An Aschbacher–OʼNan–Scott theorem for countable linear groups

The purpose of this note is to extend the classical Aschbacher–OʼNan–Scott theorem on finite groups to the class of countable linear groups. This relies on the analysis of primitive actions carried out in Gelander and Glasner (2008) [GG08]. 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.