Visual Algebra textbook & supplemental materials
In 2010, I taught abstract algebra, roughly following Visual Group Theory (VGT),
a book written by Nathan Carter, a few years after taking a course by Doug Hofstadter at the University of Indiana, by the same name. The renowned
mathematician Steven Strogatz at Cornell, calls VGT "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. However, it is, by design, a "general audience book," and so I needed to greatly supplement it to meet the needs of undergraduate math majors.
After a few iterations of my course, it had converged to something that I thought would never change. When I taught it online in 2016, I recorded a full set of 46 YouTube videos, that I really liked. Finally, in 2019, after my students and I were equally frusterated at the lack of a textbook for the material I was teaching, I bit the bullet and decided to write one. It is titled Visual Algebra, and should be completed in 2023.
At first, I didn't know how much it would be different than a "souped up version of VGT," but the beauty (or horror) of writing a book, is that it takes on a life of its own and to a place you never could have prediced. Several years and 700 pages later, I have learned so much more about (undergraudate!) group theory than I ever thought existed, I expanded the scope to over an entire year's worth of content. In the process, I developed hundereds of new visuals that I never would have ever dreamed of just a few years earlierwhen I naively thought my course had "converged". My book now as over 600 pictures, and I look back at my YouTube playlist with slight disappointment, and cannot wait until I get to rerecord a brand new series of videos that follows Visual Algebra. I have since taught Algebra 2, and Graduate Algebra 1 following this approach.
On this page, I will post relevant materials, including slides, HW, exams, and links. If you are an instructor interested in teaching abstract algebra this way, please get in touch! I would be happy to help you, give advice, and share materials.
Table of Contents, with links to corresponding lecture slides
 Groups, intuitively (52 pages. Last updated Mar 5, 2024)
 Examples of groups (110 pages. Last updated Jan 31, 2024)
 Group structure (98 pages. Last updated Mar 4, 2024)
 Maps between groups (94 pages. Last updated Mar 9, 2024)
 Actions of groups (122 pages. Last updated Apr 1, 2024)
 Extensions of groups (80 pages. Last updated Dec 18, 2023)
 Universal constructions (97 pages. Last updated Dec 18, 2023)
 Rings (86 pages. Last updated Apr 12, 2024)
 Domains (88 pages. Last updated Jan 16, 2024)
 Fields (coming soon!)
 Galois theory (coming soon!)
To do list

Finish writing Visual Algebra (mostly just Ch 11)
 Add more content to Chapter 9: quadratic integer rings and algebraic number theory
 Add more exercises to Chapters 811
 Rewrite a few earlier sections based on the book's "natural
evolution"
 Create slides for Chapters 10 and 11
 Clean up and standardize the LaTeX of existing slides
 Post LaTeX files of slides to GitHub (after book is finished)
 Record a new set of Visual Algebra YouTube lectures
(after book is finished)
My courses that use Visual Algebra (click through for materials)
 Undergraduate Algebra I (Math 4120)
 Undergraduate Algebra II (Math 4130)
 Graduate Algebra I (Math 8510)
Visual Algebra exams
Undergraduate Visual Algebra homework (Math 4120, 4130)
 HW 1: pdf 
tex
 img. Topics: Introduction to groups,
symmetries, and Cayley diagrams.
 HW 2: pdf 
tex
 img. Topics: Examples of groups, roots of
unity.
 HW 3: pdf 
tex
 img. Topics: Polytopes and groups of
permutations.
 HW 4: pdf
 tex
 img. Topics: Dicyclic, diquaternion, semidihedral,
and semiabelian groups. Automorphisms.
 HW 5: pdf
 tex
 img. Topics: Subgroups, cosets, and
normalizers.
 HW 6: pdf
 tex
 img. Topics: Normality and
conjugate subgroups.
 HW 7: pdf
 tex
 img. Topics: Quotient groups, conjugacy
classes, and centralizers.
 HW 8: pdf
 tex  img. Topics: Homomorphisms and
isomorphisms.
 HW 9: pdf
 tex
 img. Topics: Isomorphism theorems,
commutators, automorphisms, semidirect products.
 HW 10: pdf
 tex
 img. Topics: Group actions.
 HW 11: pdf
 tex
 img. Topics: Groups acting on elements,
subgroups, and cosets.
 HW 12: pdf
 tex
 img. Topics: Sylow theory.
 HW 13: pdf
 tex
 img.
Topics: Rings, ideals, and homomorphisms.
 HW 14: pdf
 tex
 img.
Topics: Finite fields, prime and primary ideals, radicals.
 HW 0: pdf 
tex
 img. Topics: Review of Algebra 1.
 HW 1: pdf 
tex
 img. Topics: Rings of fractions.
 HW 2: pdf 
tex
 img. Topics: PIDs, Euclidean domains, and
quadratic integer rings.
 HW 3: pdf 
tex
 img. Topics: Norm Euclidean domains, Sunzi's remainder
theorem.
 HW 4: pdf 
tex
 img. Topics: Polynomial rings.
 HW 5: pdf 
tex
 img. Topics: Group extensions and short
exact sequences.
 HW 6: pdf 
tex
 img. Topics: Composition series, commutators,
and solvable groups.
 HW 7: pdf 
tex
 img. Topics: Central series and nilpotent
groups.
 HW 8: pdf 
tex
 img. Topics: Field extensions.
 HW 9: pdf 
tex
 img. Topics: Galois groups.
 HW 10: pdf 
tex
 img. Topics: Galois theory.
 HW 11: pdf 
tex
 img. Topics: Free groups, group presentations.
Graduate Visual Algebra homework (Math 8510)
 HW 1: pdf 
tex 
img.
Topics: Introduction to groups and Cayley graphs.
 HW 2: pdf 
tex 
img.
Topics: Subgroups, cosets, conjugacy, normalizers,
centralizers.
 HW 3: pdf 
tex 
img.
Topics: Homomorphisms, semidirect products.
 HW 4: pdf 
tex 
img.
Topics: Automorphisms, group actions.
 HW 5: pdf 
tex 
img.
Topics: Sylow theory.
 HW 6: pdf 
tex 
img.
Topics: Extensions, short exact sequences, solvability.
 HW 7: pdf 
tex 
img.
Topics: Nilpotent groups.
 HW 8: pdf 
tex 
img.
Topics: Universal constructions, products, coproducts.
 HW 9: pdf 
tex 
img.
Topics: Categories, free groups.
 HW 10: pdf 
tex 
img. Topics: Group presentations, free products, fiber products and coproducts.
 HW 11: pdf 
tex  img.
Topics: Rings and ideals.
 HW 12: pdf 
tex
 img.
Topics: Finite fields, prime ideals, rings of fractions.
 HW 13: pdf 
tex
 img.
Topics: Divisibility and factorization, quadratic integer rings.
 HW 14: pdf 
tex
 img. Topics: Sunzi's remainder theorem, polynomial
rings.
Other resources
 My webpage.
 Webpage for my six full YouTube classes, including Visual Group Theory.
 My undergraduate Visual Algebra I course (Spring 2022 version) in
one long meta Twitter thread of my weekly summary threads, from
@VisualAlgebra.
 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 Explorer, a free software program to
accompany the
Visual Group Theory book.
 GroupNames, a tremendous resource and database for finite groups.
 LMFDB, a powerful search tool for finite groups.
 WeddsList group theory page, with lots of nice visuals, especially Cayley graphs.
 Database of ring theory
 The free open source GAP (Groups, Algorithms, Programming) software package, and a nice Mac interface called Gap.app
 An inquirybased approach to abstract
algebra, by Dana
Ernst. Free ebook which follows the "Visual Group Theory"
approach. His course materials can be
found here.