I was enrolled in a representation theory (of finite groups) course in the fall and throughout the class we focused on properties of representations and paradigms built around them. The whole time, I felt a bit let down. Not in the sense that the material was not beautiful because it very much was but in the sense that I wanted to prove things about groups somehow using representations (not facts about the representations themselves like constraints on the dimension of the subrepresentations and such).
My question is this:
What are some classical examples of properties we specifically prove about a group from representation theory that we might not be able to do otherwise?
We had a couple of problems in which we were to prove facts about very specific instances of groups based on the representations but they could all be done with group theoretic notions anyway; I'm looking for maybe a more fundamental paradigm underlying representation theory.
Perhaps the subtext to my post is this:
Why study the representation theory of finite groups?
I vaguely understand why we do it for locally compact groups since they can be much harder to handle, but for finite groups, it seems like beautiful math but not necessary for the understanding of groups.