Zeta functions of finite groups by enumerating subgroups

ABSTRACT

For a finite group G, we consider the zeta function ${\zeta }_G\left(s\right)={\sum }_H{\left|H\right|}^{-s}$, 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 ${\zeta }_G\left(s\right)={\zeta }_{G^{\prime }}\left(s\right)$. 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.

KEYWORDS:

• Enumerating subgroups of abelian p-groups
• enumerating subgroups of finite groups
• local densities of square matrices
• zeta functions of finite groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20E07 (Primary)
• 11M41
• 20K01

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.