Rationality of the probabilistic zeta function of finitely generated profinite groups

TL;DR

This paper investigates how the probabilistic zeta function of a finitely generated profinite group relates to its structural properties, showing that rationality of this function implies finiteness of maximal subgroups under certain conditions.

Contribution

It establishes a link between the rationality of the probabilistic zeta function and the finiteness of maximal subgroups for groups with composition factors of Lie type.

Findings

Rational probabilistic zeta function implies finitely many maximal subgroups.

Finiteness properties can be deduced from the probabilistic zeta function.

Results apply to groups with composition factors of Lie type in a fixed characteristic.

Abstract

We discuss whether finiteness properties of a profinite group $G$ can be deduced from the probabilistic zeta function $P_G(s)$. In particular we prove that if $P_G(s)$ is rational and all but finitely many nonabelian composition factors of $G$ are groups of Lie type in a fixed characteristic, then $G$ contains only finitely many maximal subgroups.