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

Tsachik Gelander*, Yair Glasner

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


The purpose of this note is to extend the classical Aschbacher-O'Nan-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.

Original languageEnglish (US)
Pages (from-to)58-63
Number of pages6
JournalJournal of Algebra
StatePublished - Mar 5 2013


  • Aschbacher-O'Nan-Scott theorem
  • Infinite permutation groups
  • Linear groups
  • Margulis-Soifer theorem
  • Primitive groups

ASJC Scopus subject areas

  • Algebra and Number Theory


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