Combinatorial invariants for certain classes of non-abelian groups

TL;DR

The paper investigates zero-sum invariants in finite groups, focusing on the ordered Davenport constant $\mathsf{D}_o(G)$, the small Davenport constant $\mathsf{d}(G)$, and the Gao constant $\mathsf{E}(G)$, and their links to the Noether number $\beta(G)$ and Loewy length $\mathsf{L}(G)$. It establishes a tight link between $\mathsf{D}_o(G)$ and $\mathsf{d}(G)$ for even-order non-abelian groups arising from semidirect products $G=A \rtimes_{-1} C_2$, showing $\mathsf{D}_o(G)=\mathsf{d}(A)+2$ and $\mathsf{D}_o(G)=\beta(G)$. The authors verify Gao and Li's conjecture for groups of order $2p^\alpha$, derive bounds on $\mathsf{E}(G)$ in this setting, and relate $\mathsf{D}_o(G)$ to $\mathsf{L}(G)$ for selected $2$-groups $G_1$–$G_4$, with explicit formulas for $\mathsf{L}(G_i)$. They further prove that if $G$ has a cyclic subgroup of index $p$, then $\mathsf{d}(G)+1=\mathsf{D}_o(G)=\mathsf{L}(G)=|G|/p+p-1$. Overall, the work deepens the understanding of zero-sum phenomena in non-abelian groups and connects combinatorial invariants to invariant theory and group structure.

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 $2p^α$, 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.

Theorems & Definitions (31)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Proposition 2.1