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.Mathematics Subject Classification (2010). Primary: 20F05; Secondary: 20E32, 20F65.

In this paper we use the Classification of the Finite Simple Groups, the O'Nan-Scott Theorem and Aschbacher's theorem to classify the primitive permutation groups of degree less than 4096. The results will be added to the primitive groups databases of GAP and Magma.

We study the probability of generating a finite simple group, together with its generalisation P G,soc G (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ soc G. We prove that P G,soc G (2) 53/90, with equality if and only if G is A 6 or S 6 , and establish a similar result for P G,soc G (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel nikov on profinite groups, as well as to a conjecture of Holt and Stather, follow easily from our results.

We show that the probability of generating an iterated wreath product of non-abelian finite simple groups converges to 1 as the order of the first simple group tends to infinity provided the wreath products are constructed with transitive and faithful actions. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated.

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