Zeta functions of finite groups by enumerating subgroups

TL;DR

This paper investigates the zeta functions of finite groups, showing that they can be identical for different groups, but also that they uniquely identify abelian groups, and explores the relationship between abelian and non-abelian groups with the same zeta function.

Contribution

It demonstrates that non-abelian groups can share zeta functions with abelian groups, yet zeta functions uniquely determine abelian groups, and examines the association between abelian and non-abelian groups with identical zeta functions.

Findings

Non-abelian groups can have the same zeta function as abelian groups.

Zeta functions of abelian groups uniquely determine their isomorphism class.

Certain non-abelian groups share zeta functions with specific abelian groups.

Abstract

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