#### Abstract:

Quillen’s calculation of algebraic -theory of finite fields incorporates a wide range of techniques from group cohomology to representation theory. This exposition focuses on the proof of Green’s theoerm: For any finite group and any representation of over ( is a power of some prime ), one can construct a character called the Brauer character which is a virtual complex character. We will also introduce its application in algebraic K-theory. The Brauer character of the n-dimensional standard representation of is a virtual complex character, thus it induces a map , hence a map . Taking the n-th homotopy group gives a map . This map is a key construction that allows one to identify with the homotopy groups of a better understood space.

This writing is part of the course project for MATH 506 Group Representation Theory: