### 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 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 Google Group. 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: An Intuitive Approach to Groups (Updated on 9/9/2014)
- Chapter 3: Cayley Diagrams (Updated on 9/7/2014)
- Chapter 4: An Introduction to Subgroups and Isomorphisms (Updated on 9/9/2014)
- 4.1 Subgroups
- 4.2 Isomorphisms

- Chapter 5: A Formal Approach to Groups (Updated on 10/21/2014)
- 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 (Updated on 11/4/2014)
- 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 (Updated on 11/4/2014)
- 7.1 Cosets
- 7.2 Lagrange’s Theorem
- 7.3 Normal Subgroups

- Chapter 8: Products and Quotients of Groups (Updated on 11/12/2014)
- 8.1 Products of Groups
- 8.2 Quotients of Groups

- Chapter 9: Homomorphisms and the Isomorphism Theorems (see notes from class)
- 9.1 Homomorphisms
- 9.2 The Isomorphism Theorems

- Chapter 11: An Introduction to Rings (see notes from class)
- 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