Part I Cohomology of Finite Groups

Abstract

The cohomology of finite groups deals with a general situation that occurs frequently in different concrete forms. For example, if L|K is a finite Galois extension with Galois group G, then G acts on the multiplicative group L ^× of the extension field L. In the special case of an extension of finite algebraic number fields, G acts on the ideal group J of the extension field L. The theory of group extensions provides us with the following example: If G is an abstract finite group and A is a normal subgroup, then G acts on A via conjugation. In representation theory we study matrix groups G that act on a vector space. The basic notion underlying all these examples is that of a G-module. We will now present some general considerations about G-modules, some of which the reader may already know from the theory of modules over general rings.