MODULAR REPRESENTATIONS OF LOEWY LENGTH TWO

Let G be a finite p-group, K a field of characteristic p ,a ndJ the radical of the group algebra K[G]. We study modular representations using some new results of the theory of extensions of modules. More precisely, we describe the K[G]-modules M such that J 2 M = 0 and give some properties and isomorphism invariants which allow us to compute the number of isomorphism classes of K[G]-modules M such that dimK (M) = µ(M) + 1, where µ(M) is the minimum number of generators of the K[G]-module M. We also compute the number of isomorphism classes of indecomposable K[G]-modules M such that dimK (Rad(M)) = 1.