The probabilistic zeta function of \(\text{PSL}(2,q)\), of the Suzuki groups \(^2B_2(q)\) and of the Ree groups \(^2G_2(q)\).
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 \(a_n(G)\neq 0\).
scientific article; zbMATH DE number 5541991