The probability of generating a finite soluble group. (Q2766371)

\textit{J. D. Dixon} [Math. Z. 110, 199-205 (1969; Zbl 0176.29901)] conjectured that if two elements are randomly chosen from a finite simple group \(G\), they will generate \(G\) with probability \(\to 1\) as \(|G|\to\infty\). Dixon [loc. cit.] proved this if \(G\) is an alternating group. \textit{W. M. Kantor} and \textit{A. Lubotzky} [Geom. Dedicata 36, No. 1, 67-87 (1990; Zbl 0718.20011)] proved Dixon's conjecture for classical groups and certain exceptional groups. \textit{W. M. Liebeck} and \textit{A. Shalev} [Geom. Dedicata 56, No. 1, 103-113 (1995; Zbl 0836.20068)] settled the remaining exceptional groups.NEWLINENEWLINE It is well known that a finite group is soluble if each of its 2-generator subgroups is soluble. In the paper under review the authors prove the following probabilistic version of this result. Theorem A: Let \(G\) be a finite group. (a) If the probability that two randomly chosen elements of \(G\) generate a soluble group is greater than \(11/30\), then \(G\) is soluble. (b) If the probability that two randomly chosen elements of \(G\) generate a nilpotent group is greater than \(1/2\), then \(G\) is nilpotent. (c) If the probability that two randomly chosen elements of \(G\) generate a group of odd order is greater than \(11/30\), then \(G\) has odd order.NEWLINENEWLINE By means of the proof of Dixon's conjecture the authors obtain the following Theorem B: There exists a real number \(\kappa\) strictly between \(0\) and \(1\) with the following property. Let \(\mathfrak X\) be any class of finite groups which is closed for subgroups, quotient groups and extensions, and let \(G\) be a finite group. If the probability that two randomly chosen elements of \(G\) generate a group in \(\mathfrak X\) is greater than \(\kappa\), then \(G\) is in \(\mathfrak X\). The proofs use not only the classification theorem, but also some of the detailed information which is now available concerning the maximal subgroups of finite almost simple groups.