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

A finiteness condition on the coefficients of the probabilistic zeta function