On imprimitive rank 3 permutation groups

TL;DR

This paper classifies all quasiprimitive rank 3 permutation groups, focusing on imprimitive cases, by analyzing their structure and actions, and completing the broader classification with earlier foundational work.

Contribution

It provides a complete classification of imprimitive rank 3 permutation groups that are quasiprimitive but not primitive, including infinite families and specific examples.

Findings

Identified two infinite families of such groups.

Classified all imprimitive almost simple permutation groups with specific action properties.

Determined conditions for imprimitive rank 3 groups with almost simple induced actions.

Abstract

A classification is given of rank 3 group actions which are quasiprimitive but not primitive. There are two infinite families and a finite number of individual imprimitive examples. When combined with earlier work of Bannai, Kantor, Liebler, Liebeck and Saxl, this yields a classification of all quasiprimitive rank 3 permutation groups. Our classification is achieved by first classifying imprimitive almost simple permutation groups which induce a 2-transitive action on a block system and for which a block stabiliser acts 2-transitively on the block. We also determine those imprimitive rank 3 permutation groups $G$ such that the induced action on a block is almost simple and $G$ does not contain the full socle of the natural wreath product in which $G$ embeds.