Linear group actions with at most three orbits of the largest size

Abstract

Let G be a finite group and let V be a finite completely reducible faithful G-module, possibly of mixed characteristic. The second author and Yong Yang proved in previous work that | G: G′| ≤ M, where M is the largest orbit size of G on V. They also showed that if G is a nonabelian group and | G: G′| = M, then G is nilpotent having at least two orbits of maximal size M on V. In this paper, we deal with the latter case, where G is nonabelian and | G: G′| = M, and we classify all linear group actions in which G has at most three orbits of size M on V.