Regular orbits of finite primitive solvable groups

Abstract: The case when a linear group $G$ acting primitively on the vector space $V$ is of central importance in the theory of representations of solvable groups. In short, such groups have an invariant $e$ that measures their complexity. It is known that if $e > 118$, $G$ has a regular orbit.

I was able to improve this result dramatically by classifying all the cases when the regular orbit exists. In some of my early papers, I gave a coarse classification of the existence of regular orbits for primitive solvable linear groups, and the results have been widely used by other people and myself to study related problems of arithmetic properties of group invariants. A more detailed final classification has been completed in some of my recent work along with several further applications.