The probability of generating a uniserial group

Abstract

Famously, every finite simple group $G$ can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate $G$ tends to $1$ as $|G| \to \infty$. More generally, work of Lucchini and Menegazzo (1997) implies that $G$ can be generated by a pair of elements whenever $G$ has a unique chief series. In this paper, we generalize the theorem of Liebeck and Shalev by proving that if $G$ has a unique chief series and the unique simple quotient of $G$ is $S$, then the probability that a pair of elements generate $G$ tends to $1$ as $|S| \to \infty$. As a consequence of our main theorem, for any profinite group $G$ where the open normal subgroups form a chain, the probability that a pair of elements topologically generate $G$ is positive. Along the way, we establish results on the maximal subgroup zeta function of groups with a unique minimal normal subgroup.