An upper bound for Davenport constant of finite groups

TL;DR

This paper establishes an upper bound for the Davenport constant of finite groups, linking it to the group's order and smallest prime divisor, advancing understanding of zero-sum problems in group theory.

Contribution

It provides a new upper bound for the Davenport constant of finite groups, applicable to non-abelian groups, which was previously less understood.

Findings

Proves that d(G) ≤ |G|/p + 9p^2 - 10p for finite groups G.

Extends bounds on Davenport constants beyond abelian groups.

Offers insights into zero-sum sequences in non-abelian group contexts.

Abstract

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)\leq \frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as the maximal integer $\ell$ such that there is a sequence over $G$ of length $\ell$ contains no nonempty one-product subsequence.