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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 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?

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 book)

• Title Page and Table of Contents
• Preface
• Acknowledgments
• Chapter 1: Introduction
  • 1.1 What is This Course All About?
  • 1.2 An Inquiry-Based Approach
  • 1.3 Structure of the Notes
  • 1.4 Rules of the Game
  • 1.5 Rights of the Learner
  • 1.6 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 Permutation Groups and Cayley’s Theorem
  • 4.5 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

• 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 (1:46 min)
  • 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
• 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)

• 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.
  • Theorem 2.44
  • Theorem 2.47(a)
  • Theorem 2.47(b)
  • Theorem 2.65
  • Problem 2.72(abcdef)
  • Problem 4.120

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). This portion of the exam is due by the beginning of class on Wednesday, March 2 Friday, March 4.
• Exam 2 (Part 2). This portion of the exam is due by the beginning of class on Friday, April 15.
• Final Exam (Part 2). This portion of the exam is due by 5pm on Thursday, May 5.