Cohomological uniqueness of finite groups of prime power order

Abstract

Group cohomology originates, along with homological algebra, from algebraic topology and the study of cohomology groups of certain topological spaces. Given a finite group G, we can associate to it a classifying space BG, which satisfies that its first homotopy group is isomorphic to G, and its higher order homotopy groups are trivial. We can then define the cohomology groups Hn(G, V ) of G with coefficients on an RG-module V , where R is a commutative ring, to be the cohomology groups of its classifying space BG, see [Hat02]. It is also possible to give a purely algebraic definition of the cohomology groups of G in terms of derived functors, and in fact this is the approach that we will follow throughout this thesis.