#### 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. Actually, I’ll be finishing up the notes as we go. 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. The course notes are available here.

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 in class or posting questions to the course forum on Canvas. Here’s what Paul Halmos has to say about reading mathematics.

Don’t just read it; fight it! Ask your own questions, 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?

You can find the course notes below. 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

- 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: Prerequisites
- 2.1 Basic Set Theory
- 2.2 Relations
- 2.3 Partitions
- 2.4 Functions
- 2.5 Induction

- Chapter 3: An Intuitive Approach to Groups
- Chapter 4: Cayley Diagrams
- Chapter 5: An Introduction to Subgroups and Isomorphisms
- 5.1 Subgroups
- 5.2 Isomorphisms

- Chapter 6: A Formal Approach to Groups
- 6.1 Binary Operations
- 6.2 Groups
- 6.3 Group Tables
- 6.4 Revisiting Cayley Diagrams and Our Original Definition of a Group
- 6.5 Revisiting Subgroups
- 6.6 Revisiting Isomorphisms

- Chapter 7: Families of Groups
- 7.1 Cyclic Groups
- 7.2 Dihedral Groups
- 7.3 Symmetric Groups
- 7.4 Alternating Groups

- Chapter 8: Cosets, Lagrange’s Theorem, and Normal Subgroups
- 8.1 Coset
- 8.2 Lagrange’s Theorem
- 8.3 Normal Subgroups

- Chapter 9: Products and Quotients of Groups
- 9.1 Products of Groups
- 9.2 Quotients of Groups

- Chapter 10: Homomorphisms and the Isomorphism Theorems
- 10.1 Homomorphisms
- 10.2 The Isomorphism Theorems

- Chapter 11: An Introduction to Rings
- Appendix A: Elements of Style for Proofs
- Appendix B: Fancy Mathematical Terms
- Appendix C: Definitions in Mathematics