The Erdős--Ginzburg--Ziv theorem for finite groups

Source abstract

Let $G$ be a finite group, and let $p$ be the smallest prime divisor of $|G|$. The Gao constant $\mathsf{E}(G)$ is the least integer $\ell$ such that every sequence over $G$ of length $\ell$ contains a subsequence of length $|G|$ whose terms can be ordered to have product equal to the identity. In 1961, Erdős, Ginzburg and Ziv proved that every sequence of $2n-1$ integers has an $n$-term subsequence whose sum is divisible by $n$; equivalently, $\mathsf{E}(C_n)\leq 2n-1$, where $C_n$ denotes the cyclic group of order $n$. In this paper, we prove that $\mathsf{E}(G)\leq |G|+|G|/p-1$ whenever $G$ contains no cyclic subgroup of index $p$. Consequently, every finite non-cyclic group satisfies $\mathsf{E}(G)\leq |G|+|G|/p+p-2$, with equality if and only if $G$ contains a cyclic subgroup of index $p$. This establishes the conjectured upper bound for the Gao constant and provides a complete characterization of the equality case.