Math 4120 (Modern Algebra), Summer I 2020 (online)
Math 4120 (Modern Algebra), Summer I 2020 (online)
"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 draw from several sources: 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. The second source is a free e-book called An inquiry-based approach to abstract algebra, by Dana Ernst. This follows the "Visual Group Theory" approach, but is more rigorous and proof-based. However, most of the proofs are not provided; you are supposed to fill them in. This is what the "inquiry-based" part means.
In class, we will play with the Rubik's cube. We will analyze art
freises, chemical molecules, and contra dances. At the end of the
semester, you will truly understand groups, subgroups, cosets,
products 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, and which provided the framework necessary to elegantly
solve several classic mathematical mysteries of the ancient Greeks. In
the end, you will leave with a new appreciation of the beauty, and
difficulty, of an area of mathematics you never dreamt existed.
Homepage of math
professor and former Rubik's cube world recorder
holder Macky Makisumi. He is interested in speedcubing
theory and runs the
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 discoveries! Every configuration of the Rubik's Cube Group is at most
20 "moves" from the solved state (Proven July 2010), or 26 "moves" in the quarter-turn metric (Proven August 2014).
Crystal systems of minerals (lots of pictures, and
references to group theory!)
The following are a series of lecture notes (slides) I wrote. They
originally followed the progression of the material in Visual Group
Theory, though they are quite supplemented with proofs, rigor, and
a lot of extra content.
I use these slides when I teach Math 4120 during the semester. For this online class, I made about 50 video lectures from this content by breaking each section above into many small lectures. The links to the individual lectures and accompanying slides can be found below. Or, you can view the full YouTube playlist. To avoid blurriness, these are best viewed by changing the settings to 720p (High Definition) rather than the default of 240p. This can be easily done by clicking the "wheel" on the lower right corner; right next to the "cc" button.
Lecture 7.7: The Chinese remainder theorem.
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). Dana currently teaches
abstract algebra in an Inquiry Based Learning format using the "Visual Group Theory" approach, and he
has written a wonderful accompanying set of notes (basically an e-book)
that is freely available.
Homework should be written up carefully and
concisely. Please write in complete sentences. Part of your
grade will be based on the presentation and 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.
HW 1: pdf |
tex. Topics: Introduction to groups. Due Wednesday, May 13, 2020.
HW 2: pdf |
tex. Topics: Cayley graphs & multiplication tables. Due Friday, May 15, 2020.
HW 3: pdf |
tex. Topics: Symmetric and alternating groups. Due Monday, May 18, 2020.
HW 4: pdf |
tex. Topics: Subgroups, cosets, & Cayley's theorem. Due Wednesday, May 20, 2020.
HW 5: pdf |
tex. Topics: Normal subgroups. Due Friday, May 22, 2020.
HW 6: pdf |
tex. Topics: Quotients, normalizers, & conjugacy classes. Due Tuesday, May 26, 2020.
HW 7: pdf |
tex. Topics: Homomorphisms. Due Friday, May 29, 2020.
HW 8: pdf |
tex. Topics: The isomorphism theorems; finite abelian groups. Due Monday, June 1, 2020.
HW 9: pdf |
tex. Topics: Commutators, automorphisms, & group actions. Due Wednesday, June 3, 2020.
HW 10: pdf |
tex. Topics: The orbit-stabilizer theorem; the Sylow theorems. Due Friday, June 5, 2020.
HW 11: pdf |
tex. Topics: Finite fields, extension fields, and Galois groups. Due Tuesday, June 9, 2020.
HW 12: pdf |
tex. Topics: Irreducibility; the fundamental
theorem of Galois theory. Due Friday, June 12, 2020.
HW 13: pdf |
tex. Topics: Rings, ideals, and
homomorphisms. Due Monday, June 15, 2020.
HW 14: pdf |
tex. Topics: Prime and maximal ideals; PIDs; algebraic integers. Due Thursday, June 18, 2020.