Math 4120 (Modern Algebra), Fall 2021

Math 4120 (Modern Algebra), Fall 2021

"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

Spring 2022 class webpage (slighly improved & updated materials)

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.

We will not use a textbook for this class because one does not yet exist that covers our approach. (I started writing it in October 2019, and should finish in 2022.) 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 class, we will play with the Rubik's cube. We will analyze art freises, wallpapers, chemical molecules, and Archimedian solids. At the end of the semester, you will truly understand groups and rings, subgroups, cosets, products and quotients, homomorphisms, group actions, conjugacy classes, centralizers, normalizers, semidirect products, theorems by Lagrange, Cayley, Cauchy, and Sylow, and what Évariste Galois stayed up until dawn writing the night before his untimely death in a duel at age 20, that remains one of the most celebrated achievements in all of mathematics, and which provided the framework necessary to elegantly solve several classic mathematical mysteries of the ancient Greeks. In the end, you will leave with a new appreciation of the beauty, and difficulty, of an area of mathematics you never dreamt existed.

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 original pdfs are not needed to read, understand, and grade them. Along with assignment, I will post ``scratch paper'' consisting of blank images that you are free to use, rather than re-draw by hand.
• HW 1: pdf | tex | img. Topics: Introduction to groups, symmetries, and Cayley diagrams. Due Friday, August 27, 2021.
• HW 2: pdf | tex | img. Topics: Examples of groups, roots of unity. Due Friday, September 3, 2021.
• HW 3: pdf | tex | img. Topics: Polytopes and groups of permutations. Due Friday, September 10, 2021.
• HW 4: pdf | tex | img. Topics: Dicyclic, semidihedral, and modular groups. Semi-direct products. Due Friday, September 17, 2021.
• HW 5: pdf | tex | img. Topics: Subgroups, cosets, and normalizers. Due Friday, September 24, 2021.
• HW 6: pdf | tex | img. Topics: Normality and conjugate subgroups. Due Friday, October 1, 2021.
• HW 7: pdf | tex | img. Topics: Quotient groups, conjugacy classes, and centralizers. Due Friday, October 8, 2021.
• HW 8: pdf | tex | img. Topics: Homomorphisms and isomorphisms. Due Friday, October 15, 2021.
• HW 9: pdf | tex | img. Topics: Isomorphism theorems, commutators, automorphisms, semidirect products.. Due Friday, October 22, 2021.
• HW 10: pdf | tex | img. Topics: Group actions. Due Friday, October 29, 2021.
• HW 11: pdf | tex | img. Topics: Groups acting on elements, subgroups, and cosets. Due Friday, November 5, 2021.
• HW 12: pdf | tex | img. Topics: Sylow theory. Due Friday, November 12, 2021.
• HW 13: pdf | tex | img. Topics: Rings and ideals. Due Monday, November 22, 2021.
• HW 14: pdf | tex. Topics: Prime, primary, and radical ideals. Divisibility and factorization. Due Friday, December 3, 2021.

Lecture notes (new!)

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 will eventually record YouTube lectures to go along with these, but not this semester. Most likely, during the Spring 2022 semester.

Lecture notes (old)

The following are a series of lecture notes (slides) I wrote. They originally followed the progression of the material in Visual Group Theory, though they are quite supplemented with proofs, rigor, and a lot of extra content.
After using these for a number of Math 4120 classes, they convergered to something that I really liked and thought I would never change. However, in October 2019, I decided to write a book on this, and in the process (as usually happens) I discovered so many more examples, visual ideas, and unique perspectives, than I had ever dreamt could have existed. I have almost 400 pages written, and 5 (of 8 chapters) and hope to finish in 2022.

When I first taught Math 4120 online, I made about 50 video lectures from the above content. I did this by breaking each section above into many small lectures. The links to the individual lectures and accompanying slides can be found below. Or, you can view the full YouTube playlist.

Section 1: Groups, intuitively. (6 lectures, 2 hrs 10 min.)

• Lecture 1.1: What is a group? [YouTube (16:15) | Slides]
• Lecture 1.2: Cayley graphs. [YouTube (33:27) | Slides]
• Lecture 1.3: Groups in science, art, and mathematics. [YouTube (30:36) | Slides]
• Lecture 1.4: Group presentations. [YouTube (18:12) | Slides]
• Lecture 1.5: Multiplication tables. [YouTube (19:23) | Slides]
• Lecture 1.6: The formal definition of a group. [YouTube (12:48) | Slides]
Section 2: Examples of groups. (4 lectures, 1 hr 27 min.)
Section 3: Structure of groups. (7 lectures, 2 hrs 52 min.)
Section 4: Maps between groups. (6 lectures, 3 hrs 27 min.)
• Lecture 4.1: Homomorphisms and isomorphisms. [YouTube (47:18) | Slides]
• Lecture 4.2: Kernels. [YouTube (31:32) | Slides]
• Lecture 4.3: The fundamental homomorphism theorem. [YouTube (32:52) | Slides]
• Lecture 4.4: Finitely generated abelian groups. [YouTube (24:47) | Slides]
• Lecture 4.5: The isomorphism theorems. [YouTube (46:19) | Slides]
• Lecture 4.6: Automorphisms. [YouTube (24:34) | Slides]
• Lecture 4.7: Semidirect products. [YouTube (??:??) | Slides (coming soon)]
Section 5: Group actions. (7 lectures, 3 hrs 45 min)
Section 6: Field and Galois theory. (8 lectures, 3 hr 52 min)
• Lecture 6.1: Fields and their extensions. [YouTube (26:34) | Slides]
• Lecture 6.2: Field automorphisms. [YouTube (35:41) | Slides]
• Lecture 6.3: Polynomials and irreducibility. [YouTube (38:21) | Slides]
• Lecture 6.4: Galois groups. [YouTube (34:13) | Slides]
• Lecture 6.5: Galois group actions and normal field extensions. [YouTube (26:28) | Slides]
• Lecture 6.6: The fundamental theorem of Galois theory. [YouTube (31:29) | Slides]
• Lecture 6.7: Ruler and compass constructions. [YouTube (22:46) | Slides]
• Lecture 6.8: Impossibility proofs. [YouTube (17:12) | Slides]
Section 7: Ring theory.
• Lecture 7.1: Basic ring theory. [YouTube (32:36) | Slides]
• Lecture 7.2: Ideals, quotient rings, and finite fields. [YouTube (34:20) | Slides]
• Lecture 7.3: Ring homomorphisms. [YouTube (45:53) | Slides]
• Lecture 7.4: Divisibility and factorization. [YouTube (39:38) | Slides]
• Lecture 7.5: Euclidean domains and algebraic integers. [YouTube (30:09) | Slides]
• Lecture 7.6: Rings of fractions. [YouTube (??:??) | Slides]
• Lecture 7.7: The Chinese remainder theorem. [YouTube (??:??) | Slides (coming soon)]