The equality between the Erd\H{o}s-Ginzburg-Ziv constant and the short product-one constant for finite nonabelian groups

Abstract

Let $G$ be a finite group, and let $\exp(G)$ denote its exponent. The Erd\H{o}s-Ginzburg-Ziv constant $s(G)$ is the least integer forcing a product-one subsequence of length $\exp(G)$, while the short product-one constant $\eta(G)$ is the least integer forcing a nonempty product-one subsequence of length at most $\exp(G)$. The natural nonabelian extension of a conjecture [W. Gao, \emph{On zero-sum subsequences of restricted size II}, Discrete Math. 2003] on the Erd\H{o}s-Ginzburg-Ziv constant in finite abelian groups predicts that $s(G)=\eta(G)+\exp(G)-1.$ We confirm this equality for every finite nonabelian group $G$ having a cyclic subgroup of index $p$, where $p$ is the smallest prime divisor of $|G|$. As further consequences, we determine all generalized Erd\H{o}s-Ginzburg-Ziv constants $s_{m\exp(G)}(G)$ for this family of groups.