The probabilistic zeta function of \(\text{PSL}(2,q)\), of the Suzuki groups \(^2B_2(q)\) and of the Ree groups \(^2G_2(q)\).
type: Item
instance of: scholarly article
title: The probabilistic zeta function of \(\text{PSL}(2,q)\), of the Suzuki groups \(^2B_2(q)\) and of the Ree groups \(^2G_2(q)\).
zbMATH Open document ID: 1167.20013
author: Massimiliano Patassini
published in: Pacific Journal of Mathematics
publication date: 2009-04-08T00:00:00Z
review text: Let \(G\) be a finite group. There is a Dirichlet polynomial \(P_G(s)\) associated with \(G\), with the property that for all \(t\in\mathbb{N}\), \(P_G(t)\) is the probability that \(t\) random elements of \(G\) generate \(G\). It is defined as \(P_G(s)=\sum a_n(G)n^{-s}\) where \(a_n(G)=\sum_{|G:H|=n}\mu_G(H)\) (here \(\mu_G\) is the Möbius function of the subgroup lattice of \(G\), and \(H\) ranges over all subgroups). The reciprocal of \(P_G(s)\) is the probabilistic zeta function of \(G\). There are several open questions about the behavior of this Dirichlet polynomial. The author investigates these questions in the particular case when \(G\) is one of the following simple groups of Lie type: \(\text{PSL}(2,q)\), the Suzuki groups \(^2B_2(q)\) and the Ree groups \(^2G_2(q)\). For such \(G\) he proves that if \(H\) is a finite group satisfying \(P_G(s)=P_H(s),\) then \(H/\text{Frat\,}H\cong G\) (it was conjectured by \textit{E. Damian} and the reviewer [J. Algebra 313, No. 2, 957-971 (2007; Zbl 1127.20052)] that this holds for any finite simple group \(G\)). Moreover he shows that if \(P_G(s)\) is reducible in the ring of Dirichlet polynomials, then \(G\cong\text{PSL}(2,p)\) with \(p\) a Mersenne prime. Finally he proves that \(P_G(-1)\neq 0\) and this implies that the coset poset of \(G\) is non contractible, as conjectured by \textit{K. S. Brown} [J. Algebra 225, No. 2, 989-1012 (2000; Zbl 0973.20016)]. The author computes explicitly the Dirichlet polynomials for \(\text{PSL}(2,q)\) and this makes it possible to test directly certain properties one might wonder about. For example this is used to disprove the conjecture (due to the reviewer) that if \(G\) is a simple group, then \(|G|\) is the lowest common multiple of the \(n\in\mathbb{N}\) with \(a_n(G)\neq 0\).
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 5541991
zbMATH Keywords: probabilistic zeta functions
zbMATH Keywords: finite groups
zbMATH Keywords: Dirichlet polynomials
zbMATH Keywords: Möbius function
zbMATH Keywords: subgroup lattices
zbMATH Keywords: simple groups of Lie type
zbMATH Keywords: Suzuki groups
zbMATH Keywords: Ree groups
zbMATH Keywords: coset posets
MaRDI profile type: Publication
full work available at URL: pjm.2009.240.185
OpenAlex ID: W2052691508
DOI: 10.2140/PJM.2009.240.185
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description: scientific article; zbMATH DE number 5541991