Non-isomorphic metacyclic $p$-groups of split type with the same group zeta function

Abstract

For a finite group $G$, let $a_n(G)$ be the number of subgroups of order $n$ and define $\zeta_G(s)=\sum_{n\ge 1} a_n(G)n^{-s}$. Examples are known of non-isomorphic finite groups with the same group zeta function. However, no general criterion is known for when two finite groups have the same group zeta function. Fix integers $m,n\ge 1$ and a prime $p$, and consider the metacyclic $p$-groups of split type $G(p,m,n,k)$ defined by $ G(p,m,n,k)=\langle a,b \mid a^{p^{m}}=b^{p^{n}}=\mathrm{id}, b^{-1}ab=a^{k}\rangle$. For fixed $m$ and $n$, we characterize the pairs of parameters $k_1,k_2$ for which $\zeta_{G(p,m,n,k_1)}(s)=\zeta_{G(p,m,n,k_2)}(s)$.