Tetsuro Okuyama

Subgroups and almost split sequences of a finite group

Let G be a finite group and hbe a field of characteristic p > 0. The isomorphism classes of KG-modules, relative to direct sums, form a free abelian group n(G) called the representation ring or the Green ring of G, where the multiplications are defined to be the tensor products of modules. In their paper [ 11 Benson and Parker introduced two inner products on a(G) and showed that these inner products are nonsingular. In fact they proved the result by giving “the orthogonality relations” with respect to these inner products (Theorems 3.4 and 3.5 of [ 11). In this paper, by using their inner products we study the inductions from subgroups and the restrictions to subgroups of the almost split sequences of modules. In Section 2 we prove “Nakayama relations” for some families of indecomposable modules, which is a natural extension of Nakayama relations for projective indecomposable modules and simple modules (Chapter III, Theorem 2.6 of [3]). In Section 3 we consider the relative Grothendiek ring a,(G) with respect to a family X of subgroups of G (for the definition see below) and shall show that if each subgroup in X has a cyclic Sylow p-subgroup, then a,(G) is a free abelian group. We prove the result by giving a free basis of a,(G).