The probabilistic zeta function of finite simple groups

Let G be a finite group; there exists a uniquely determined Dirichlet polynomial P-G(s) such that if t is an element of K then PG W gives the probability of generating G with t randomly chosen elements. We show that it may be recognized from the knowledge of PG (s) whether G/ Frat G is a simple group. (c) 2007 Elsevier Inc. All rights reserved.