Stabilizers and Fitting Subgroups

Moreto and Wolf have shown that if G is a solvable group and V is a finite, completely reducible, and faithful G-module, then there is an element \(v \in V\) with a “large” orbit, in the sense that the centralizer of v is contained in a “small” Fitting subgroup of G. We discuss the subtle proof of Moreto and Wolf, which will have several applications later in the book. We also discuss a stronger result by the second author.