An Aschbacher--O'Nan--Scott theorem for countable linear groups

TL;DR

This paper extends the Aschbacher--O'Nan--Scott theorem to countable linear groups, revealing that the number of primitive actions varies by group type, with some types admitting infinitely many actions.

Contribution

It provides a classification of primitive actions for countable linear groups, showing a stark contrast in the number of actions based on group type, extending finite group theory results.

Findings

Linear groups of almost simple type have uncountably many primitive actions.

Affine and diagonal groups have only one primitive action.

The results are significant for rigid groups like simple and arithmetic groups.

Abstract

The purpose of this note is to extend the classical Aschbacher--O'Nan--Scott theorem for finite groups to the class of countable linear groups. This relies on the analysis of primitive actions carried out in a previous paper. Unlike the situation for finite groups, we show here that the number of primitive actions depends on the type: linear groups of almost simple type admit infinitely (and in fact unaccountably) many primitive actions, while affine and diagonal groups admit only one. The abundance of primitive permutation representations is particularly interesting for rigid groups such as simple and arithmetic ones.