Math 4120 (Modern Algebra), Spring 2014
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)