Zeta functions of finite groups by enumerating subgroups

For a finite group G, we consider the zeta function ζ_G(s) = ∑_H ǀHǀ^-s, where H runs over the subgroups of G. First we give simple examples of abelian p-group G and non-abelian p-group G^′ of order p^m, m≥3 for odd p (resp. 2^m, m≥4) for which ζ_G(s) = ζG'(s). Hence we see there are many non-abelian groups whose zeta functions have symmetry and Euler product, like the case of abelian groups. On the other hand, we show that ζ_G(s) determines the isomorphism class of G within abelian groups, by estimating the number of subgroups of abelian p-groups. Finally we study the problem which abelian p-group is associated with a non-abelian group having the same zeta function.