Zeta functions of finite groups by enumerating subgroups

Abstract

For a finite group $G$, we consider the zeta function $ζ_G(s) = \sum_{H} \abs{H}^{-s}$, where $H$ runs over the subgroups of $G$. First we give simple examples of abelian $p$-group $G$ and non-abelian $p$-group $G^{\prime }$ of order $p^m,m\ge 3$ for odd $p$ (resp. $2^m,m\ge 4$) for which ${\zeta }_G(s)={\zeta }_{G^{\prime }}(s)$. 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 ${\zeta }_G(s)$ 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.

16 pages To corrected some typos and enlarge the final remark

References in corpus (1)

• Orthogonal polynomials associated with root systems