Notes for Oral Examination Presentation

k field, algebraically closed G finite group Definitions (1) A representation of G over k of degree n is a homomorphism of groups ρ : G→GLn(k), or equivalently, a kG-module structure for an n-dimensional k vector space M , where kG is the group algebra. (2) A representation M is irreducible if it is a simple kG-module, and indecomposable if M 6= N1 ⊕ N2 for any non-trivial submodules N1, N2. Notice that irreducible =⇒ indecomposable. Representation Theory of Finite Groups is thus the study of groups acting on vector spaces, ie, sending abstract group elements to concrete matrices. It has been used to prove grouptheoretic results of Frobenius and Burnside. Classically there are two cases.