The Gao-Zhuang conjecture for the Heisenberg group

Abstract: Let $G$ be a finite nonabelian group. The small Davenport constant $\mathsf d(G)$ of $G$ is the largest integer $\ell$ such that there exists a product-one-free sequence over $G$ of length $\ell$, while the Gao constant $E(G)$ of $G$ is the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a product-one subsequence of length exactly $|G|$. A long-standing conjecture of Zhuang and Gao \cite{ZG2005} asserts that $E(G)=\mathsf d(G)+|G|$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^{3}=\operatorname{UT}_3(\mathbb} F_p)$ be the Heisenberg group of order $p<sup>3$ 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 $H_{p<sup>3}$ for every odd prime $p$. Recently, Volkmann proved that $\mathsf d(H_{p<sup>3})=3p-3$. In this paper, we determine the Gao constant of $H_{p<sup>3}$ and prove that $$E(H_{p^3})=\mathsf d(H_{p<sup>3})+|H_{p<sup>3}|=p<sup>3+3p-3$$.