Primitive permutation groups with a suborbit of length 5 and vertex-primitive graphs of valency 5

TL;DR

This paper classifies finite primitive permutation groups with a suborbit of length 5 and applies this to classify vertex-primitive graphs of valency 5, also analyzing related almost simple groups with specific maximal subgroups.

Contribution

It provides a complete classification of primitive groups with a suborbit of length 5 and vertex-primitive graphs of valency 5, including the structure of certain almost simple groups.

Findings

Classification of primitive permutation groups with a suborbit of length 5

Classification of vertex-primitive graphs of valency 5

Identification of maximal subgroups isomorphic to Alt(5) or Sym(5)

Abstract

We classify finite primitive permutation groups having a suborbit of length 5. As a corollary, we obtain a classification of finite vertex-primitive graphs of valency 5. In the process, we also classify finite almost simple groups that have a maximal subgroup isomorphic to $\mathrm{Alt}(5)$ or $\mathrm{Sym}(5)$.