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Given a finite group G, the Dirichlet polynomial of G is The multiplicative counterpart of this polynomial is called the Probabilistic Zeta function of the group G. We prove that if H is a finite group such that P G (s) = P H (s), then G and H have the same non-Frattini abelian chief factors.

Let G be a classical group and suppose that G does not contains non-trivial graph automorphisms. In this paper we prove that the order complex of the coset poset of G is non-contractible. In order to prove it, we show that PG(−1) does not vanish, where PG(s) is the Dirichlet polynomial associated to the group G.

Given a finite group G and a normal subgroup N of G, the Dirichlet polynomial of G given G/N is $${P_{G,N}}(s) = \sum\limits_{\scriptstyle H \le G \hfill \atop \scriptstyle NH = G \hfill} {{{{\mu _G}(H)} \over {|G:H{|^s}}}} .$$ In this paper, we assume that G is a primitive monolithic group with nonabelian socle soc(G) ≅ S n for some simple group S of Lie type. Under some assumptions on the Lie rank of S, we prove that P G,soc(G)(s) is irreducible in the ring of finite Dirichlet series. Moreover, we show that the Dirichlet polynomial P S (s) = P S,S (s) of a simple group S of Lie type is reducible if and only if S is isomorphic to A 1(p), where p is a Mersenne prime such that log2(p + 1) ≡ 3 (mod 4).

We find a method to recognize the characteristic of a simple group of Lie type G from its Dirichlet polynomial P G ( s ) . This is enough to complete the proof of the following statement: if G is a simple group, H is a finite group and P G ( s ) = P H ( s ) , then H / Frat ( H ) ≅ G .

We study the Dirichlet polynomial PG(s) of the groups G = PSL(2,q), 2 B2(q), and 2G2(q). For such G we show that if H is a group satisfying PH(s) = PG(s), then H / Frat(H)?G. We also prove that, when q is not a prime number, PG(s) is irreducible in the ring of Dirichlet polynomials. Finally, we prove that the coset poset of G is noncontractible.

We prove several results detecting cyclicity or nilpotency of a finite group G in terms of inequalities involving the orders of the elements of G and the orders of the elements of the cyclic group of order |G|. We prove that, among the groups of the same order, the number of cyclic subgroups is minimal for the cyclic group, and the product of the orders of the elements is maximal for the cyclic group.