I will define what it means for a group to be solvable and what the derived length of a solvable group is. I will then give some general motivations for why this is an important property.
Next I will give a crash course on what an irreducible character of a finite group is and what the degree of such a character is. I will then mention some results that help one calculate these degrees, and finally, I will mention several results that relate the set of these degrees to the solvability and derived length of a group.
The prerequisite will be knowledge of groups corresponding to the course algebra, since I will mention everything else I need (proofs will be kept to a minimum since otherwise I would not have time to get to the interesting results).
Even if you are following or have previously taken a course on character theory, I promise I will mention at least one result you have not seen before.