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.
- 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
- 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
Explorer, a free software program to accompany Visual Group
- Guidelines for good
Su. (4 pages)
- Group theory and
the Rubik's cube,
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
- 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.
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.
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
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