On a conjecture of Zhuang and Gao

Let G be a multiplicatively written finite group. We denote by E(G) the smallest integer t such that every sequence of t elements in G contains a product-one subsequence of length |G|. In 1961, Erdős, Ginzburg and Ziv proved that E(G) ≤ 2|G|−1 for every finite ablian group G and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that E(G) = d(G) + |G|, where d(G) is the small Davenport constant. In this paper, we confirm the conjecture for the case when G = 〈x, y|x = y = 1, xyx = y〉, where p is the smallest prime divisor of |G| and gcd(p(r − 1),m) = 1.