Augmentations, nilpotent ideals, and semisimplicity

Abstract

In the first section we explore a few of the most fundamental properties of group algebras and their modules. One of our main points is the proof of Maschke’s Theorem, which tells us that if the characteristic of k does not divide the order of G, then every exact sequence of kG-modules splits and every kG-module is both projective and injective. Therefore the application of homological algebra is interesting only in the complementary case that the characteristic of the coefficient field k divides the order of the group G. We begin with an assortment of loosely related facts.