The Probabilistic Zeta Function of Alternating and Symmetric Groups

The probability that a finite group G is generated by s elements is given by a truncated Dirichlet series in s, denoted by P(G,s). We give an explicit criterion that allows one to recognize whether the factor group G/Frat(G) is simple by only looking at the coefficients of P(G,s). In order to get such a criterion, we prove that the series derived from P(G,s) by removing the even-indexed terms has only a simple zero at s=1