Some properties of the probabilistic zeta function of finite simple groups

Abstract

In the factorial ring of Dirichlet polynomials we explore the connections between how the Dirichlet polynomial P G (s) associated with a finite group G factorizes and the structure of G. If P G (s) is irreducible, then G/Frat G is simple. We investigate whether the converse is true, studying the factorization in the case of some simple groups. For any prime p ≥ 5 we show that if P G (s) = P Alt(p) (s), then G/Frat G ∼ = Alt(p) and P Alt(p) (s) is irreducible. Moreover, if P G (s) = P PSL(2,p) (s), then G/Frat G is simple, but P PSL(2,p) (s) is reducible whenever p = 2 t − 1 and t = 3 mod 4.

Recognizing the Alternating Groups From Their Probabilistic Zeta Function

“…The ring R of Dirichlet polynomials is a factorial domain and an important role in the factorization of P G (s) in R is played by the normal subgroups of G. We recall a result in this direction that has been employed already in [4] …”

“…In [4] we showed that if n is a prime number, then P Alt(n) (s) is irreducible and G/ Frat G Alt(n). Hence we shall assume that n is not a prime number.…”

“…In this paper we prove this conjecture when G 1 = Alt(n). The case of alternating groups of prime degree was considered in [4]; moreover it has been proved that the polynomial P Alt(n) (s) is irreducible when n is a prime number. It is still an open question whether this result holds for any n.…”

On the irreducibility of the dirichlet polynomial of a simple group of lie type

“…The above theorem was proved for G ∼ = A 1 (p), p prime, in [7] and for G ∼ = A 1 (t), 2 B 2 (t 2 ) and 2 G 2 (t 2 ) in [25]. Now, we turn to a more general setting.…”

“…Classical groups, class S S π S (S) (l, t) A l (t) t l+1 , t l (2, 4), (5, t), (7,2), (17,3), (19,2) and (3, 2 k …”