QUARTERLY JOURNAL OF MATHEMATICS, vol.61, no.1, pp.43-68, 2010 (SCI-Expanded)
In this paper, we answer this question completely. We show that zeta(G) is non-zero if and only if G is an elementary abelian p-group or G is isomorphic to Z/9, Z/4 x Z/4 or (Z/2)(n) x Z/4 for some integer n >= 0. We obtain this result by first showing that zeta(G) is zero when G is a non-abelian group, then by calculating zeta(G) for specific abelian groups. The key ingredient in the proof is an observation by Mandell which says that the Ext class of the subset complex delta (G) is equal to the (twisted) Euler class of the augmentation module of the regular representation of G.