In the second semester, we cover Chapters 6 through 8, along with at least one of Chapter 9 or 10. Chapter 11 is not a part of either course, but I included it for students (graduate or undergraduate) who want to pursue a research project and need an introduction that builds on what came before.
As of this writing, some of those chapters still need significant debugging, so don’t take anything you read there too seriously. Not much of the material can be omitted. Within each section, many examples are used and reused; this applies to exercises, as well. Textbooks often avoid this, to give instructors more flexibility; I don’t care about other instructors’ points of view,
so I don’t mind putting into the exercises problems that I return to later in the notes. We try to concentrate on a few critical examples, re-examining them in the light of each new topic. One consequence is that rings cannot be taught independently from groups using these notes. To give you a heads-up, the following material will probably be omitted.
• I like the idea of placing elliptic curves (Section 2.4) early in the class. Previous editions of these notes had them in the section immediately after the introduction of groups! It gives students immediate insight into how powerful abstraction can be. Unfortunately, I haven’t yet been able to get going fast enough to get them done.
• Groups of automorphisms (Section 4.4) are generally considered optional.
• I have not in the past taught solvable groups (Section 3.6), but hope to do so eventually.
•, I have sometimes not made it past alternating groups (Section 5.5). Considering that I used to be able to make it to the RSA algorithm (Section 6.5), that does not mean we won’t get there, especially since I’ve simplified the beginning. That was before I added the stuff on monoids, though. . .
• The discussion of the 15-puzzle is simplified from other places I’ve found it, nonstandard, and optional website home page for converter.