Math 4120 (Modern Algebra), Summer I 2016 (online)

"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

About the class

Group theory is the study of symmetry, and 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 use 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.

In class, we will play with the Rubik's cube. We will study patterns and symmetry and use free mathematical software such as Sage and Group Explorer. We will analyze art freises, chemical molecules, and contra dances. At the end of the semester, you will truly understand groups, subgroups, cosets, product 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. In the end, you will leave with a new appreciation of the beauty, and difficulty, of an area of mathematics you never dreamt existed.

Class essentials

Course Calendar
Course Syllabus
Proctor test policies
Proctor approval form
Visual Group Theory, by Nathan Carter. (Required textbook). Steven Stogatz calls it One of the best introductions to group theory -- or to any branch of higher math -- I've ever read

Resources

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).
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 discovery as of July 2010: Every configuration of the Rubik's Cube Group is at most 20 moves from the solved state.
Crystal systems of minerals (lots of pictures, and references to group theory!)
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.

Lectures

Links to the individual lectures are listed below. Or, you can view the full YouTube playlist here.

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.)

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]

Section 3: Structure of groups. (7 lectures, 2 hrs 52 min.)

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]

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)

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]

Section 6: Field theory 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). These lecture notes (Chapters 1-7, and the beginning of Chapter 8) began as modifications of ones Dana wrote, though they have significantly diverged, and now do not resemble much of Dana's original slides.

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.

Homework 1: pdf | tex. Due Thursday, May 12, 2016
Homework 2: pdf | tex. Due Friday, May 13, 2016
Homework 3: pdf | tex. Due Monday, May 16, 2016
Homework 4: pdf | tex. Due Wednesday, May 18, 2016
Homework 5: pdf | tex. Due Friday, May 20, 2016
Homework 6: pdf | tex. Due Monday, May 23, 2016
Homework 7: pdf | tex. Due Friday, May 27, 2016
Homework 8: pdf | tex. Due Monday, May 30, 2016
Homework 9: pdf | tex. Due Wednesday, June 1, 2016
Homework 10: pdf | tex. Due Friday, June 3, 2016
Homework 11: pdf | tex. Due Tuesday, June 7, 2016
Homework 12: pdf | tex. Due Friday, June 10, 2016
Homework 13: pdf | tex. Due Monday, June 13, 2016
Homework 14: pdf | tex. Due Thursday, June 16, 2016

Exams

• 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)