On the probabilistic zeta function for finite groups

[摘要] Let G be a finite group, and define the function P(G,s) := (H less than or equal to G)Sigma mu(H,G)/[G;H](s), where mu is the Mobius function on the subgroup lattice of G. The function P(G, s) is the multiplicative inverse of a zeta function for G, as described by Mann and Boston. Boston conjectured that P'(G,I) = 0 if G is a nonabelian simple. We will prove a generalization of this conjecture, showing that P'(G,1)= 0 unless G/O-p(G) is cyclic for some prime p. (C) 1998 Academic Press.

[发布日期] 1998-12-15