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The probability of generating a finite simple group. (English) Zbl 0836.20068

In the paper [Math. Z. 110, 199-205 (1969; Zbl 0176.29901)] the reviewer proved that a pair of randomly chosen elements from the alternating \(A_n\) generates \(A_n\) with probability tending to 1 as \(n\to\infty\). He conjectured that a similar result is true for the other infinite classes of finite simple groups. This conjecture was verified for the classical simple groups and for small rank exceptional groups by W. M. Kantor and A. Lubotzky [Geom. Dedicata 36, 67-87 (1990; Zbl 0718.20011)]. The authors of the present paper have finally settled the remaining cases to show that the following slightly stronger statement is true. Let \(G_0\) be a finite simple group, and let \(G\) be a group such that \(G_0\leq G\leq\operatorname{Aut}(G_0)\). Then the probability that two randomly chosen elements of \(G\) generate a subgroup containing \(G_0\) tends to 1 as \(|G|\to\infty\). The proof requires, of course, the classification of finite simple groups as well as detailed properties of the groups in question.

MSC:

20G40 Linear algebraic groups over finite fields
20D06 Simple groups: alternating groups and groups of Lie type
20P05 Probabilistic methods in group theory
20F05 Generators, relations, and presentations of groups
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