Separating versus ordinary Noether numbers

TL;DR

This paper investigates the relationship between separating and ordinary Noether numbers for finite groups, identifying conditions where they coincide or differ, and computes exact values for small groups.

Contribution

It establishes conditions under which separating and ordinary Noether numbers are equal and determines exact values for groups of order up to 16.

Findings

Separating and ordinary Noether numbers coincide for groups with a cyclic subgroup of index at most 2.

For certain non-abelian groups, the separating Noether number is strictly less than the ordinary Noether number.

Exact values of separating Noether numbers are computed for all groups of order up to 16.

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.