The largest orbit sizes of linear group actions and abelian quotients

Abstract

Let G be a finite group and let V be a finite and faithful G-module such that V is completely reducible, possibly of mixed characteristic, or V is a G-module over a field of characteristic p with Op(G)=1. Suppose that M is the largest orbit size in the action of G on V. It is known that the index of the commutator subgroup G′ in G is bounded by M, that is, |G:G′|≤M. In this paper, we classify all linear group actions in which |G:G′|=M. It turns out that our classification is a vast generalization of a classic 1967 result by D. Passman. We also include an application of the main results to Brauer's k(B)-problem.