On the irreducibility of the Dirichlet polynomial of an alternating group (Q2847028)

The Dirichlet polynomial of a finite group \(G\) is NEWLINE\[NEWLINE P_G(s) = \sum\nolimits_{H \leq G} \mu_G(H) | G : H |^{-s}, NEWLINE\]NEWLINE where \(\mu_G\) denotes the Möbius function of the subgroup lattice of~\(G\) and \(s\) is a complex variable. It is well-known that, for every non-negative integer~\(k\), the probability that \(k\) random elements of \(G\) generate \(G\) equals~\(P_G(k)\). With a view toward infinite groups and the classical Riemann zeta function, \(\zeta_G(s) = 1/P_G(s)\) is called the probabilistic zeta function of~\(G\); the study of such zeta functions was initiated by \textit{A. Mann} [Forum Math. 8, No. 4, 429--459 (1996; Zbl 0852.20019)].NEWLINENEWLINEThe ring \(\mathcal{R}\) of all Dirichlet polynomials over \(\mathbb{Z}\) is a factorial domain and, naturally, one takes interest in possible factorisations of \(P_G(s)\) in~\(\mathcal{R}\). Any chief series \(1 = G_0 \triangleleft \dots \triangleleft G_n = G\) yields a decomposition: NEWLINE\[NEWLINE P_G(s) = \prod_{i=0}^{k-1} P_{G/G_i,G_{i+1}/G_i}(s), \tag{\dag}NEWLINE\]NEWLINE where, for \(N \trianglelefteq G\), NEWLINE\[NEWLINE P_{G,N}(s) = \sum\nolimits_{H \in \mathcal{H}_{G,N}} \mu_G(H) | G : H |^{-s} \quad \text{with} \quad \mathcal{H}_{G,N} = \{H \leq G \mid HN = G \}. NEWLINE\]NEWLINE We remark that \(P_{G,N}(s) = 1\) if and only if \(N\) is contained in the Frattini subgroup~\(\Phi(G)\); consequently, it suffices to control the non-Frattini chief factors in the product decomposition~\((\dag)\). Moreover, if \(P_G(s)\) is irreducible in~\(\mathcal{R}\), then the Frattini quotient \(G/\Phi(G)\) is simple. On the other hand, even if \(G\) is simple, \(P_G(s)\) need not be irreducible.NEWLINENEWLINEIn the paper under review, the author establishes that the Dirichlet polynomials of alternating groups are irreducible, generalising partial results of \textit{E. Damian} et al. [Pac. J. Math. 215, No. 1, 3--14 (2004; Zbl 1113.20063)].NEWLINENEWLINE\smallskipNEWLINENEWLINE\noindent Theorem 1. The Dirichlet polynomial \(P_{\mathrm{Alt}(k)}(s)\) is irreducible in \(\mathcal{R}\) for \(5 \leq k \leq 4.2 \cdot 10^{16}\) and for \(k \geq (e^{e^{15}} + 2)^3\). Assuming the Riemann hypothesis, \(P_{\mathrm{Alt}(k)}(s)\) is irreducible in \(\mathcal{R}\) for all \(k \geq 5\).NEWLINENEWLINE\smallskipNEWLINENEWLINEThe factors in the product decomposition \((\dag)\) can be described in terms of Dirichlet polynomials of monolithic primitive groups; recall that a finite group \(G\) is monolithic primitive if it has a maximal subgroup \(M\) with trivial core \(\bigcap_{g \in G} M^g = 1\) and if its socle \(\mathrm{soc}(G)\) consists of one minimal normal subgroup (rather than a the product of two minimal normal subgroups).NEWLINENEWLINE\smallskipNEWLINENEWLINE\noindent Theorem 2. Let \(G\) be a monolithic primitive finite group with socle \(\mathrm{G} \cong \mathrm{Alt(k)}^n\), where \(k \geq 5\) and \(n \geq 1\). If \(k=6\), assume that \(N_G(S)/C_G(S)\) embeds naturally into \(\mathrm{Sym}(6)\), where \(S \cong \mathrm{Alt}(6)\) is a simple component of \(\mathrm{soc}(G)\). Then \(P_{G,\mathrm{soc}(G)}(s)\) is irreducible in \(\mathcal{R}\) for \(5 \leq k \leq 4.2 \cdot 10^{16}\) and for \(k \geq (e^{e^{15}} + 2)^3\). Assuming the Riemann hypothesis, \(P_{G,\mathrm{soc}(G)}(s)\) is irreducible in \(\mathcal{R}\) for all \(k \geq 5\).NEWLINENEWLINE\smallskipNEWLINENEWLINE\noindent Theorem 3. Let \(G\) and \(H\) be finite groups such that \(P_G(s) = P_H(s)\). If all composition factors of \(G\) and \(H\) are alternating groups satisfying the hypotheses in Theorem 2, then \(G\) and \(H\) have the same non-Frattini chief factors.NEWLINENEWLINE\smallskipNEWLINENEWLINEThe starting point for proving the irreducibility of a Dirichlet polynomial \(f(s)\) is a criterion described by Damian, et al. [loc. cit.]. The proof proceeds by an inductive argument, involving certain `admissible' pairs of integers related to the distribution of prime numbers (Conjecture~7). The explicit bounds for the degrees \(k\) of alternating groups in the theorems above arise from number theoretic considerations regarding gaps between consecutive primes and the occurrence of primes in certain intervals. Using the Riemann hypothesis, these bounds can be improved so that a direct computer-assisted verification (using GAP) becomes feasible.