On simple modules and normal subgroups of p-solvable groups

Let N be a normal subgroup of a p-solvable group G and let M be a simple FN-module, were F is an algebraically closed field of characteristic p. Next, denote by IRR0(FG| M) the set of all simple FG-modules V lying over M such that the p-part of dim(F) V is as small as possible. In this paper, | IRR0(FG| M)| and the vertices of modules in IRR0(FG| M) are determined. The p-blocks of G to which modules in IRR0(FG| M) belong are also determined.