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.



less elementary group theory
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.


