A branch of algebra that concerns itself with representations, or homomorphisms from groups into the general linear group over some vector space. It happens that a lot of the interesting information of a representation can be distilled into a single statistic, the trace of the associated matrices. These traces are called the character of the representation.

In case the group is finite, and the representations are over a field that doesn't divide the order of the group, Maschke's theorem holds. This states that every representation is completely reducible, meaning that the corresponding module of the representation (yes, the action of the group on the vector space also is an algebra, as linear maps can be added and subtracted) can be decomposed into the direct sum of irreducible modules.

Representation theory proves some results that have no known elementary proof. Until fairly recently, a famous theorem of Burnside was such an example, namely the fact that every group whose order is the product of the power of two primes is soluble.