"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

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

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.

- Course syllabus
- Course schedule
- Visual Group Theory, by Nathan Carter.
Steven Stogatz calls it
*One of the best introductions to group theory -- or to any branch of higher math -- I've ever read* - An inquiry-based approach to abstract algebra, by Dana Ernst. Free e-book which follows the "Visual Group Theory" approach.

- What is abstract algebra? (Wikipedia entry)
- A short article on abstract algebra, by Steven Strogatz, written for a general (non-mathematical) audience that appeared in the New York Times.
- Abstract Algebra: Theory and Applications, a free open-source textbook, by Tom Judson.
- Group
Explorer, a free software program to accompany
*Visual Group Theory* - Guidelines for good mathematical writing, by Francis Su. (4 pages)
- Group theory and the Rubik's cube, by Janet Chen (39 pages).
- Homepage of math professor and former Rubik's cube world recorder holder Macky Makisumi. He is interested in speedcubing theory and runs the website Cubefreak.
- Gödel, Escher, Bach: An Eternal Golden Braid
is a wonderful, playful, Pulitzer-Prize winning book exploring the
common themes and symmetries underlying mathematics, art, and
music. It was written by Doug Hofstadter, who Nathan Carter cites as an
influence in his writing of
*Visual Group Theory*(both were at Indiana University). - New discoveries! Every configuration of the Rubik's Cube Group is at most 20 "moves" from the solved state (Proven July 2010), or 26 "moves" in the quarter-turn metric (Proven August 2014).
- Crystal systems of minerals (lots of pictures, and references to group theory!)
- Articles on Group Theory and its Application to Chemistry from LibreTexts, a ChemWiki hosted at UC Davis.
- Tilings in everyday places, by Dror Bar-Natan of the University of Toronto.

- Section 1: Groups, intuitvely
*(61 pages. Last updated Sept 6, 2017)* - Section 2: Examples of groups
*(37 pages. Last updated Sept 6, 2017)* - Section 3: The structure of groups
*(63 pages. Last updated Sept 14, 2017)* - Section 4: Maps between groups
*(50 pages. Last updated Oct 25, 2017)* - Section 5: Groups acting on sets
*(56 pages. Last updated Oct 31, 2017)* - Section 6: Field and Galois theory
*(59 pages. Last updated Sept 6, 2017)* - Section 7: Ring theory
*(46 pages. Last updated Sept 6, 2017)*

**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]

- Lecture 2.1: Cyclic and abelian groups [YouTube (30:45) | Slides]
- Lecture 2.2: Dihedral groups [YouTube (16:14) | Slides]
- Lecture 2.3: Symmetric and alternating groups [YouTube (29:50) | Slides]
- Lecture 2.4: Cayley's theorem [YouTube (11:00) | Slides]

- Lecture 3.1: Subgroups. [YouTube (13:43) | Slides]
- Lecture 3.2: Cosets. [YouTube (19:11) | Slides]
- Lecture 3.3: Normal subgroups. [YouTube (14:00) | Slides]
- Lecture 3.4: Direct products. [YouTube (23:16) | Slides]
- Lecture 3.5: Quotients. [YouTube (40:38) | Slides]
- Lecture 3.6: Normalizers. [YouTube (22:46) | Slides]
- Lecture 3.7: Conjugacy classes. [YouTube (39:19) | Slides]

- 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)]

- Lecture 5.1: Groups acting on sets. [YouTube (32:35) | Slides]
- Lecture 5.2: The orbit-stabilizer theorem. [YouTube (27:41) | Slides]
- Lecture 5.3: Examples of group actions. [YouTube (44:05) | Slides]
- Lecture 5.4: Fixed points and Cauchy's theorem. [YouTube (14:00) | Slides]
- Lecture 5.5:
*p*-groups. [YouTube (22:13) | Slides] - Lecture 5.6: The Sylow theorems. [YouTube (48:37) | Slides]
- Lecture 5.7: Finite simple groups. [YouTube (36:34) | Slides]

- 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]

- 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

- 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.

- Study
guide

- Practice
Midterm 2 (Spring 2014)

- Practice
Midterm 1 (Fall 2013)

- Practice
Midterm 2 (Fall 2013)

- Practice
Midterm 1 (Fall 2010)

- Practice
Midterm 2 (Fall 2010)