Crowns and factorization of the probabilistic zeta function of a finite group.
Let \(G\) be a finite group and \(t\) a non-negative integer. Then let \(P_G(t)\) be the probability that \(t\) random elements generate \(G\). P. Hall proved that \(P_G(t)=\sum_{H\leq G}(\mu(H,G)/[H:G]^t)\), where \(\mu\) is the Möbius function of the subgroup lattice of \(G\). Hence \(P_G(t)\) can be exhibited as a finite Dirichlet series \(\sum_{n\in\mathbb{N}}a_nn^{-t}\) with \(a_n\in\mathbb{Z}\) and \(a_n=0\) unless \(n\) divides \(|G|\). So, in view of Hall's formula, we can speak of \(P_G(s)\) for an arbitrary complex number \(s\). The ring of finite Dirichlet series with coefficients in \(\mathbb{Z}\) has the property of unique factorization. Boston suggested that the study of the irreducible factors of \(P_G(s)\) could be useful for a better understanding of the properties of this function and for possible number-theoretic interpretation. These factors are called the generalized Euler factors. The authors employ the concept of crown to study the generalized Euler factors of the probabilistic zeta function of finite groups. It is obtained an expression of \(P_G(s)\) as a product indexed by the non-Frattini chief factors in the series. It is proved also, that if \(\lambda(G)\) is the number of non-Frattini chief factors in a chief series of any finite group \(G\) then for a given real number \(0<\alpha<1\) there exists a constant \(C_\alpha\) such that for any finite group \(G\) and any positive integer \(u\geq C_\alpha\max[1,\log(\lambda(G))]\) we have \(P_G(d(G)+u)\geq\alpha\), where \(d(G)\) is the smallest cardinality of a generating set of \(G\).
scientific article; zbMATH DE number 1963613
scientific article