Topics in Algebra - Advice for Seminar

Question

I have a seminar in Algebra next week, and I need to lecture. The professor gave us the freedom to choose a subject and I need suggestions. I have taken a course on group theory and a course on Galois theory. Although a lot of people love Galois theory, I didn't enjoy it as much as I enjoyed group theory. So can anyone suggest a topics in group theory that I can lecture in 60-80 minutes which has interesting results? (I mean it's a one time lecture, so by interesting results I mean that it will lead to something) Thanks!

Answer 1

I would start with an introduction to topological groups, since as you said that the students already had basic notions on groups. So why not extend this notion and add some complexity? Maybe it is also interesting to know that one can add such structures to such (in the beginning) simple seeming structures. A good introduction to topological groups with examples may be found in the book Introduction to topological manifolds by John M. Lee. The section also has a basic introduction to group actions which you may skip and just say something about some examples, maybe something on the homeomorphism $$\mathbb{R}^n/\mathbb{Z}^n \approx \mathbb{S}^1 \times \dots \mathbb{S}^1 =: \mathbb{T}^n$$ Now you can also say something on Lie groups and group actions by them which can be found in the book Introduction to smooth manifolds also by John M. Lee which is also a rich application. This would be the topological part. I am also interested in harmonic analysis and there topological groups are used for generally defining convolution and approximate identities (we equip the topological group with a certain Haar measure). You may found something on this subject in the book classical fourier analysis by Loukas Grafakos. So to summarize:

You may introduce topological groups which adds some complexity to an already established notion. There are (as far as I can say at the moment) two main fields of applications:

• Advanced analysis, i.e. Fourier analysis;
• General topology, maybe also algebraic topology;

which are certainly interesting on its own and give the students a view of what subjects or themes are upcoming.

I hope this helps.

Answer 2

You could give an introduction to the representation theory of the symmetric group. Roughly speaking it's about the group homomorphisms from a symmetric group $S_n$ to some $GL_m(K)$ (in the classical case one takes $K=\mathbb{C}$). It has some connections to combinatorics (Frobenius character formula, Pieri's rule, $\dots$).

Added: Another topic that might interest you is the study of symmetry groups. Artin treats this nicely in chapter 5 in his book 'Algebra' (but maybe you have already seen all these classification theorems). He does everything for the symmetry group of the plane.