Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky

We prove that a randomly chosen involution and a randomly chosen additional element of a finite simple group G generate G with probability → 1 as |G| → ∞. This confirms a conjecture of Kantor and Lubotzky. Applications and related results are derived. For example, we show that, except for the Suzuki groups and finitely many possible other exceptions, all finite simple groups can be generated by two elements, one of which has order 3. We also obtain sharp estimates on the probability P(G) of generating a finite simple group of exceptional Lie type G by two randomly chosen elements. This complements analogous estimates of Babai and Kantor for alternating and classical groups. Denoting by m(G) the minimal index of a proper subgroup of a finite simple group G, we conclude, in particular, that m(G)(1 - P(G)) is bounded between two positive absolute constants.