The classification of almost simple $\tfrac{3}{2}$-transitive groups

TL;DR

This paper classifies finite 3/2-transitive permutation groups, showing they are either affine or almost simple, and provides a detailed classification of the almost simple case, including an arithmetic analysis of groups of Lie type.

Contribution

It proves that all 3/2-transitive groups are either affine or almost simple and classifies the almost simple groups, advancing understanding of their structure.

Findings

3/2-transitive groups are either affine or almost simple

Almost simple 3/2-transitive groups are classified

Primitive groups of Lie type have subdegrees divisible by p, with specific exceptions

Abstract

A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less obvious ones. We prove that 3/2-transitive groups are either affine or almost simple, and classify the latter. One of the main steps in the proof is an arithmetic result on the subdegrees of groups of Lie type in characteristic $p$: with some explicitly listed exceptions, every primitive action of such a group is either 2-transitive, or has a subdegree divisible by $p$.