Visual Algebra textbook & supplemental materials

In 2010, I taught abstract algebra, roughly following Visual Group Theory (VGT), a book written by Nathan Carter, a few years after taking a course by Doug Hofstadter at the University of Indiana, by the same name. The renowned mathematician Steven Strogatz at Cornell, calls VGT "One of the best introductions to group theory -- or to any branch of higher math -- I've ever read." VGT has 300 color illustrations, and focuses on the intuition behind the difficult concepts in group theory. However, it is, by design, a "general audience book," and so I needed to greatly supplement it to meet the needs of undergraduate math majors.

After a few iterations of my course, it had converged to something that I thought would never change. When I taught it online in 2016, I recorded a full set of 46 YouTube videos, that I really liked. Finally, in 2019, after my students and I were equally frusterated at the lack of a textbook for the material I was teaching, I bit the bullet and decided to write one. It is titled Visual Algebra, and should be completed in 2023.

At first, I didn't know how much it would be different than a "souped up version of VGT," but the beauty (or horror) of writing a book, is that it takes on a life of its own and to a place you never could have prediced. Several years and 700 pages later, I have learned so much more about (undergraudate!) group theory than I ever thought existed, I expanded the scope to over an entire year's worth of content. In the process, I developed hundereds of new visuals that I never would have ever dreamed of just a few years earlier---when I naively thought my course had "converged". My book now as over 600 pictures, and I look back at my YouTube playlist with slight disappointment, and cannot wait until I get to re-record a brand new series of videos that follows Visual Algebra. I have since taught Algebra 2, and Graduate Algebra 1 following this approach.

On this page, I will post relevant materials, including slides, HW, exams, and links. If you are an instructor interested in teaching abstract algebra this way, please get in touch! I would be happy to help you, give advice, and share materials.

Table of Contents, with links to corresponding lecture slides

  1. Groups, intuitively (52 pages. Last updated Mar 5, 2024)
  2. Examples of groups (110 pages. Last updated Jan 31, 2024)
  3. Group structure (98 pages. Last updated Mar 4, 2024)
  4. Maps between groups (94 pages. Last updated Mar 9, 2024)
  5. Actions of groups (122 pages. Last updated Apr 1, 2024)
  6. Extensions of groups (80 pages. Last updated Dec 18, 2023)
  7. Universal constructions (97 pages. Last updated Dec 18, 2023)
  8. Rings (86 pages. Last updated Apr 12, 2024)
  9. Domains (88 pages. Last updated Jan 16, 2024)
  10. Fields (coming soon!)
  11. Galois theory (coming soon!)
To do list My courses that use Visual Algebra (click through for materials)
  1. Undergraduate Algebra I (Math 4120)
  2. Undergraduate Algebra II (Math 4130)
  3. Graduate Algebra I (Math 8510)
Visual Algebra exams Undergraduate Visual Algebra homework (Math 4120, 4130) Graduate Visual Algebra homework (Math 8510) Other resources

  All work is licensed under the Creative Commons BY-NC-SA 4.0 License.