The infinite simple group V of Richard J. Thompson: presentations by permutations

Collin Bleak*, Martyn Quick

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)
30 Downloads (Pure)


We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group V, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions.  This perspective provides an intuitive explanation towards the simplicity of V and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups.  We find a natural infinite presentation for V as a group generated by these "transpositions," which presentation bears comparison with Dehornoy's infinite presentation and which enables us to develop two small presentations for V: a human-interpretable presentation with three generators and eight relations, and a Tietze-derived presentation with two generators and seven relations.
Original languageEnglish
Pages (from-to)1401-1436
Number of pages36
JournalGroups, Geometry, and Dynamics
Issue number4
Early online date7 Dec 2017
Publication statusPublished - 2017


  • Thompson's groups
  • Simple groups
  • Presentations
  • Generators and relations
  • Permutations
  • Transpositions


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