On subgroups of prime power index

All finite groups G are determined that admit a subgroup K of index p(a), p a prime, under the assumption that G has an irreducible and faithful GF(q) -module in characteristic p whose dimension over GF(q) is at most a. As an application to the theory of permutation groups, the maximal transitive sub groups of the primitive affine permutation groups are determined. The above -mentioned classification is generalized by dropping the assumption that p\ q. In both cases surprisingly nice results are obtained.