Classifying Primitive Solvable Permutation Groups of Rank 5 and 6

TL;DR

This paper classifies finite primitive solvable permutation groups of rank 5 and 6, extending previous classifications for lower ranks, and provides a complete understanding of these specific group actions.

Contribution

It offers a complete classification of primitive solvable permutation groups with rank 5 and 6, advancing the understanding of their structure and properties.

Findings

Classified all primitive solvable groups of rank 5 and 6

Extended previous classifications for ranks 4 and below

Provides structural insights into these groups

Abstract

Let $G$ be a finite solvable permutation group acting faithfully and primitively on a finite set $\Omega$. Let $G_0$ be the stabilizer of a point $\alpha \in \Omega$ The rank of $G$ is defined as the number of orbits of $G_0$ in $\Omega$, including the trivial orbit $\{\alpha\}$. In this paper, we completely classify the cases where $G$ has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.