Combinatorial invariants for certain classes of non-abelian groups

TL;DR

This paper investigates zero-sum invariants in finite non-abelian groups, establishing new relations among invariants like the ordered Davenport constant, Gao's constant, and the Noether number, and confirming key conjectures.

Contribution

It introduces new connections between various zero-sum invariants in non-abelian groups and proves conjectures for specific group classes.

Findings

Relation between ordered Davenport constant and small Davenport constant

Confirmation of Gao and Li's conjecture for groups of order 2p^α

Connection between ordered Davenport constant and Loewy length in certain 2-groups

Abstract

This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the ordered Davenport constant and the small Davenport constant for a finite non-abelian group of even order, which in turn gives a relation with the Noether number. Additionally, we confirm a conjecture of Gao and Li for a non-abelian group of order $2p^{\alpha}$, where $p$ is a prime. Furthermore, we prove a conjecture that connects the ordered Davenport constant to the Loewy length for certain classes of finite $2$-groups.