Separating versus ordinary Noether numbers
Mátyás Domokos, Barna Schefler
TL;DR
This work investigates the separating Noether number $\beta_{ ext{sep}}^K(G)$ for finite groups in the non-modular setting, comparing it to the ordinary Noether number $\beta^K(G)$. It proves exact values for groups with a cyclic subgroup of index two and for $\mathrm{D}_{2n}\times \mathrm{C}_2$, and determines $\beta_{ ext{sep}}^K(G)$ for all groups of order at most $16$, revealing that $\beta_{ ext{sep}}^K(G)$ can coincide with that of a proper direct factor in non-abelian cases. The paper also identifies the smallest non-abelian groups where $\beta_{ ext{sep}}^K(G)<\beta^K(G)$ and develops methods to reduce to multiplicity-free representations using Helly-dimension concepts and automorphism techniques. A detailed discussion of base-field dependence, monotonicity failures, and methodological insights is provided, along with an outline of a computational framework for estimating $\beta_{ ext{sep}}^K(G)$ in practice. Overall, the results clarify the nuanced relationship between separating invariants and generators across concrete finite groups and lay groundwork for further exploration of $\beta_{ ext{sep}}^K(G)$ in broader classes.
Abstract
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.