Course Collaboration Project—Part 3 (Content)

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:

  1. Introduction to Groups
  2. Groups
  3. Finite Groups; Subgroups
  4. Cyclic Groups
  5. Permutation Groups
  6. Isomorphisms
  7. Cosets and Lagrange’s Theorem
  8. External Direct Products
  9. Normal Subgroups and Factor Groups
  10. Group Homomorphisms
  11. Fundamental Theorem of Abelian Groups
  12. Introduction to Rings
  13. Integral Domains
  14. Ideals and Factor Rings
  15. 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:

  1. 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.
  2. I will de-emphasize direct products and the proof of the Fundamental Theorem of Abelian Groups. This should save some time.
  3. I will try to tie the ring theory to high school ideas as much as possible to ground it.

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