general quadratic polynomial in terms of its coefficients. There are also similar formulas for solutions of the general cubic and quartic polynomials. In these three cases, the roots can be expressed in terms of the coefficients using only basic algebra and radicals. We then say that the general quadratic, cubic, and quartic polynomials are solvable by radicals. The question then becomes: Is the general quintic polynomial solvable by radicals? Abel was the first to prove that it is not. In turn, Galois provided a general method of determining when …

Cayley Graphs Of Groups And Their Applications, Anna Tripi

MSU Graduate Theses

Cayley graphs are graphs associated to a group and a set of generators for that group (there is also an associated directed graph). The purpose of this study was to examine multiple examples of Cayley graphs through group theory, graph theory, and applications. We gave background material on groups and graphs and gave numerous examples of Cayley graphs and digraphs. This helped investigate the conjecture that the Cayley graph of any group (except Z_2) is hamiltonian. We found the conjecture to still be open. We found Cayley graphs and hamiltonian cycles could be applied to campanology (in particular, to the …

Specifications Grading In A First Course In Abstract Algebra, Mike Janssen

Faculty Work Comprehensive List

Specifications grading offers an alternative to more traditional, points-based grading and assessment structures. In place of partial credit, students are assessed pass/fail on whether or not they have achieved the learning outcomes being assessed on a given piece of work according to certain specifications, with limited opportunities for revision of non-passing work. This talk will describe the learning outcomes and specifications grading system I used in my Fall 2016 abstract algebra course, as well as student responses.

Jay Leno And Abstract Algebra, Adam Glesser, Martin Bonsangue

Journal of Humanistic Mathematics

The Jay Leno skit Jaywalking, showing ordinary people struggling to answer basic questions, is both entertaining and applicable to teaching. This article describes how an instructor can strengthen students' conceptual understanding by creating an element of confusion, or "cognitive dissonance," in the students' minds using Jaywalking-style interactions in the classroom.

Geometric Constructions From An Algebraic Perspective, Betzabe Bojorquez

Electronic Theses, Projects, and Dissertations

Many topics that mathematicians study at times seem so unrelated such as Geometry and Abstract Algebra. These two branches of math would seem unrelated at first glance. I will try to bridge Geometry and Abstract Algebra just a bit with the following topics. We can be sure that after we construct our basic parallel and perpendicular lines, bisected angles, regular polygons, and other basic geometric figures, we are actually constructing what in geometry is simply stated and accepted, because it will be proven using abstract algebra. Also we will look at many classic problems in Geometry that are not possible …

Convexity Properties Of The Diestel-Leader Group Γ_3(2), Peter J. Davids

Honors Projects

The Diestel-Leader groups are a family of groups first introduced in 2001 by Diestel and Leader in [7]. In this paper, we demonstrate that the Diestel-Leader group Γ₃(2) is not almost convex with respect to a particular generating set S. Almost convexity is a geometric property that has been shown by Cannon [3] to guarantee a solvable word problem (that is, in any almost convex group there is a finite-step algorithm to determine if two strings of generators, or “words”, represent the same group element). Our proof relies on the word length formula given by Stein and Taback …

The Kronecker-Weber Theorem: An Exposition, Amber Verser

Lawrence University Honors Projects

This paper is an investigation of the mathematics necessary to understand the Kronecker-Weber Theorem. Following an article by Greenberg, published in The American Mathematical Monthly in 1974, the presented proof does not use class field theory, as the most traditional treatments of the theorem do, but rather returns to more basic mathematics, like the original proofs of the theorem. This paper seeks to present the necessary mathematical background to understand the proof for a reader with a solid undergraduate background in abstract algebra. Its goal is to make what is usually an advanced topic in the study of algebraic number …

An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan

Honors Theses

Though it may seem non-intuitive, abstract algebra is often useful in the study of number theory. In this thesis, we explore some uses of abstract algebra to prove number theoretic statements. We begin by examining the structure of unique factorization domains in general. Then we introduce number fields and their rings of algebraic integers, whose structures have characteristics that are analogous to some of those of the rational numbers and the rational integers. Next we discuss quadratic fields, a special case of number fields that have important applications to number theoretic problems. We will use the structures that we introduce …

Students' Perceptions Of Sense Of Community In Abstract Algebra: Contributing Factors And Benefits, Hortensia Soto-Johnson, Nissa Yestness, Casey Dalton

Mathematical Sciences Faculty Publications

In this phenomenological study, we explore how multiple assessments contribute to creating a sense of community (SOC) in an undergraduate abstract algebra course. Strike (2004) describes community as a process rather than a feeling and outlines four characteristics of community: coherence, cohesion, care, and contact. In this report, we describe contributing factors to and perceived benefits of SOC that students provided in an open-ended interview. Our findings indicate students viewed the teacher and the classroom environment as the primary sources of creating a SOC. Our findings also suggest students believed the SOC of the classroom increased classroom interaction and opened …