Finite group modular field extensions, Green theory and absolutely indecomposable and simple modules (Q6551369)

Let \(G\) be a finite group, let \(K\) be a field of prime characteristic \(p\) and let \(k\) be a finite subfield of \(K\).\N\NIn the paper under review, the author develops an equivalence relation between the set of isomorphism types of indecomposable (simple) \(kG\)-modules and he relates the equivalence classes to the set of isomorphism types of indecomposable (simple) \(G\)-modules. In this way, he obtains a classification of the isomorphism types of simple \(G\)-modules and a new formula for the number of such types in each equivalence class.\N\NIn the case where \(k=\mathrm{GF}(p)\) and \(K\) is an algebraic closure of \(k\), the author shows that (see Remark 3.6):\N\[\N|\mathrm{ITS}(KG)|=\sum_{W \in \mathrm{ITS}(kG)} \big | \Aut(\mathrm{End}_{kG}(W)) \big|,\N\]\Nwhere \(\mathrm{ITS}(FG)\) denotes the set of distinct isomorphism types of simple \(FG\)-modules over the field \(F\).