Modular representations of Loewy length two

Let G be a finite p -group, K a field of characteristic p , and J 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 dim K ( 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 dim K ( Rad ( M ) ) = 1 .