The Erd\H{o}s-Ginzburg-Ziv theorem constant of finite groups

TL;DR

This paper investigates the Erdős-Ginzburg-Ziv constant for finite groups, confirming a conjecture for certain non-cyclic groups and characterizing those that attain the conjectured bound.

Contribution

It proves Gao and Li's conjecture for non-cyclic groups with orders not divisible by four and characterizes groups where the constant reaches the bound.

Findings

Confirmed the conjecture for non-cyclic groups with order not divisible by four.

Characterized groups with the Erdős-Ginzburg-Ziv constant equal to 1.5 times their order.

Extended understanding of the Erdős-Ginzburg-Ziv theorem to broader classes of finite groups.

Abstract

Let $G$ be a multiplicatively written finite group of order $n$. The Erd\H{o}s-Ginzburg-Ziv Theorem constant of the group $G$, denoted $\mathsf E(G)$, is defined as the smallest positive integer $\ell$ with the following property: for any given sequence $(g_1,\ldots,g_{\ell})$ over $G$, there exist $n$ distinct integers $i_1,\ldots,i_n\in \{1,\ldots,\ell\}$ such that the product of $g_{i_1},\ldots,g_{i_n}$, in some order, is the identity element of $G$. The Erd\H{o}s-Ginzburg-Ziv Theorem constant originates from the celebrated additive theorem proved by Erd\H{o}s, Ginzburg and Ziv in 1961, which amounts to proving $\mathsf E(G)\leq 2|G|-1$ holds in case that $G$ is abelian. It is also well-known that $\mathsf E(G)=2|G|-1$ holds for all finite cyclic groups. In 2010, Gao and Li [J. Pure Appl. Algebra] conjectured that $\mathsf E(G)\leq \frac{3|G|}{2}$ for every finite non-cyclic group $G$. In this paper, we confirm the conjecture for all non-cyclic groups $G$ whose order is not divisible by four, and characterize the groups achieving the equality $\mathsf E(G)=\frac{3|G|}{2}$ as those with a cyclic subgroup of index two.