… We show that two random elements of a finite simple group G generate G with probability
1 as IG[ ~ c~. This settles a conjecture of Dixon. … Twenty years later, using the
classification of finite simple groups, Kantor and Lubotzky [KL] dealt with the case where G
is a classical group or a small rank exceptional group. In this paper we complete the proof of
Dixon's conjecture by handling the remaining exceptional groups. In fact, our result (as well
as the results of [Di] and [KL]) is slightly more general, in that it deals with almost …

We study the probability of generating a finite simple group, together with its generalisation
PG, 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 PG, soc G (2)⩽ 53/90,
with equality if and only if G is A 6 or S 6, and establish a similar result for PG, soc G (3).
Positive answers to longstanding questions of Wiegold on direct products, and of Mel'nikov
on profinite groups, follow easily from our results.