Course Materials
An Inquiry-Based Approach to Abstract Algebra
We will not be using a textbook this semester, but rather a task-sequence adopted for IBL. The task-sequence that we are using was written by me. Any errors in the notes are no one’s fault but my own. In this vein, if you think you see an error, please inform me, so that it can be remedied. I reserve the right to modify them as we go, but I will always inform you of any significant changes. The notes will be released incrementally. Each link below is to a PDF file.
- An Inquiry-Based Approach to Abstract Algebra (complete set of notes)
- Title Page
- Chapter 1: Introduction
- 1.1 What is Abstract Algebra?
- 1.2 An Inquiry-Based Approach
- 1.3 Rules of the Game
- 1.4 Structure of the Notes
- 1.5 Some Minimal Guidance
- Chapter 2: An Intuitive Approach to Groups
- Chapter 3: Cayley Diagrams
- Chapter 4: An Introduction to Subgroups and Isomorphisms
- 4.1 Subgroups
- 4.2 Isomorphisms
- Chapter 5: A Formal Approach to Groups
- 5.1 Binary Operations
- 5.2 Groups
- 5.3 Group Tables
- 5.4 Revisiting Cayley Diagrams and Our Original Definition of a Group
- 5.5 Revisiting Subgroups
- 5.6 Revisiting Isomorphisms
- Chapter 6: Families of Groups
- 6.1 Cyclic Groups
- 6.2 Dihedral Groups
- 6.3 Symmetric Groups
- 6.4 Alternating Groups
- Chapter 7: Cosets, Lagrange’s Theorem, and Normal Subgroups
- 7.1 Cosets
- 7.2 Lagrange’s Theorem
- 7.3 Normal Subgroups
- Chapter 8: Products and Quotients of Groups
- 8.1 Products of Groups
- 8.2 Quotients of Groups
- Chapter 9: Homomorphisms and the Isomorphism Theorems
- 9.1 Homomorphisms
- 9.2 The Isomorphism Theorems
- Chapter 10: An Introduction to Rings
- 10.1 Definitions and Examples
- 10.2 Ring Homomorphisms
- 10.3 Ideals and Quotient Rings
- 10.4 Maximal and Prime Ideals
- Appendix A: Prerequisites
- A.1 Basic Set Theory
- A.2 Relations
- A.3 Partitions
- A.4 Functions
- A.5 Induction
- Appendix B: Elements of Style for Proofs
- Appendix C: Fancy Mathematical Terms
- Appendix D: Definitions in Mathematics
I will not be covering every detail of the notes and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions. Here’s one of my favorite quotes about reading mathematics.
Don’t just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
Miscellaneous Materials
- Student Contract
- Setting the Stage (first day activity)
Exams
Below are links to each exam.
- Exam 1 (take-home) (PDF). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due Monday, October 3)
- Exam 2 (take-home) (PDF). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. For the last question, you will need the following pictures of quilts that correspond to group tables for groups of order 8. (Due Monday, October 31 by 4pm)
- Exam 3 (take-home) (PDF). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due Tuesday, December 6 by 5pm)
- Final Exam (take-home) (PDF). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due by 5pm on Thursday, December 15)
