The probability of generating a uniserial group

TL;DR

This paper extends known results on the probability of generating finite groups by pairs of elements, focusing on groups with a unique chief series and their asymptotic behavior as the size of simple quotients grows.

Contribution

It generalizes the theorem of Liebeck and Shalev to groups with a unique chief series and a specified simple quotient, analyzing the probability of generation as the simple quotient size increases.

Findings

Probability tends to 1 for groups with a unique chief series as simple quotient size grows.

Established results on the maximal subgroup zeta function for groups with a unique minimal normal subgroup.

Proved positive probability of topological generation in certain 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 $|G| \to \infty$. 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.