Math 4130 (Modern Algebra II), Spring 2023
Math 4130 (Modern Algebra II), Spring 2023
"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
"If you are someone who prefers large vistas and powerful theories, then it is essential to be able to test general results by applying them to simple examples....where one can do concrete calculations, sometimes with elaborate formulas, that help to make the general theory understandable. They keep your feet on the ground...A good example is a thing of beauty. It shines and convinces. It gives insight and understaning. It provides the bedrock of belief." --Sir Michael Atiyah, in Advice to a young mathematician.
About the class
Group theory is the study of symmetry, and it 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. Yet, it is usually taught with little to no visuals, which is a travesty. The primary reason for this is that every establish books does it this way, and instructors tend to teach it the way the learned it. Without available resources, this will never change. And I'm working to change this. My vision is that in 20 years, students will find it incredulous that this subject used to be taught non-visually. Just like how no one would teach calculus without graphs. Visuals are as essential in group theory as pictures are in fields such as crystallography or art history.
We will not use a textbook for this class because one does not yet
exist. (I started writing it in October 2019, and should finish it this year) A few fantastic sources that helped inspire this class include a
2009 general-audience 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. Another
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, wallpapers, chemical molecules, and Archimedian solids. At
the end of the semester, you will truly understand groups and rings,
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.
Class essentials
Resources
- The entire prereq, Math 4120, (Spring 2022 version) in
one long meta Twitter thread of my weekly summary threads, from
@VisualAlgebra.
- My 46-video Visual Group Theory YouTube playlist. There is some overlap with our materials.
- Homepage of Math 4120, Algebra I (Fall 2022), the previous semester.
- Homepage of Math 8510, the graduate-level version of this class.
- YouTube link to a talk I gave titled What is...a Cayley diagram? at the virtual What is...a seminar?, December 2021.
- YouTube link to a talk I gave titled A visual tour of the beauty of group theory, at the Talk math with your friends seminar, October 2021.
- Group theory, abstraction, and the 196,883-dimensional monster, a video by the phenomenal Grant Sanderson, aka 3blue1brown.
- 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.
- GroupNames, a tremendous resource and database for finite groups.
- LMFDB, a powerful search tool for finite groups.
- Database of ring theory
- Group Explorer, a free software program to
accompany Visual Group Theory
- Permutation Group Visualizer app, written by
Nat Alison
- The free open source GAP (Groups, Algorithms, Programming) software package, and a nice Mac interface called Gap.app
- Guidelines for good
mathematical writing,
by Francis
Su. (4 pages)
- Francis Su's book Mathematics for Human Flourising, which won the 2021 Euler Book Prize.
- Group theory and
the Rubik's cube,
by Janet
Chen (39 pages).
- Homepage of math
professor and former Rubik's cube world recorder
holder Macky Makisumi. He is interested in speedcubing
theory and runs the
website Cubefreak.
- 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 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!)
- Articles on Group
Theory and its Application to Chemistry from LibreTexts, a ChemWiki hosted at UC Davis.
- Tilings in everyday places, by Dror Bar-Natan of the University of Toronto.
- Fundamental axioms of mathematics, by Federico Ardilla
- The Lesson of Grace in Teaching, by Francis Su. Also available as a YouTube lecture.
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 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. Along with assignment,
I will post "supplemental material" consisting of blank images that you are
free to use, rather than redraw by hand.
- HW 0: pdf |
tex
| img. Topics: Review of Algebra 1. Due
Monday, January 23, 2023.
- HW 1: pdf |
tex
| img. Topics: Rings of fractions. Due
Friday, January 27, 2023.
- HW 2: pdf |
tex
| img. Topics: PIDs, Euclidean domains, and
quadratic integer rings. Due Friday, February 3, 2023.
- HW 3: pdf |
tex
| img. Topics: Norm Euclidean domains, Sunzi's remainder
theorem. Due Friday, February 10, 2023.
- HW 4: pdf |
tex
| img. Topics: Polynomial rings. Due
Friday, February 17, 2023.
- HW 5: pdf |
tex
| img. Topics: Group extensions and short
exact sequences. Due Friday, February 4, 2023.
- HW 6: pdf |
tex
| img. Topics: Composition series, commutators,
and solvable groups. Due Friday, March 3, 2023.
- HW 7: pdf |
tex
| img. Topics: Central series and nilpotent
groups. Due Friday, March 10, 2023.
- HW 8: pdf |
tex
| img. Topics: Field extensions.
Due Wednesday, March 29, 2023.
- HW 9: pdf |
tex
| img. Topics: Galois groups.
Due Friday, April 7, 2023.
- HW 10: pdf |
tex
| img. Topics: Galois theory.
Due Friday, April 21, 2023.
- HW 11: pdf |
tex
| img. Topics: Free groups, group presentations.
Due Friday, April 28, 2023.
Lecture notes
See an explanation below for the story behind these, and why they are a new and improved version of what I thought had converged to something that I would never change! I will eventually record YouTube lectures to go along with these, but not this semester.
- Chapter 1: Groups, intuitvely (52 pages. Last updated Jan 319, 2023)
- Chapter 2: Examples of groups (110 pages. Last updated Jan 31, 2024)
- Chapter 3: Group structure (93 pages. Last updated Sept 19, 2023)
- Chapter 4: Maps between groups (86 pages. Last updated Sept 14, 2023)
- Chapter 5: Actions of groups (128 pages. Last updated Sept 11, 2023)
- Chapter 6: Extensions of groups (80 pages. Last updated Dec 18, 2023)
- Chapter 7: Universal constructions (97 pages. Last updated Dec 18, 2023)
- Chapter 8: Rings (85 pages. Last updated Decr 18, 2023)
- Chapter 9: Domains (88 pages. Last updated Jan 8, 2024)
- Chapter 10: Fields and Galois theory (67 pages, in progress. Last updated Mar 10, 2023)
Exams