Regular orbits of finite primitive solvable groups, the final classification

TL;DR

This paper classifies all small exceptions where finite primitive solvable groups acting on vector spaces lack regular orbits, extending understanding of group actions in algebra.

Contribution

It provides a complete classification of the small cases where such groups do not have regular orbits, filling a gap in the theory of group actions.

Findings

Most primitive solvable groups have regular orbits on the vector space.

Identifies specific small cases where regular orbits do not exist.

Completes the classification of these exceptional cases.

Abstract

Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$, and $G$ is not metacyclic. Then $G$ always has a regular orbit on $V$ except for a few "small" cases. We completely classify these cases in this paper.