Gao’s conjecture for finite groups

Establish that the Erdős–Ginzburg–Ziv constant of every finite group satisfies s(G) = n(G) + exp(G) − 1.

Background

The paper studies the short product-one constant n(G), the Erdős–Ginzburg–Ziv constant s(G), and related zero-sum invariants for a family of finite non-abelian groups with a cyclic subgroup of index two. The general equality s(G) = n(G) + exp(G) − 1 is identified as Gao’s conjecture. The paper proves this equality for the specific groups under consideration, but the quoted statement formulates the conjecture for every finite group.

References

In [13], Gao conjectured that s(G) = n(G) + exp(G) - 1 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