ABSTRACT
For a finite group G, we consider the zeta function , 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 . 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.
2010 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgement
The author expresses her thanks to Hoshi who checked the groups of order 27 by using GAP. She is thankful also to the members of WINJ, since WINJ7 was a good opportunity to think about this theme.