The Gao-Zhuang conjecture for the Heisenberg group

Let $G$ be a finite nonabelian group. The small Davenport constant $\mathsf{d}(G)$ of $G$ is the largest integer $\mathrm{\ell }$ such that there exists a product-one-free sequence over $G$ of length $\mathrm{\ell }$, while the Gao constant $E(G)$ of $G$ is the least integer $\mathrm{\ell }$ such that every sequence over $G$ of length at least $\mathrm{\ell }$ contains a product-one subsequence of length exactly $\mathrm{\mid }G\mathrm{\mid }$. A long-standing conjecture of Zhuang and Gao \cite{ZG2005} asserts that $E(G)=\mathsf{d}(G)+\mathrm{\mid }G\mathrm{\mid }$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^3}={\mathrm{UT}}_3({\mathbb{F}}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. Godara and Sarkar proved the Zhuang–Gao equality for the nonabelian group of order $27$ and exponent $3$, and asked whether the same equality holds for $H_{p^3}$ for every odd prime $p$. Recently, Volkmann proved that $\mathsf{d}(H_{p^3})=3p-3$. In this paper, we determine the Gao constant of $H_{p^3}$ and prove that $E(H_{p^3})=\mathsf{d}(H_{p^3})+\mathrm{\mid }{H}_{p^3}\mathrm{\mid }=p^3+3p-3$.