For mostly my benefit, I will discuss the content for my abstract algebra course. I will also determine what I will emphasize and de-emphasized. The unfortunate fact is that I only have one semester; 36 class periods; 2520 minutes.
The chapters that are typically covered are:
- Introduction to Groups
- Finite Groups; Subgroups
- Cyclic Groups
- Permutation Groups
- Cosets and Lagrange’s Theorem
- External Direct Products
- Normal Subgroups and Factor Groups
- Group Homomorphisms
- Fundamental Theorem of Abelian Groups
- Introduction to Rings
- Integral Domains
- Ideals and Factor Rings
- Ring Homomorphisms
If there is time left, I will cover Polynomial Rings, Factorization of Polynomials, and Divisibility of Integral Domains. There will not be time left. If things go well in the first part of the semester, maybe I would do some Sylow Theory. In fact, I might have a tough time keeping myself from doing Sylow Theory, regardless of the amount of time we have.
This is a full semester. If I lectured all semester long, I think that I would be able to finish with just a little bit of time left. Since I am not going to lecture, this means that decisions will have to be made. Here are my basic ideas:
- I am planning on starting the semester by introducing several hands-on examples of groups: several cyclic groups, several dihedral groups, a couple of symmetric groups, and the alternating group on 4 letters. I also hope to introduce the quaternions in an easy-to-understand way. This should make the first four chapters much easier to understand. By the end of the semester, I hope that my students are experts in 8-9 different groups.
- I will de-emphasize direct products and the proof of the Fundamental Theorem of Abelian Groups. This should save some time.
- I will try to tie the ring theory to high school ideas as much as possible to ground it.