The Erd\H{o}s-Ginzburg-Ziv theorem constant 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.