less elementary group theory

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.)

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