For a finite group G, we consider the zeta function ζG(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 pm, m≥3 for odd p (resp. 2m, m≥4) for which ζG(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 ζG(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.
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