The modular representation algebra of a finite group

1.1. Notation and terminology. G is a finite group, with unit element e. k is a field of characteristic p. By a G-module M is meant a (/c, G)-module. Elements of G act as right operators on M, and me m (m M). The k-dimension dim M of M is assumed finite. For example, F F(k, G) is the regular G-module, i.e., the group algebra of G over k, regarded as G-module, and ]ca is the unit G-module, i.e., the field ]c, made into a "trivial" G-module, i.e.,Kx K (ek, xeG). For any G-moduleM, {M} is the class of all G-modules isomorphic to M. V (i runs over a suitable index set I) is a set of representatives of the classes {V} of indecomposable G-modules. The number of these indecomposable classes is finite if and only if either p 0, or p is a finite prime such that the Sylow p-subgroups of G are cyclic (D. G. Higman [5]). F (j 1, n) is a set of representatives of the classes {FA of irreducible G-modules. The number n of these is always finite. If k is algebraically closed, n is equal to the number of p-regular classes of G (R. Brauer, see [1], [2]). If M’, M" are G-modules, M’ M" denotes their direct sum. If M is a G-module, and s a nonnegative integer, sM denotes the direct sum of s isomorphic copies of M.