For every nontrivial finite abelian group G, let mathsf d(G) denote the maximum length of a zero-sum-free sequence over G. Let nu(G) be the least integer ell such that every zero-sum-free sequence T over G with |T|≥ ell satisfies: the set of all nonzero elements of G that are not sums of nonempty subsequences of T is contained in a proper coset of a subgroup of G. Gao's conjecture asserts that nu(G)=mathsf d(G)-1 for every nontrivial finite abelian group G.

Gao’s conjecture on the invariant ν(G)

Gao’s conjecture asserts that the zero-sum invariant satisfies $\nu(G)=\mathsf{d}(G)-1$ for every nontrivial finite abelian group. Before the latest claim, the equality was known only for several substantial families, while the general question remained open.

Known results

Cyclic groups and $p$ -groups: $\nu(G)=\mathsf{d}(G)-1$ .
Groups of rank at most $2$ : the equality is established.
Further cases include $\nu(C_2^2\oplus C_{2n})=\mathsf{d}(C_2^2\oplus C_{2n})-1$ and, for odd $n>70$ , the analogous equality for $C_2^4\oplus C_{2n}$ .
No group was previously known with $\nu(G)=\mathsf{d}(G)$ ; the general conjecture was described as out of reach.

September 2026 claimed disproof

On September 15, 2026, a report linked Alfred Geroldinger, Guoqing Wang, and Wenkai Yang’s arXiv preprint On a classical zero-sum invariant II: Disproof of a long-standing conjecture, which claims a counterexample to Gao’s universal equality. The retrieved metadata does not specify the counterexample, and the preprint is unrefereed.

Current status (as of September 2026): Gao’s conjecture is subject to a claimed counterexample in an unrefereed preprint, so the universal equality is not yet independently verified as false; the established positive cases remain settled.

Gao’s conjecture on the invariant ν(G)

Primary source

Additional references

Progress summary

Solutions 0