Abstract

. We investigate the topological properties of the poset of proper cosets xH in a finite group G. Of particular interest is the reduced Euler characteristic, which is closely related to the value at \Gamma1 of the probabilistic zeta function of G. Our main result gives divisibility properties of this reduced Euler characteristic. 1. Introduction For a finite group G and a non-negative integer s, let P (G; s) be the probability that a randomly chosen ordered s-tuple from G generates G. Philip Hall [16] gave an explicit formula for P (G; s), exhibiting the latter as a finite Dirichlet series P n ann \Gammas , with an 2 Z and an = 0 unless n divides jGj. For example, P (A 5 ; s) = 1 \Gamma 5 5 s \Gamma 6 6 s \Gamma 10 10 s + 20 20 s + 60 30 s \Gamma 60 60 s : In view of Hall's formula, we can speak of P (G; s) for an arbitrary complex number s. The reciprocal of this function of s is sometimes called the zeta function of G; see [5, 19]. The present paper arose fr...

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