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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!

Charles Carmichael
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    What level seminar are you talking about, here? What are the students' backgrounds? – Noah Schweber Dec 26 '16 at 21:22
  • Lie groups, groups actions on smooth manifolds. Everyone will be paying attention. – IAmNoOne Dec 26 '16 at 21:44
  • My answer is at the forefront of group theory applied to physics. I've provided two links as references, and used direct quotes from both. Unfortunately, people have downvoted my answer so I will most likely delete it. Such is the nature of ME. –  Dec 26 '16 at 23:10
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    @NoahSchweber - I realized that my questions lacks info. It's a seminar in algebra in bachelor's degree and we are assumed to have background in linear algebra, group theory and galois theory. But obviously we are expected to lecture about more advanced topics. But I am not really interested in galois theory, and therefore I am looking for a subject in group theory that I can lecture in 60-80 minutes and it will lead to something. Introducing an application of group theory in another field would interesting as well. – Charles Carmichael Dec 27 '16 at 09:45
  • You should edit your question and add this info into it. That way people can taylor an answer that really suits you. Could you include as well what background the other students have in other subject? Furthermore, what style of seminar are you expected to give? Do you have to give all the details or should you more talk about the big picture? – Severin Schraven Dec 27 '16 at 10:37
  • Yes, you are right, I will do that now. It's more about the big picture, because I am introducing a new subject and have 60-80 minutes. Therefore it's more about the big picture. I can also show the use of algebra in another field as well. for example a student gave a lecture about differential groups and some applications on differential equations. But it was very general. – Charles Carmichael Dec 28 '16 at 19:47

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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.

Severin Schraven
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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.

TheGeekGreek
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  • Thanks for the feedback! The students have taken a course in group theory as well just like me (I should have mentioned that) so they already know that subject know these application. Applications of group actions in general topology sound interesting though, I will look into it. Is there something specific that you can recommend? – Charles Carmichael Dec 26 '16 at 19:52
  • @TheGeekGreek You got me curious. What would you need on the algebraic side except the basic definition of a group action? I only know these things from a purely analytic (respectively topological) point of view. – Severin Schraven Dec 26 '16 at 21:33
  • @SeverinSchraven Yes, this is correct. I was at first not entirely sure if I should write such an answer but since the question is vague and there are no tight restrictions I decided to give a more analytical/topological than algebraical answer. – TheGeekGreek Dec 26 '16 at 21:38
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    @TheGeekGreek I was just curious whether I missed something. I think your subjects are highly interesting, but the algebraic content is a bit low for an algebra seminar :) (unless you want to dig into algebraic topology, but then the audience would need to have a solid background in basic manifold theory. Then one could give an introduction to homological algebra). B.t.w. another interesting field based on group actions is dynamical systems. – Severin Schraven Dec 26 '16 at 21:46