Zhuang–Gao conjecture for finite groups

Establish that the Gao constant of every finite group satisfies E(G) = d(G) + |G|.

Background

The paper also studies the Gao constant E(G) and the small Davenport constant d(G). The general equality E(G) = d(G) + |G| is identified as the Zhuang–Gao conjecture; the paper notes that this conjecture has been proved for abelian groups. The quoted statement concerns the broader class of all finite groups, while the paper establishes the equality for its particular family of non-abelian groups.

References

In [26], Zhuang and Gao conjectured that E(G) = d(G) + |G| for every finite group.

— Some zero-sum problems over $\langle x,y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle$  (2501.03338 - Ribas, 6 Jan 2025) in Section 2, immediately before Section 3

For finite abelian groups, Gao proved the fundamental identity $E(G)=\mathsf d(G)+|G|$. Motivated by this result, Zhuang and Gao conjectured that the same equality holds for every finite group. In contrast with the abelian case, the nonabelian problem involves both the selection of a subsequence and the ordering of its terms, and the conjecture is known only for particular classes of groups.

— The Gao-Zhuang conjecture for the Heisenberg group  (2608.23319 - Qu et al., 24 Aug 2026) in Section 1, Introduction