Here is a classic theorem of Herstein: $G$ is a finite group, $M$ a maximal subgroup, which is abelian. Then $G$ is solvable.
The proof is pretty easy, but it uses character theory (specifically, Frobenius' theorem on Frobenius groups). Is there a character-theory-free proof?
To get things going, note that we can reduce to the case where:
i) $M$ is core-free and a Hall subgroup of $G$;