Math 4120 (Algebra I), Fall 2026

Math 4120 (Algebra I), Fall 2026


"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." --Bertrand Russell

Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. --Hermann Weyl

"If you are someone who prefers large vistas and powerful theories, then it is essential to be able to test general results by applying them to simple examples....where one can do concrete calculations, sometimes with elaborate formulas, that help to make the general theory understandable. They keep your feet on the ground...A good example is a thing of beauty. It shines and convinces. It gives insight and understaning. It provides the bedrock of belief." --Sir Michael Atiyah, in Advice to a young mathematician.

About the class

Group theory is the study of symmetry, and it is one of the most beautiful areas in all of mathematics. It arises in puzzles, visual arts, music, nature, the physical and life sciences, computer science, cryptography, and of course, all throughout mathematics. Yet, it is usually taught with little to no visuals, which is a travesty. The primary reason for this is that every establish books does it this way, and instructors tend to teach it the way the learned it. Without available resources, this will never change. And I'm working to change this. My vision is that in 20 years, students will find it incredulous that this subject used to be taught non-visually. Just like how no one would teach calculus without graphs. Visuals are as essential in group theory as pictures are in fields such as crystallography or art history.

For this course, we will use my textbook Visual Algebra, which I finished writing this summer, but is not yet published. It will be available on Canvas. A few fantastic sources that helped inspire this class include a 2009 general-audience book called Visual Group Theory (VGT), by Nathan Carter. The renowned mathematician Steven Strogatz at Cornell, calls it 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. Another source is a free e-book called An inquiry-based approach to abstract algebra, by Dana Ernst. This follows the "Visual Group Theory" approach, but is more rigorous and proof-based. However, most of the proofs are not provided; you are supposed to fill them in. This is what the "inquiry-based" part means.

In this class, we will discover abstract algebra through beautiful visuals, rigorous mathematics, and concrete examples, such as the Rubik's cube, frieze and wallpaper patterns, roots of unity, matrices, permutations, polytopes, and chemical molecules. The two main visual themes running throughout the course will be Cayley graphs and subgroup lattices. Guided by these pictures and examples, we will develop the structure and theory of groups: subgroups and cosets, products and quotients, homomorphisms and isomorphisms, group actions, conjugacy, automorphisms, semidirect products, and the celebrated theorems of Lagrange, Cayley, Cauchy, and Sylow. We will then turn to rings and fields, studying ideals, quotient rings, ring homomorphisms, maximal ideals, and finite fields. By the end of the semester, you will leave with a new appreciation for the beauty, power, and difficulty of an area of mathematics that, just a few years ago, you may never have imagined existed.

Class essentials

Resources

Homework

Homework should be written up carefully and concisely. Please write in complete sentences. Part of your grade will be based on the presentation and clarity of your answers. Enough of the problem statements should be copied down so that your homework solutions are self-contained and the textbook is not needed to read, understand, and grade them. Along with assignment, I will post "supplemental material" consisting of blank images that you are free to use, rather than redraw by hand. The LaTeX files are also posted, but require a custom style file: visualalgebra.sty.

Lecture notes

See an explanation below for the story behind these, and why they are a new and improved version of what I thought had converged to something that I would never change! I am in the process of recording YouTube lectures to go along with these, and they can be found here.

Exams


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