Some interesting finite groups arise as symmetry groups of geometric objects. Since rotations are linear and thus can be represented by matrices, a symmetry group can be thought of as a multiplicative group of matrices. Representation theory studies the converse problem: given an abstract finite group, how closely can it be represented by a group of matrices?
The main textbook for the course will be "Character Theory of Finite Groups" by I. Martin Isaacs, published by Dover for $12.95. We'll cover representations (especially in characteristic 0), characters, orthogonality relations, Burnside's theorem on solvability of groups whose order is divisible by only two primes, the existence of normal complements in Frobenius groups, and Brauer induction, which allows characters to be recognized by their restrictions to hyper-elementary subgroups.
There probably won't be time, but if there is, we may refer to "Representations and cohomology; I: Basic representation theory of finite groups and associative algebras", by David J. Benson, published by Cambridge University Press for $33, to get an introduction to a modern point of view on representations in positive characteristic via homological algebra and K-theory. Volume II of his text goes on to treat the construction of a projective algebraic "support" variety from a group representation.
Prerequisite: Math 401-402.