On the invariant E(G) for groups of odd order

Abstract

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf{E}(G)$ the smallest integer $t$ such that every sequence of $t$ elements in $G$ contains a product-one subsequence of length $|G|$. In 1961, Erdős, Ginzburg and Ziv proved that $\mathsf{E}(G)\le 2|G|-1$ for every finite solvable group $G$ and this result is well known as the Erdős-Ginzburg-Ziv Theorem. In 2010, Gao and Li improved this result to $\mathsf{E}(G)\le \frac{7|G|}{4}-1$ and they conjectured that $\mathsf{E}(G)\le \frac{3|G|}{2}$ holds for any finite non-cyclic group. In this paper, we confirm the conjecture for all finite non-cyclic groups of odd order.

14 pages, to appear in Acta Arithmetica