The probability of generating finite and profinite groups

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 $|G| \to \infty$. In this paper, we generalize this theorem of Liebeck and Shalev. Work of Lucchini and Menegazzo (1997) implies that a finite group $G$ can be generated by a pair of elements if it has a unique chief series. As a consequence of our main theorem, the probability that a pair of elements generate such a group $G$ tends to $1$ as $|S| \to \infty$, where $S$ is the unique simple quotient of $G$. We also prove that a profinite group $G$ with finitely many chief series has a topological generating set of size $d < \infty$, and for any such $d$, the probability that a $d$-tuple of elements topologically generates $G$ is positive; moreover, we can take $d = 2$ if $G$ has a unique chief series. Along the way, we show that the chief factors of a finite group with a unique chief series are highly constrained, and we also analyze the maximal subgroup zeta function of a finite group with a unique minimal normal subgroup.