My aim in this article is to show how simple probabilistic arguments can be applied to prove group theoretical results. While some of these results are formulated in probabilistic language, some have purely group theoretical formulations (see Theorems 2 and 3). I am going to give only a sample of such results, referring to [Sh] for a more exhaustive survey. Not all of the results here occur in [Sh], though, and Propositions 1 and 2 have not been published before.

Our subject apparently begins with E. Netto, who, more than a century ago, wrote: “If we arbitrarily select two or more substitutions of n elements, it is to be regarded as extremely probable that the group of lowest order which contains these is the symmetric group, or at least the alternating group” [N1, p.76]. Later, in the English version of his book, he added: “In the case of two substitutions the probability in favour of the symmetric group may be as about 3/4, and in favour of the alternating, but not symmetric, group as about 1/4” [N2, p.90]. These statements were made precise by J. D. Dixon [D], who proved that, as n → ∞, the probability that two elements of Sn generate either Sn or An tends to 1. Dixon made then a generalized conjecture: let us write P(G, k) for the probability that k random elements generate the group G, then Dixon conjectured that, letting S range over all finite simple groups, P(S, 2) → 1, as |S| → ∞.