Classifying solvable primitive permutation groups of low rank

Abstract

Suppose that a finite solvable permutation group G acts faithfully and primitively on a finite set Ω. Let $G_0$ be the stabilizer of a point $\alpha \in \Omega$ and the rank of G be the number of distinct orbits of $G_0$ in Ω (including the trivial orbit $\left\{\alpha \right\}$). Then G always has rank greater than four except for in a few cases. We completely classify these cases in this paper.