The probability of generating a permutation group. (Q1881572)

For every finite group \(G\), let \(P_G(t)\) denote the probability that \(t\) elements chosen at random from the group generate \(G\). Clearly \(P_G(t)\leq P_{G/K}(t)\) whenever \(K\) is a normal subgroup of \(G\). It is known that the minimal number \(d(G)\) of generators of a group \(G\) is at most \(\lfloor n/2\rfloor\) for every permutation group \(G\) of degree \(n\) [see \textit{A. McIver} and \textit{P. M. Neumann}, Q. J. Math., Oxf. II. Ser. 38, 473-488 (1987; Zbl 0627.20014) and \textit{P. J. Cameron, R. Solomon} and \textit{A. Turull}, J. Algebra 127, No. 2, 340-352 (1989; Zbl 0683.20004)]. The main result of the present paper is the following (where \(G/O^2(G)\) is the largest \(2\)-factor of \(G\)): \[ \lim_{n\to\infty}\sup_{G\leq\text{Sym}(n)}\{P_{G/O^{2}(G)}(\lfloor n/2\rfloor)-P_G(\lfloor n/2\rfloor)\}=0. \] From this is deduced that \[ \lim_{n\to\infty}\inf_{G\leq\text{Sym}(n)}P_G(\lfloor n/2\rfloor)=\prod_{i=1}^\infty(1-\tfrac 1{2^i})=0.2887\dots. \] The proof is based on methods of an earlier paper [\textit{E. Detomi, A. Lucchini} and \textit{F. Morini}, Isr. J. Math. 132, 29-44 (2002; Zbl 1042.20046)] and a result of independent interest: there is a constant \(\alpha<\tfrac 12\) such that if \(G\leq\text{Sym}(n)\) then \(d(G/R(G))\leq\alpha n\) where \(R(G)\) is the solvable radical of \(G\). The permutation groups of degree \(n\) which require \(\lfloor n/2\rfloor\) generators are also described (they are certain \(\{2,3\}\)-groups whose orbits are all of length at most \(8\)).