Non-isomorphic metacyclic $p$-groups of split type with the same group zeta function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912797503258624 |
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| author | Nogata, Yuto |
| author_facet | Nogata, Yuto |
| contents | For a finite group $G$, let $a_n(G)$ be the number of subgroups of order $n$ and define $ζ_G(s)=\sum_{n\ge 1} a_n(G)n^{-s}$. 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 $ζ_{G(p,m,n,k_1)}(s)=ζ_{G(p,m,n,k_2)}(s)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24546 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-isomorphic metacyclic $p$-groups of split type with the same group zeta function Nogata, Yuto Group Theory 20D15, 20E07 For a finite group $G$, let $a_n(G)$ be the number of subgroups of order $n$ and define $ζ_G(s)=\sum_{n\ge 1} a_n(G)n^{-s}$. 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 $ζ_{G(p,m,n,k_1)}(s)=ζ_{G(p,m,n,k_2)}(s)$. |
| title | Non-isomorphic metacyclic $p$-groups of split type with the same group zeta function |
| topic | Group Theory 20D15, 20E07 |
| url | https://arxiv.org/abs/2512.24546 |