Probabilistic generation of finite groups with a unique minimal normal subgroup. (Q2840161)

Let \(G\) be a finite group and denote by \(d(G)\) the smallest cardinality of a set of generators for \(G\). Let \(P_G(d)\) be the probability of generating \(G\) with \(d\) elements. If \(N\) is a normal subgroup of \(G\) and \(d\geq d(G/N)\), define \(P_{G,N}(d)=P_G(d)/P_{G/N}(d)\) to be the conditional probability that \(d\) randomly chosen elements of \(G\) generate \(G\), given that they generate \(G\) modulo \(N\). Let \(L\) be a monolithic primitive group with unique minimal normal subgroup \(N\). If \(L\) is cyclic, then \(L\) has order \(p\) for some prime \(p\) and \(P_{L,N}(d)=P_L(d)=1-\frac{1}{p^d}\). If \(L\) is noncyclic, then \(d(L)=\max\{2,d(L/N)\}\) and \(P_{L,N}(d(L))\) tends to 1 when \(|N|\) tends to infinity. In particular, there exists an absolute constant \(c\) such that \(P_{L,N}(d)\geq c\) whenever \(d\geq d(L)\) [\textit{A. Lucchini} and \textit{F. Morini}, Pac. J. Math. 203, No. 2, 429-440 (2002; Zbl 1064.20072)].NEWLINENEWLINE The aim of this paper is to give a more precise estimate of this constant \(c\). The authors prove that if \(L\) is a monolithic primitive group with socle \(N\) and \(d\geq d(L)\), then \(P_{L,N}(d)\geq\frac{1}{2}\). Precisely, they prove that when \(N=\text{soc}(L)\) is abelian, then \(c=\frac{1}{2}\) if \(L\) is cyclic; otherwise \(P_{L,N}(d)\geq\frac{2}{3}\). In the case when \(N\) is non-abelian the authors extend a result of \textit{N. E. Menezes} et al. [Isr. J. Math. 198, 371-392 (2013; Zbl 1291.20068)] and prove that if \(L\) is a monolithic group with non-abelian socle \(N\) and \(d\geq d(L)\), then \(P_{L,N}(d)\geq\frac{53}{90}\). In particular \(P_{L,N}(d)\geq 0.8\), whenever \(L\not\in\{A_5,A_6,A_7,A_8,\text{PSL}(2,7),\text{PSL}(2,11),S_5,S_6\}\).