Probabilistic generation of finite almost simple groups (Q6608708)

A finite group \(G\) is said to be \(\frac{3}{2}\)-generated if every non-trivial element belongs to a generating pair. \textit{T. Breuer} et al., J. Algebra 320, No. 2, 443--494 (2008; Zbl 1181.20013) proved that \(G\) is \(\frac{3}{2}\)-generated if and only if every proper quotient of \(G\) is cyclic. \textit{W. M. Kantor} and \textit{A. Lubotzky} [Geom. Dedicata 36, No. 1, 67--87 (1990; Zbl 0718.20011)] asked whether there was a probabilistic version of \(\frac{3}{2}\)-generation. This is not the case for alternating groups: if \(x \in A_{n}\) moves only a bounded number of points, the probability that \(x\) and a random element of \(A_{n}\) generate a transitive group goes to 0 as \(n \rightarrow \infty\).\N\NThe main result of the paper under review is Theorem 1.1: There exists an absolute constant \(\varepsilon > 0\) such that the following holds. Let \(S\) be a finite simple group of Lie type of large enough order and let \(x,y \in \Aut(S)\) with \(x \not =1\). Then the probability that \(x\) and a random element of \(Sy\) generate \(\langle S,x,y \rangle\) is at least \(\varepsilon\).\N\NAs a consequence of the main theorem, the authors prove Corollary 1.3: Let \(G\) be a profinite group and let \(g\in G\). Then, the following are equivalent: (i) the probability that \(g\) and a random element of \(G\) generate a prosolvable group is positive; (ii) there exists \(C \geq 1\) such that \(g\) centralizes all but at most \(C\) non-abelian chief factors of \(G/N\) for every open normal subgroup \(N\) of \(G\).\N\NA key ingredient in the proof of Theorem 1.1 is a result of independent interest. The authors consider the proportion of elements in a classical group of dimension \(n\) over the field of size \(q\) which fix no subspace of dimension at most \(t\). For \(q\) and \(t\) fixed, they prove that the limit as \(n \rightarrow \infty\) exists and is strictly between \(0\) and \(1\).