Math 4120 (Modern Algebra), Spring 2014

"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.

Resources

Course Syllabus
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.
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
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, a ChemWiki hosted at UC Davis.
Tilings in everyday places, by Dror Bar-Natan of the University of Toronto.

Lecture notes

The following are a series of lecture notes (slides) I wrote. The first 8 roughly follow the first 8 chapters of Visual Group Theory. The next two (9 and 10) follow Chapter 9, and the 11th follows Chapter 10 in VGT. The last three are beyond the scope of VGT.

Chapter 1: What is a group?
Chapter 2: Cayley graphs
Chapter 3: Groups in science, art, and mathematics
Chapter 4: Algebra and group presentations
Chapter 5: Five families of groups
Chapter 6: Subgroups
Chapter 7: Products and quotients
Chapter 8: Homomorphisms
Chapter 9: Group actions
Chapter 10: The Sylow theorems
Chapter 11: Galois theory
Chapter 12: Ruler and compass constructions
Chapter 13: Basic ring theory
Chapter 14: Divisibility and factorization

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 diverged quite a bit and in most chapters, 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 the 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 Friday, January 17, 2014
Homework 2: pdf | tex. Due Friday, January 24, 2014
Homework 3: pdf | tex. Due Friday, January 31, 2014
Homework 4: pdf | tex. Due Friday, February 7, 2014
Homework 5: pdf | tex. Due Friday, February 14, 2014
Homework 6: pdf | tex. Due Monday, February 24, 2014
Homework 7: pdf | tex. Due Monday, March 3, 2014
Homework 8: pdf | tex. Due Friday, March 7, 2014
Homework 9: pdf | tex. Due Friday, March 14, 2014
Homework 10: pdf | tex. Due Friday, March 28, 2014
Homework 11: pdf | tex. Due Friday, April 4, 2014
Homework 12: pdf | tex. Due Monday, April 14, 2014
Homework 13: pdf | tex. Due Monday, April 21, 2014
Homework 14: pdf | tex. Due Friday, April 25, 2014

Exams

• Midterm 1
• Midterm 2
• Practice Midterm 1 (Fall 2013)
• Practice Midterm 2 (Fall 2013)
• Practice Midterm 1 (Fall 2010)
• Practice Midterm 2 (Fall 2010)