The infinite simple group V of Richard J. Thompson : presentations by permutations
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We show one can naturally describe elements of R. Thompson's infinite finitely presented 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.
Quick , M & Bleak , C P 2015 , ' The infinite simple group V of Richard J. Thompson : presentations by permutations ' Mathematics arXiv .