Classifying Solvable Primitive Permutation Groups of Low Rank

TL;DR

This paper completes the classification of finite, solvable, primitive permutation groups of rank less than 5, confirming previous results for ranks 2 and 3, and explicitly constructing rank 4 groups.

Contribution

It finalizes the classification of all solvable primitive permutation groups of rank less than 5, including explicit constructions for rank 4 groups.

Findings

Confirmed classifications for ranks 2 and 3.

Explicitly constructed rank 4 groups.

Provided a complete classification for ranks less than 5.

Abstract

Suppose that $G$ is a finite, transitive, solvable permutation group acting on a set $S$ with $n$ elements. Let $G_0$ be the stabilizer of a point $\alpha \in \Omega$. Define the rank of a permutation group, denoted $r(G),$ as the number of distinct orbits of $G_0$ in $S$ (including the trivial orbit $\{\alpha\}$). Huppert \cite{Huppert} and Foulser \cite{Foulser} classified all finite, solvable, permutation groups of rank two and three respectively, and Foulser restricted the rank four groups to a small list of possibilities. This paper completes the classification of all groups of rank less than $5$ by explicitly confirming these past results and computationally constructing the groups of rank $4$.