Recognizing the characteristic of a simple group of Lie type from its probabilistic zeta function.
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 \(a_n(G)=\sum_{|G:K|=n}\mu_G(K)\) (here \(\mu_G\) is the Möbius function of the subgroup lattice of \(G\), and \(K\) ranges over all subgroups). The reciprocal of \(P_G(s)\) is the probabilistic zeta function of \(G\). Since \(\mu_G\) vanishes outside the sublattice generated by maximal subgroups, it is clear that \(P_G(s)=P_{G/\text{Frat}(G)}(s)\) where \(\text{Frat}(G)\) denotes the Frattini subgroup of \(G\). \textit{E. Damian} and the reviewer [J. Algebra 313, No. 2, 957-971 (2007; Zbl 1127.20052)] proved that if \(P_G(s)=P_H(s)\) and \(H\) is a simple group, then \(G/\text{Frat}(G)\) is simple. They conjectured that a stronger result holds: if \(P_G(s)=P_H(s)\) and \(H\) is a simple group, then \(G/\text{Frat}(G)\cong H\). This conjecture is easy for \(H\) cyclic and was proved for \(H\) alternating and sporadic. In the case of groups of Lie type the following is known: let \(G_1\) and \(G_2\) be two groups of Lie type defined over fields with the same characteristic; if \(P_{G_1}(s)=P_{G_2}(s)\), then \(G_1\cong G_2\) [\textit{E. Damian, A. Lucchini}, Rend. Semin. Mat. Univ. Padova 115, 51-69 (2006; Zbl 1167.20334)]. In this paper the author completes the proof of the conjecture, finding a method to recognize the characteristic of a simple group of Lie type \(G\) from its Dirichlet polynomial \(P_G(s)\).
scientific article; zbMATH DE number 5969540
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