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

Let \(\Phi \) $\mathrm{\Phi }$ 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 \(\Phi \) $\mathrm{\Phi }$ , and relate the equivalence classes to the set of isomorphism types of indecomposable (resp. simple) \(\Phi G\) $\mathrm{\Phi }G$ -modules. When \(\Phi \) $\mathrm{\Phi }$ is the algebraic closure of a field F of order p, we study indecomposable (resp. simple) \(\Phi G-\) $\mathrm{\Phi }G-$ modules and obtain a classification of the isomorphism types of simple \(\Phi G\) $\mathrm{\Phi }G$ -modules and a new formula for the number of such types in each equivalence class.