less elementary group theory

Question

Most of the group theory that is taught in introductory graduate classes is of the form $$(\mbox{number theory} + \mbox{ group actions} + \mbox{ orbit-stabilizer thm}) + \mbox{group axioms} \Rightarrow \mbox{theorems}$$ So what is the equivalent of "(number theory + group actions + orbit-stabilizer thm)" in the more advanced parts of group theory?

To clarify based on some comments: The techniques I learned in a graduate group theory class were just the orbit-stabilizer thm + some number theory + Lagrange's thm. Adding in some more constructions like semi-direct products allows one to make some inroads into some less elementary parts of group theory, i.e. we get some classification theorems for groups of small order with the help of Sylow theorems which is really just clever number theory + orbit-stabilizer theorem. So I would like to expand my toolbox a little bit by seeing what other tools are used in more advanced group theory but are still applicable to the elementary parts of group theory like classifying groups of small order.

Tidbits collected from comments: Kevin McGerty makes some excellent points about the extension of the theory from actions on sets which allow number theoretic arguments to actions on vector spaces which increase the sophistication and depth of the theory. The move from mere sets to vector spaces allows the use of linear algebra as another tool which in turn allows some tools from homological algebra to enter into the game.

Answer 1

As someone who has spent far too much time thinking about classifying groups of small order, one thing you'd want to do is understand some more results about p-groups. You already probably learned one of the main results, every p-group has a nontrivial center, which is proved by the number theory type arguments you discussed. One of the key ideas in understanding p-groups is the Burnside basis theorem that says that any generating set for $G/\Phi(G)$ is a generating set for G. Here $\Phi(G)$ is the Frattini subgroup which is generated by all commutators and all pth powers. You can think of this as a version of Nakayma's lemma for p-groups and think that p-groups are similar to commutative rings.

Answer 2

Another move you can make is to study the actions of your group on topological or metric spaces. This is more or less the starting point for geometric group theory. It's perhaps less relevant to the study of finite group, but it's central to the modern study of infinite discrete groups.

Answer 3

This is a very vague question (bordering on misuse of the site, since I don't think it really has an answer), but at least part of the answer is linear representation theory, which has a bit more depth than just group actions. Not only is this an interesting subject in and of itself, but it's also absolutely essential for many of the most exciting theorems of group theory, like the whole edifice around the classification of finite simple groups.

If you want to learn more, I recommend Serre's book on the subject.

Answer 4

I'm going to first re-recommend Robinson's Course in the Theory of Groups. There are many interesting branches of group theory like geometric group theory that tie into other subjects, but you should probably make sure you know group theory proper first. And the only real way to know what topics are important is to get a book.

Answer 5

A great book to pick up about finite group theory is Isaacs's "Finite Group Theory". It does a ton more than you would probably see in the standard graduate algebra class. The only drawback is it does no character theory. But of course Isaacs also wrote a book called "Character Theory of Finite Groups". If you're getting both, might as well go for the trinity and get his "Algebra" as well.

Steve

Answer 6

If you're interested in stuff around the classification of finite simple groups -- for instance, using Sylow's theorems and slight extensions lets you classify simple groups with a small number of prime factors, but this quickly becomes very technical and then out of reach completely (I don't believe it's known whether there are an infinite number of finite simple groups with exactly six prime factors) -- I highly recommend an historical overview by Ronald Solomon from a few years back. There are a few things about it that are a little disappointing -- Solomon devotes only a few short paragraphs to the discovery of the sporadic groups, for example -- but it still makes for a fascinating read, even the parts (pretty much everything after 1960 or so) where the work becomes highly technical.

Answer 7

The impression I get is that a large chunk finite group theory can be built up from the beginner's toolset: orbit-stabiliser, the isomorphism theorems, and a lot of fiddling around with conjugation, normalisers and centralisers, and induction on the order of the group. You can achieve a lot with surprisingly little.

Character theory (over the complex numbers) is probably the non-'elementary' tool that sees the heaviest use. For instance, one often wants to solve the equation $x y = z$, where $z$ is given and $x$ and $y$ must come from specified conjugacy classes. It turns out that there is a formula for the number of solutions in terms of characters. So instead of trying to find an explicit $(x,y)$, one can try to estimate the value of the formula and prove that the answer is non-zero. (Typically, the trivial character makes a large positive contribution, and the aim is to show that all the other characters make small contributions.)

Answer 8

re-develop your course employing category theory vision: Once you construct objects construct arrows. Example, if you have defined groups inmediately define homomorphisms, if you define subgroups and quotients, define kernels...etc, arrive to the homomorphism's fundamental theorems...