Separating versus ordinary Noether numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911280061743104 |
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| author | Domokos, Mátyás Schefler, Barna |
| author_facet | Domokos, Mátyás Schefler, Barna |
| contents | Let $G$ be a finite group and $K$ a field containing an element of multiplicative order $|G|$. It is shown that if $G$ has a cyclic subgroup of index at most $2$, then the separating Noether number over $K$ of $G$ coincides with the Noether number over $K$ of $G$. The same conclusion holds when $G$ is the direct product of a dihedral group and the $2$-element group. On the other hand, the smallest non-abelian groups $G$ are found for which the separating Noether number over $K$ is strictly less than the Noether number over $K$. Along the way the exact value of the separating Noether number is determined for all groups of order at most $16$. The results show in particular that unlike the ordinary Noether number, the separating Noether number of a non-abelian finite group may well be equal to the separating Noether number of a proper direct factor of the group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Separating versus ordinary Noether numbers Domokos, Mátyás Schefler, Barna Commutative Algebra Group Theory Representation Theory Primary 13A50, Secondary 13P15, 20C15 Let $G$ be a finite group and $K$ a field containing an element of multiplicative order $|G|$. It is shown that if $G$ has a cyclic subgroup of index at most $2$, then the separating Noether number over $K$ of $G$ coincides with the Noether number over $K$ of $G$. The same conclusion holds when $G$ is the direct product of a dihedral group and the $2$-element group. On the other hand, the smallest non-abelian groups $G$ are found for which the separating Noether number over $K$ is strictly less than the Noether number over $K$. Along the way the exact value of the separating Noether number is determined for all groups of order at most $16$. The results show in particular that unlike the ordinary Noether number, the separating Noether number of a non-abelian finite group may well be equal to the separating Noether number of a proper direct factor of the group. |
| title | Separating versus ordinary Noether numbers |
| topic | Commutative Algebra Group Theory Representation Theory Primary 13A50, Secondary 13P15, 20C15 |
| url | https://arxiv.org/abs/2511.17719 |