Group Representations

Abstract

The goal of group representation theory is to study groups via their actions on vector spaces. Consideration of groups acting on sets leads to such important results as the Sylow theorems. By studying actions on vector spaces even more detailed information about a group can be obtained. This is the subject of representation theory. Our study of matrix representations of groups will lead us naturally to Fourier analysis and the study of complex-valued functions on a group. This in turn has applications to various disciplines like engineering, graph theory, and probability, just to name a few.