Crowns and factorization of the probabilistic zeta function of a finite group.
type: Item
instance of: scholarly article
title: Crowns and factorization of the probabilistic zeta function of a finite group.
zbMATH Open document ID: 1072.20031
DOI: 10.1016/S0021-8693(03)00275-8
author: Eloisa Detomi
author: Andrea Lucchini
published in: Journal of Algebra
publication date: 2003-08-13T00:00:00Z
review text: 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 \(G\).
reviewed by: N. M. Kurnosenko
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 1963613
zbMATH Keywords: factorizations
zbMATH Keywords: probabilistic zeta functions
zbMATH Keywords: crowns
zbMATH Keywords: generalized Euler factors
zbMATH Keywords: subgroup lattices
zbMATH Keywords: random elements
zbMATH Keywords: chief factors
zbMATH Keywords: numbers of generators
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1963613