My name is Aram Lindroth. I’m a junior at DA, and while I find all of my academic classes interesting, I’m particularly interested in math. During the fall semester I took a Global Online Academy course on Number Theory alongside my year-long math class at DA (Geometries, Topologies, and Shapes), and this semester I’m replacing that GOA course with this independent study. I’ve been interested in group theory for quite a while, so I’m coming into the study with at least some background, and I’m hoping that through this study I’ll be able to solidify my knowledge and fill in the gaps.
This blog is where I post updates and solutions to exercises from this independent study, which primarily focuses on group theory but also includes a brief introduction to rings. The primary text for the study is Joseph Gallian’s Contemporary Abstract Algebra, in which most of the exercises that I discuss on this blog can be found. Nathan Carter’s Visual Group Theory is used as a supplemental resource that presents the content in a different style.
The original proposal for this study is included below.
INDEPENDENT STUDY PROPOSAL (Spring 2019)
Title: Introduction to Abstract Algebra
Advisor: Dr. Prudhom
Purpose: The purpose of this independent study will be to investigate basic abstract algebra, beginning with groups, and eventually moving to rings near the end of the semester. Accordingly, the driving questions of the study, listed roughly in the order that they will be addressed, include:
- What are groups, and why are they defined in the way that they are?
- How are groups useful, both in mathematics and non-mathematical fields (eg. chemistry, computer science)?
- How do groups behave, and what are their key properties?
- What are rings, and what are their key properties? How are they similar to and different from groups?
Resources: Gallian, Joseph A. Contemporary Abstract Algebra. 8th ed., Brooks/Cole Cengage Learning, 2013.
Preliminary Monthly Plan:
January (only 2 weeks, starting with 2nd semester/3rd quarter)
- Chapters 1-3
- Definition and fundamental properties of groups
- Dihedral groups
- Subgroups and subgroup tests
- Chapters 4-6
- Cyclic groups, subgroups of cyclic groups
- Permutation groups
- Isomorphisms, Cayley’s theorem, automorphisms
- Chapters 7-10
- Cosets, Lagrange’s Theorem
- Direct products
- Normal subgroups and quotient groups
- Homomorphisms, isomorphism theorems
- Chapters 11, 24, 12
- Fundamental Theorem of Finite Abelian Groups
- Sylow Theorems
- Introduction to rings
- Chapters 13-15
- Integral domains, fields
- Ideals, factor rings
- Ring homomorphisms
At each step, I will complete exercises from the textbook in order to assess my understanding and to show my progress through the material. I also anticipate that these exercises, and questions that arise from them, will form the basis of weekly meetings with Dr. Prudhom.
Final Product: Possible final products include a presentation on a specific topic to the math club and/or a written paper that would explain and teach a specific topic covered in a manner similar to a textbook, but with my own examples and explanations. I plan to choose my specific final product after beginning the study, once I have enough experience with the material to know what form of product would best demonstrate my learning.
Regardless of what form my final product takes, I also plan to use what I learn through this independent study to fulfill the requirements of the Shapes Project, the second-semester independent research project in my current math course (Geometries, Topologies, and Shapes). This will involve leading the class in a discussion on the topic at some point during the second semester, as well as a written reflection at the end of the semester.