MthS 4120 (Modern Algebra), Fall 2013

MthS 4120 (Modern Algebra), Fall 2013


"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 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. 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 assignments

• Practice Midterm 1
• Practice Midterm 2

Lecture notes

I will write ten sets of lecture notes (slides), one for each chapter of Visual Group Theory. The first two sets will be posted here, but the others contain copyrighted material, so I will 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 of groups
Chapter 6: Subgroups
Chapter 7: Products and quotients
Chapter 8: Homomorphisms
Chapter 9: Group actions & Sylow theory
Chapter 10: Galois theory

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) are modifications of ones Dana wrote, with the first few chapters being fairly similar, but the later chapters being quite different. I made the slides from the remaining chapters myself, though using material from Visual Group Theory.