Regular Orbits of Linear Groups with an Application to the k(GV)-Problem, 1

Let p be a prime, G a finite group of order coprime to p, and V a faithful pG-module. Suppose G has a normal quasi-simple irreducible subgroup H. We show that either G has a regular orbit on vectors, or H is an alternating group and V a minimal module, or H, n, and p are contained in a short explicit list. Together with the work of Robinson and Thompson, this leads to a solution of the k(GV)-problem for primes p > 211.