Abstract. 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.

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

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 log 2 (p + 1) ≡ 3 (mod 4).

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