Davenport constant and its variants for some non-abelian groups

TL;DR

This paper introduces two new variants of the Davenport constant for non-abelian groups, computes these constants for specific groups, and explores their differences, providing insights and partial results related to a conjecture of J. Bass.

Contribution

It defines and computes new variants of the Davenport constant for non-abelian groups, highlighting differences from the abelian case and advancing understanding of related conjectures.

Findings

Constants can differ in non-abelian groups unlike abelian groups.

Computed Davenport constants for specific non-abelian groups.

Provided evidence and conjectures for metacyclic groups.

Abstract

We define two variants $e(G)$, $f(G)$ of the Davenport constant $d(G)$ of a finite group $G$, that is not necessarily abelian. These naturally arising constants aid in computing $d(G)$ and are of potential independent interest. We compute the constants $d(G)$, $e(G)$, $f(G)$ for some nonabelian groups G, and demonstrate that, unlike abelian groups where these constants are identical, they can each be distinct. As a byproduct of our results, we also obtain some cases of a conjecture of J. Bass. We compute the $k$-th Davenport constant for several classes of groups as well. We also make a conjecture on $f(G)$ for metacyclic groups and provide evidence towards it.