A Fork in the Road

This is the point where we have enough background to go in a lot of different directions. There is a lot more to be learned about groups. From here we could move on to group actions, Sylow Theory, and the various different ways to combine groups and make new groups. However, we have also learned a sufficient amount of group theory to move onto the theory of integral domains, rings, and then fields which are all new structures that are similar to groups but with more rules. This would be the direction that you'd want to go if you were interested in studying polynomials and algebraic topology. A third option would be to work on linear algebra, which is the study of vector spaces, matrices, and linear functions.

My personal area of research (at this point) is in character theory, which is a very cool and useful way of looking at groups and extracting all of their information. Character theory, at least from what I can tell so far, is a cool new 21^st century tool that we've developed to study groups after running dry the well of standard group theory. I think I'm going to try to move in a direction that gets us toward understanding character theory. The problem with that, though, is that the path to character theory is nowhere near a straight line. When I learned it myself it was a frustrating process of patching together bits and pieces of a lot of different topics and it is my goal to keep you from that experience. The problem is, though, that even though I'm going to try and keep you away from the frustrating process, its still going to take the combination of a couple different branches of mathematics. First, we need a significant amount of group theory - including homomorphisms, group actions, conjugacy classes, and simple groups. Then we need to learn something called representation theory which is the study of functions that take arbitrary groups into groups of matrices. That means that before we can learn representation theory we need to understand these matrix groups which requires some linear algebra.

So, as you can see, there is quite a lot of work ahead of us before we get to character theory. And on top of that, I did not choose this path because its better or more important than any of the other topics - I only chose it because its what I'm familiar with, so even if we get to character theory there's quite a bit ahead of that. For now, though, I'm going to continue pushing along with group theory.