# The Noether numbers and the Davenport constants of the groups of order less than 32

## TL;DR

This paper completes the computation of Noether numbers for all groups under 32 elements, revealing their properties and relationships with Davenport constants, and introduces algorithms for these calculations.

## Contribution

It provides the first complete calculation of Noether numbers for small groups and explores their relation to Davenport constants, answering a longstanding question.

## Key findings

- Noether number is attained on a multiplicity free representation.
- Noether number is strictly monotone on subgroups and factor groups.
- Identifies a group where Noether number exceeds the large Davenport constant.

## Abstract

The computation of the Noether numbers of all groups of order less than thirty-two is completed. It turns out that for these groups in non-modular characteristic the Noether number is attained on a multiplicity free representation, it is strictly monotone on subgroups and factor groups, and it does not depend on the characteristic. Algorithms are developed and used to determine the small and large Davenport constants of these groups. For each of these groups the Noether number is greater than the small Davenport constant, whereas the first example of a group whose Noether number exceeds the large Davenport constant is found, answering partially a question posed by Geroldinger and Grynkiewicz.