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

- Course
Syllabus
- What is abstract algebra? (Wikipedia entry)
- A short article on abstract algebra, by Steven
Strogatz, written for a general (non-mathematical) audience
- Abstract Algebra:
Theory and Applications, a free open-source textbook, by Tom
Judson. (Required textbook)
- 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*
- Main webpage for Sage, a comprehensive open-source mathematics
software suite: http://www.sagemath.org/
- The public Sage Notebook server (what we will use): http://www.sagenb.org/
- The Sage complete reference manual, a two-page quick reference guide, and a 15-page manual
specifically for groups: Group Theory and Sage: A Primer, written by Rob
Beezer.
- 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.

### 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).