TITLE: less elementary group theory
QUESTION [12 upvotes]: 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.

REPLY [2 votes]: 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.)