A finiteness condition on the coefficients of the probabilistic zeta function

TL;DR

This paper investigates how the coefficients of the probabilistic zeta function relate to the finiteness properties of profinite groups, establishing conditions under which the group has finitely many maximal subgroups.

Contribution

It proves that if the probabilistic zeta function is rational and most non-abelian composition factors are isomorphic to PSL(2,p), then the group has finitely many maximal subgroups.

Findings

Rationality of the probabilistic zeta function implies finiteness of maximal subgroups under certain conditions.

Most non-abelian composition factors being PSL(2,p) leads to finiteness results.

The paper links algebraic properties of the zeta function to group-theoretic finiteness properties.

Abstract

We discuss whether finiteness properties of a profinite group $G$ can be deduced from the coefficients of the probabilistic zeta function $P_G(s)$. In particular we prove that if $P_G(s)$ is rational and all but finitely many non abelian composition factors of $G$ are isomorphic to $PSL(2,p)$ for some prime $p$, then $G$ contains only finitely many maximal subgroups.