Finite Group Modular Field Extensions, Green Theory and Absolutely Indecomposable and Simple Modules

Abstract

Let \(\Phi \) be a field of prime characteristic p and let G be a finite group. We develop an equivalence relation between the set of isomorphism types of indecomposable (simple) KG-modules, where K is any finite subfield of \(\Phi \), and relate the equivalence classes to the set of isomorphism types of indecomposable (resp. simple) \(\Phi G\)-modules. When \(\Phi \) is the algebraic closure of a field F of order p, we study indecomposable (resp. simple) \(\Phi G-\)modules and obtain a classification of the isomorphism types of simple \(\Phi G\)-modules and a new formula for the number of such types in each equivalence class.