The equality $s(G)=η(G)+\exp(G)-1$ fails for $G=C_2^7\oplus C_6$

Abstract

Let $G$ be a finite abelian group. The Erdős-Ginzburg-Ziv constant $\mathsf{s}(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a zero-sum subsequence of length $\exp(G)$. The short zero-sum constant $η(G)$ is defined similarly, with the requirement that the zero-sum subsequence is nonempty and has length at most $\exp(G)$. It is known that $\mathsf{s}(G)\ge η(G)+\exp(G)-1.$ A conjecture proposed by W. Gao in 2003, which has been widely studied in zero-sum theory, predicts that $$ \mathsf{s}(G)=η(G)+\exp(G)-1 $$ for every nontrivial finite abelian group $G$. In this paper, we prove that this conjecture fails for the group $C_2^7\oplus C_6$.