Zhuang–Gao conjecture

Zhuang and Gao conjectured in 2005 that every finite group satisfies $E(G)=d(G)+|G|$, where $E(G)$ is the Gao constant and $d(G)$ is the small Davenport constant.

Known results

• The equality is known for abelian groups and several nonabelian families, including dihedral, dicyclic, and selected semidirect-product groups.
• A 2021 paper proves it for the metacyclic family $G=\langle x,y\mid x^p=y^m=1,\ x^{-1}yx=y^r\rangle$, under stated arithmetic hypotheses, with $d(G)=m+p-2$.
• Godara and Sarkar established the case of the nonabelian group of order $27$ and exponent $3$ in 2023.

August 2026 Heisenberg-group result

An August 2026 paper, The Gao-Zhuang conjecture for the Heisenberg group, reports the full identity $E(H_{p^3})=d(H_{p^3})+|H_{p^3}|$ for $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$, an infinite family of nonabelian $p$-groups. This is a family-specific advance, not a universal resolution.

Current status (as of August 2026): The identity is claimed for all Heisenberg groups $H_{p^3}$, while the universal finite-group conjecture remains open.