Math 8510 (Abstract Algebra I), Fall 2022

Math 8510 (Abstract Algebra), Fall 2022

"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

This will be a standard first-semester graduate course in abstract algebra, but there will be an extra effort to encorporate visuals to go along with the concepts. We will cover groups, homomorphisms, actions, extensions, universal properties, and rings. In the end, you will leave with a new appreciation of the beauty (and difficulty) of algebra, in a way that will develop your learning skills, which will be applicable to your other graduate courses and research.

Class essentials

Resources

A more recent version of this class (Fall 2023).
Homepage of Math 4120 (Fall 2022), undergraduate abstract algebra, and the course summary (Spring 2022 version) in one long meta Twitter thread of my weekly summary threads, from @VisualAlgebra.
Homepage of Math 4130, Algebra II (Spring 2023).
My 46-video Visual Group Theory YouTube playlist. There is some overlap with our materials.
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.
Group Explorer, a free software program to accompany Visual Group Theory
GroupNames, a tremendous resource and database for finite groups.
LMFDB, a powerful search tool for finite groups.
Database of ring theory
The free open source GAP (Groups, Algorithms, Programming) software package, and a nice Mac interface called Gap.app
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.

Lecture notes

Chapter 1: Groups, intuitvely (27 pages. Last updated Aug 19, 2022)
Chapter 2: Examples of groups (60 pages. Last updated Sep 9, 2022)
Chapter 3: Group structure (56 pages. Last updated Nov 1, 2022)
Chapter 4: Maps between groups (81 pages. Last updated Nov 1, 2022)
Chapter 5: Actions of groups (122 pages. Last updated Feb 23, 2023)
Chapter 6: Extensions and universal constructions (155 pages. Last updated Mar 2, 2023)
Chapter 7: Rings (157 pages. Last updated Feb 6, 2023)
Chapter 8: Field and Gaolis theory

Homework

Homework should be written up carefully and concisely, and needs to be typeset in LaTeX (except for HW 1). 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 some assignments, I will post "scratch paper" consisting of blank images that you are free to use, rather than redraw by hand or remake with TikZ.

HW 1: pdf | tex | img. Topics: Introduction to groups and Cayley graphs. Due Fri, Sept 2, 2022.

HW 2: pdf | tex | img. Topics: Subgroups, cosets, conjugacy, normalizers, centralizers. Due Fri, Sept 9, 2022.

HW 3: pdf | tex | img. Topics: Homomorphisms, semidirect products. Due Fri, Sept 16, 2022.

HW 4: pdf | tex | img. Topics: Automorphisms, group actions. Due Fri, Sept 23, 2022.

HW 5: pdf | tex | img. Topics: Sylow theory. Due Fri, Sept 30, 2022.

HW 6: pdf | tex | img. Topics: Extensions, short exact sequences, solvability. Due Fri, Oct 7, 2022.

HW 7: pdf | tex | img. Topics: Nilpotent groups. Due Fri, Oct 14, 2022.

HW 8: pdf | tex | img. Topics: Universal constructions, products, coproducts. Due Fri, Oct 21, 2022.

HW 9: pdf | tex | img. Topics: Categories, free groups. Due Fri, Oct 28, 2022.

HW 10: pdf | tex | img. Topics: Group presentations, free products, fiber products and coproducts. Due Fri, Nov 4, 2022.

HW 11: pdf | tex | img. Topics: Rings and ideals. Due Mon, Nov 14, 2022.

HW 12: pdf | tex | img. Topics: Finite fields, prime ideals, rings of fractions. Due Mon, Nov 21, 2022.

HW 13: pdf | tex | img. Topics: Divisibility and factorization, quadratic integer rings. Due Fri, Dec 2, 2022.

HW 14: pdf | tex | img. Topics: Sunzi's remainder theorem, polynomial rings. Due Fri, Dec 9, 2022.