Math 4120 (Modern Algebra), Fall 2017

# Math 4120 (Modern Algebra), Fall 2017

"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

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.

We will not use a tradtional textbook for this class. Rather, we will draw from several sources: a 2009 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. Though the proof-writing is not the primary focus in the book, we will use our new-found intuition to write mathematical proofs. The second 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, chemical molecules, and contra dances. At the end of the semester, you will truly understand groups, 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.

### Lecture notes

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.
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. To avoid blurriness, these are best viewed by changing the settings to 720p (High Definition) rather than the default of 240p. This can be easily done by clicking the "wheel" on the lower right corner; right next to the "cc" button.

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)]

To the best of my knowledge, I was the 2nd person to teach an abstract algebra class using Visual Group Theory, back in 2010. The first was taught by Dana Ernst at Plymouth State University (now at Northern Arizona). Dana currently teaches abstract algebra in an Inquiry Based Learning format using the "Visual Group Theory" approach, and he has written a wonderful accompanying set of notes (basically an e-book) that is freely available.

### 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.
• HW 1: pdf | tex. Topics: Introduction to groups. Due Thursday, August 31, 2017.
• HW 2: pdf | tex. Topics: Cayley graphs & multiplication tables. Due Thursday, September 7, 2017.
• HW 3: pdf | tex. Topics: Cyclic, abelian, symmetric, and alternating groups. Due Thursday, September 14, 2017.
• HW 4: pdf | tex. Topics: Subgroups, cosets, & Cayley's theorem. Due Thursday, September 21, 2017.
• HW 5: pdf | tex. Topics: Normal subgroups. Due Thursday, September 28, 2017.
• HW 6: pdf | tex. Topics: Quotients, normalizers, & conjugacy classes. Due Thursday, October 5, 2017.
• HW 7: pdf | tex. Topics: Homomorphisms. Due Thursday, October 12, 2017.
• HW 8: pdf | tex. Topics: The isomorphism theorems; finite abelian groups. Due Tuesday, October 24, 2017.
• HW 9: pdf | tex. Topics: Commutators, automorphisms, & group actions. Due Tuesday, October 31, 2017.
• HW 10: pdf | tex. Topics: The orbit-stabilizer theorem; the Sylow theorems. Due Tuesday, November 7, 2017.
• HW 11: pdf | tex. Topics: Finite fields, extension fields, and Galois groups. Due Tuesday, November 14, 2017.
• HW 12: pdf | tex. Topics: Irreducibility; the fundamental theorem of Galois theory. Due Tuesday, November 21, 2017.
• HW 13: pdf | tex. Topics: Rings, ideals, and homomorphisms. Due Monday, December 4, 2017.
• HW 14: pdf | tex. Topics: Prime and maximal ideals; PIDs; algebraic integers. Due Friday, December 8, 2017.