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.