Course Materials
An Inquiry-Based Approach to Abstract Algebra
This free and open-source textbook was written by me (Dana C. Ernst) and is designed to be used with an inquiry-based learning (IBL) approach to a first-semester undergraduate abstract algebra course. The textbook starts with groups (up to the First Isomorphism Theorem) and finishes with an introduction to rings (up to quotients by maximal and prime ideals). While the textbook covers many of the standard topics, the focus is on building intuition and emphasizes visualization. The source files are located on GitHub.
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?
The notes will be released incrementally. Each link below is to a PDF file. If you’ve found an error or have suggestions for improvements, please let me know.
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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 Introduction to Groups
- 2.1 A First Example
- 2.2 Binary Operations
- 2.3 Groups
- 2.4 Generating Sets
- 2.5 Group Tables
- 2.6 Cayley Diagrams
- Chapter 3: Subgroups and Isomorphisms
- 3.1 Subgroups
- 3.2 Subgroup Lattices
- 3.3 Isomorphisms
- Chapter 4: Families of Groups
- 4.1 Cyclic Groups
- 4.2 Dihedral Groups
- 4.3 Symmetric Groups
- 4.4 Alternating Groups
- Chapter 5: Cosets, Lagrange’s Theorem, and Normal Subgroups
- 5.1 Cosets
- 5.2 Lagrange’s Theorem
- 5.3 Normal Subgroups
- Chapter 6: Products and Quotients of Groups
- 6.1 Products of Groups
- 6.2 Quotients of Groups
- Chapter 7: Homomorphisms and the Isomorphism Theorems
- 7.1 Homomorphisms
- 7.2 The Isomorphism Theorems
- Chapter 8: An Introduction to Rings
- 8.1 Definitions and Examples
- 8.2 Ring Homomorphisms
- 8.3 Ideals and Quotient Rings
- 8.4 Maximal and Prime Ideals
- Appendix A: Elements of Style for Proofs
- Appendix B: Fancy Mathematical Terms
- Appendix C: Definitions in Mathematics
Miscellaneous Materials
- Student Contract (PDF)
- Setting the Stage (PDF)
- Videos on Growth Mindset and Productive Failure
- Productive Failure/Struggle Skateboard Video (1:46 min)
- Productive Failure/Struggle BMX Video (1:39 min)
- Michael Jordan Failure Commercial (0:32 min)
- Mindsets: Fixed Versus Growth (2:19 min)
- Growth Mindset Animation (3:50 min)
- Grit: the power of passion and perseverance | Angela Lee Duckworth (6:12 min)
- KhanAcademy interview with Carol Dweck about growth mindset (3:06 min)
- Make Mistakes | Michael Starbird (2:11 min)
- Learning Like a Jungle Tiger | Trevor Ragan (5:23 min)
- Ira Glass on the Creative Process (1:54 min)
- Pictures of board work that we ran out of time to discuss. Note: You should not assume that the solutions/proof posted below are valid.
Exams
Below are links to the take-home portions of each exam. 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.
- Exam 1 (take-home portion) (PDF). (Due Wednesday, February 14)
- Exam 2 (take-home portion) (PDF). For the last question, you will need the following pictures of quilts that correspond to tables for groups of order 8. (Due Friday, March 16)
- Exam 3 (take-home portion) (PDF). (Due Wednesday, April 25)
- Final Exam (take-home portion) (PDF). (Due by 10am on Friday, May 11)
