Math 4120 (Modern Algebra), Summer I 2016 (online)

Math 4120 (Modern Algebra), Summer I 2016 (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 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.

Class essentials



Links to the individual lectures are listed below. Or, you can view the full YouTube playlist here.

Section 1: Groups, intuitively. (6 lectures, 2 hrs 10 min.) Section 2: Examples of groups. (4 lectures, 1 hr 27 min.) Section 3: Structure of groups. (7 lectures, 2 hrs 52 min.) Section 4: Maps between groups. (6 lectures, 3 hrs 27 min.) Section 5: Group actions. (7 lectures, 3 hrs 45 min) Section 6: Field theory and Galois theory. (8 lectures, 3 hr 52 min) Section 7: Ring theory.
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 significantly diverged, and now do not resemble much of Dana's original slides.


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.

Homework 1: pdf | tex. Due Thursday, May 12, 2016
Homework 2: pdf | tex. Due Friday, May 13, 2016
Homework 3: pdf | tex. Due Monday, May 16, 2016
Homework 4: pdf | tex. Due Wednesday, May 18, 2016
Homework 5: pdf | tex. Due Friday, May 20, 2016
Homework 6: pdf | tex. Due Monday, May 23, 2016
Homework 7: pdf | tex. Due Friday, May 27, 2016
Homework 8: pdf | tex. Due Monday, May 30, 2016
Homework 9: pdf | tex. Due Wednesday, June 1, 2016
Homework 10: pdf | tex. Due Friday, June 3, 2016
Homework 11: pdf | tex. Due Tuesday, June 7, 2016
Homework 12: pdf | tex. Due Friday, June 10, 2016
Homework 13: pdf | tex. Due Monday, June 13, 2016
Homework 14: pdf | tex. Due Thursday, June 16, 2016


Study guide
Practice Midterm 2 (Spring 2014)
Practice Midterm 1 (Fall 2013)
Practice Midterm 2 (Fall 2013)
Practice Midterm 1 (Fall 2010)
Practice Midterm 2 (Fall 2010)