The probability of generating a uniserial group
Scott Harper, Martyn Quick
TL;DR
The paper broadens the scope of probabilistic generation from finite simple groups to finite uniserial groups, showing that the pair-generation probability $P(G)$ tends to $1$ when the unique simple quotient $S$ grows, and more generally establishing a quantitative bound $P_d(G,N)$ that approaches $1$ as $|N|$ grows. The authors develop a refined maximal-subgroup zeta function framework, $\\zeta_{G,N}(s)$, and prove sharp bounds using the auxiliary function $\alpha(T)$ defined on finite simple groups, enabling precise decay control across abelian and nonabelian minimal normal subgroups and across almost simple reductions. A key technical advance is the width analysis of chief factors in uniserial groups, providing lower bounds on the multiplicities of non-Frattini factors and showing that widths typically increase, with explicit exceptions explained via representation-theoretic arguments. These results yield positive topological generation probabilities for profinite uniserial groups and yield a broad class of constructions (e.g., affine groups and iterated wreath products) that preserve uniseriality, thus extending probabilistic generation phenomena to new algebraic settings with potential implications for random generation in dense infinite groups. The work combines probabilistic group theory, representation theory, and detailed subgroup structure analysis to produce a cohesive theory of generation in uniserial and profinite contexts, with concrete bounds and constructions.
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.