Proving facts about groups with representation theory.
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.
To address the stated subtext of your post: many people, myself included, take the position that groups are important because they act on things. A representation is just a group action on a vector space (by linear operators). And whenever you have a group action, even if it isn't on a vector space, there is often a closely related representation lurking nearby.
For example, if $G$ is acting on a finite set $S$ then there is an induced representation on $\mathbb{F}^S$ for any field $\mathbb{F}$. More generally, if $G$ is acting on a space $X$ then there is an induced representation of $G$ on the algebra $\mathcal{O}(X)$ of functions on $X$, whatever we mean by "function" in this situation (e.g. perhaps $X$ is a topological space on which $G$ acts by homeomorphisms, and $\mathcal{O}(X)$ is the algebra of continuous real-valued functions on $X$).
To elaborate on this last point: suppose $X$ is a set on which $G$ acts. For any set $Y$, there is an induced action on the set $Y^X$ of maps from $X$ to $Y$: for $g\in G$ and $f:X\to Y$, the map $g\cdot f$ is defined by $$(g\cdot f)(x) = f(g^{-1}x).$$
If we equip $X$ with some additional structure, like a topology, we may insist that the action of $G$ preserve this structure. This guarantees that the action of $G$ on maps will take structure-preserving maps to structure-preserving maps. As an example, if $X$ and $Y$ are topological spaces and $G$ acts on $X$ by homeomorphisms, then the action of $G$ on $Y^X$ sends continuous maps to continuous maps. In particular, taking $Y=\mathbb{R}$, we have an action of $G$ on the algebra of real-valued continuous functions.
Two examples of purely group-theoretic theorems proved using representation theory are:
Burnside's theorem, which states that a finite group having no more than 2 distinct prime divisors must be solvable. (Much later, a proof without representation theory was found in the 1970's.)
Theorem about Frobenius groups: Assume a finite group $G$ contains a subgroup $H$ such that for every $x \in G - H$, the intersection $H \cap x^{-1}Hx = \{1\}$. Then the set of elements of $G$ not in any conjugate of $H$, together with the identity, form a normal subgroup of $G$. No proof not using representation theory is known.
Just a few unsorted comments to add to what's been said above...
The classification of finite simple groups relied heavily on modular representation theory, so there is that. Many of the sporadics are only described reasonably through their representations.
Symmetric groups have nicely describable representations that correspond to a combinatorics construct called Young Tableaux. You can also determine certain things about conjugacy classes in symmetric groups using pretty basic representation theory, for example rational conjugacy classes have a basic, group theoretic description.
More generally, there are a lot of concepts in finite group theory which generalize to representation theory and are often easier to understand there. Transfer, for example, can be realized as the determinant of an induced representation over $\mathbb{F}_2[G]$.
This is related to Brad's answer: in particular, we are often concerned about groups acting on other groups, particularly subgroups of some supergroup. If the group being acted upon is abelian, we can use representation theory over finite fields to describe what's going on. This is particularly important in the study of solvable groups, for which the minimal normal subgroups are elementary abelian.
A lot of computational group theory is done using representations, so this is one way representation theory is useful for its own sake. You don't want to be doing computations Cayley's theorem style in a group with a large number of elements.