The separating Noether number of small groups
M. Domokos, B. Schefler
TL;DR
The paper completes the determination of separating Noether numbers for all non-abelian groups with order less than $32$, under the standard base-field assumption that $K$ contains an element of order $|G|$ (with some entries requiring $ ext{char}(K)=0$). It develops and applies a framework based on relative invariants, multigradings, and Davenport-type sequences to bound and sometimes equal the separating Noether number $eta_{ m sep}^K(G)$ with the classical Noether number $eta^K(G)$, often by constructing explicit representations and orbit-separation witnesses. The results cover ten non-abelian groups in the range, including $(C_3 imes C_3) times_{-1}C_2$, $C_7 times C_3$, the binary tetrahedral group, and several semidirect products and direct products with $C_2$, presenting both upper bounds via Noether theory and sharp lower bounds via tailored points in representations. These findings extend prior work on separating invariants, provide explicit degrees needed to separate orbits in concrete representations, and have potential implications for effective computation and applications in invariant theory and related areas.
Abstract
The present paper completes the computation of the separating Noether numbers for the groups with order strictly less than $32$. Most of the results are proved for the case of a general (possibly finite) base field containing an element whose multiplicative order equals the size of the group.