2. Teaching
3. MAT411
4. Materials

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 and Table of Contents
• Preface
• 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
    • Proof of Theorem 4.34
    • Proof of Theorem 4.35
    • Solution to Problem 4.116
  • 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
    • Proof of Theorem 6.20
  • 6.2 Quotients of Groups
    • Solution to Problem 6.33
    • Solution to Problem 6.34
• 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

• Abstract Algebra videos
  • What is Abstract Algebra? | Socratica (3:21 min)
  • Abstract Algebra: The definition of a Group | Socratica (3:10 min)
  • Abstract Algebra: Motivation for the definition of a group | Socratica (2:04 min)
  • Group Definition (expanded) - Abstract Algebra | Socratica (11:14 min)
• Setting the Stage (PDF)
• Living Proof
• The Secret to Raising Smart Kids | Carol Dweck (article on Scientific American)
• 7 Things Growth Mindset is Not (article on Turn Around for Children)
• Videos on Growth Mindset and Productive Failure
  • Grit: the power of passion and perseverance | Angela Lee Duckworth (6:12 min)
  • How to Become a Better Learner | Learner Lab (13:30 min)
  • Learning Like a Jungle Tiger | Trevor Ragan (5:23 min, similar to previous video)
  • Productive Failure/Struggle Skateboard Video 1 (1:25min)
  • Productive Failure/Struggle Skateboard Video 2 (0:40 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)
  • KhanAcademy interview with Carol Dweck about growth mindset (3:06 min)
  • Make Mistakes | Michael Starbird (2:11 min)
  • Ira Glass on the Creative Process (1:54 min)
  • Productive Failure | Manu Kapur | TEDxLugano (17:28 min)
• The State of Being Stuck | Ben Orlin (blog post on Math with Bad Drawings)
• The Chinese Farmer | Alan Watts (2:26 min)
• Do Schools Kill Creativity? | Ken Robinson (19:22 min)
• The Learner Lab Podcast
• Do Hard Things | The Learner Lab (2:41 min)
• The Subtle Art of the Mathematical Conjecture (article on Quanta Magazine)
• Lessons from My Math Degree That Have Nothing to Do with Math (6 min read on Medium)
• Is Mathematics Invented or Discovered? | Roger Penrose (4:14 min)
• What is Mathematics? The Most Misunderstood Subject (short article by Dr. Robert H. Lewis, Professor of Mathematics, Fordham University)

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 (Part 2) (PDF). (Due by 11:59PM on Wednesday, September 23 Friday, September 25)
• Exam 2 (Part 2) (PDF). (Due by 11:59PM on Monday, November 2)
• Exam 3 (Option 2) (PDF). (Due by 11:59PM on Monday, November 23)