2. Teaching
3. MAT 411
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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.

• 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)
• The Chinese Farmer | Alan Watts (2:26 min)
• The Secret to Raising Smart Kids | Carol Dweck (article on Scientific American)
• Do Schools Kill Creativity? | Ken Robinson (19:22 min)
• Lessons from My Math Degree That Have Nothing to Do with Math (6 min read on Medium)
• Pictures of board work that we ran out of time to discuss. Note: You should not assume that the solutions/proofs posted below are valid or sufficient.
  • Problem 2.9
  • Problem 2.10
  • Problem 2.11
  • Problem 3.15
  • Problem 3.17

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, September 26)
• Exam 2 (take-home portion) (PDF). (Due Monday, October 29)
• Exam 3 (take-home portion) (PDF). (Due Wednesday, November 21)
• Final Exam (take-home portion) (PDF). (Due by 5pm on Thursday, December 13)