Abstract

Brauer’s Main Theorems are results in the modular representation theory of finite groups that link the blocks of a finite group G with those of its p-local subgroups. Over characteristic not dividing the group order, all finite-dimensional modules are projective and the group algebra is semisimple. This unsurprisingly does not hold for a field k of characteristic p dividing |G|, and we will introduce certain p-subgroups Q of G called vertices as measures of ‘how far from projective’ modules are, then extend this into the concept of defect groups D for the blocks of the group algebra. We will see that kG-module structure can be related to that of N_G(Q) and N_G(D) using the Green correspondence and Brauer’s Main Theorems, through small concrete examples as well as theoretical applications. If there’s time we’ll also outline Brauer-Dade theory for cyclic blocks, where the simples and indecomposable projectives can be described neatly using graphs known as Brauer trees.