MthSc 412 (Modern Algebra), Fall 2010

MthSc 412 (Modern Algebra), Fall 2010


"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." --Bertrand Russell

"Show me a guy who's afraid to look bad, and I'll show you a guy you can beat every time." --Lou Brock

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 new 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, or of our class, we will use our new-found intuition to write mathematical proofs.

In class, we will play with the Rubik's cube. We will draw with colored pencils, use scissors to cut shapes from colored paper, 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

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.

• Homework assignments

Lecture notes

There are ten sets of lecture notes (slides), one for each chapter of Visual Group Theory. The first two sets are posted here, but the others contain copyrighted material, so I put them on Blackboard.

Chapter 1: What is a group?
Chapter 2: What do groups look like?
Chapter 3: Why study groups?
Chapter 4: Algebra at last
Chapter 5: Five families
Chapter 6: Subgroups
Chapter 7: Products and quotients
Chapter 8: The power of homomorphisms
Chapter 9: Sylow theory
Chapter 10: Galois theory

To the best of my knowledge, this is the 2nd abstract algebra class ever to be taught using Visual Group Theory. 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) are modifications of ones Dana wrote, with the first few chapters being very similar, but the later chapters being quite different. I made the slides from the remaining chapters myself, though using material from Visual Group Theory.

If you would like a copy of these lecture notes, or the LaTeX source files, send me an email. Especially if you are considering teaching a class using Visual Group Theory (which I strongly recommend, if you have that choice). I actually use the Beamer layover features in lecture, but I post the handout versions for convenience. (Note: you need to have the TikZ package installed for the slides to compile).