Nonabelian Erdős–Ginzburg–Ziv conjecture

For every finite nonabelian group $G$ , let $\exp(G)$ be the exponent of $G$ . Define $s(G)$ to be the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a product-one subsequence of length exactly $\exp(G)$ , and define $\eta(G)$ to be the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty product-one subsequence of length at most $\exp(G)$ . Here a subsequence $g_1,\ldots,g_k$ is product-one if its terms can be ordered so that $g_{\pi(1)}\cdots g_{\pi(k)}=1_G$ . The conjecture asserts that $s(G)=\eta(G)+\exp(G)-1$ .

Progress summary

Partially solved

A new preprint proves the conjectured equality for a broad but restricted family of nonabelian groups, while the full conjecture remains open.

Gao and Li formulated the conjecture in 2010: for every finite noncyclic group $G$ , the Erdős–Ginzburg–Ziv constant should satisfy $\mathsf{E}(G)\leq 3|G|/2$ . The general finite nonabelian case is not settled.

Known results

Gao, Li, and Qu (2023) proved the conjectured bound for all groups of odd order.
Earlier work handled additional special families, including groups of order $2p^t$ under an abelian-subgroup hypothesis.
For finite noncyclic groups with $4\nmid |G|$ , $\mathsf{E}(G)\leq 3|G|/2$ , with equality exactly when $G$ has a cyclic subgroup of index $2$ .

August 16, 2026 restricted-family proof

Qu, Wang, and Li prove the equality $s(G)=\eta(G)+\exp(G)-1=\mathsf{d}(G)+\exp(G)$ when $G$ has a cyclic subgroup $H$ of index $p$ , with $p$ the smallest prime divisor of $|G|$ . They also determine $s_{m\exp(G)}(G)=\eta(G)+m\exp(G)-1$ for every $m\geq1$ . This advances the conjecture substantially but leaves other finite nonabelian groups open.

Current status (as of August 2026): A substantial restricted family is settled by a new preprint, but the conjecture for all finite nonabelian groups remains open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

The equality between the Erdős-Ginzburg-Ziv constant and the short product-one constant for finite nonabelian groups — arXiv — Yongke Qu, Guoqing Wang, Yuanlin Li

additive-combinatorics group-theory

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