Extremal product-one free sequences and |G|-product-one free sequences of a metacyclic group

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 abelian 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| for every finite group, where d(G) is the small Davenport constant. Very recently, we confirmed this conjecture for the case when G=〈x,y|xp=ym=1,x−1yx=yr〉 where p is the smallest prime divisor of |G| and gcd(p(r−1),m)=1. In this paper, we study the associated inverse problems on d(G) and E(G). Our main results characterize the structure of any product-one free sequence with extremal length d(G), and that of any |G|-product-one free sequence with extremal length E(G)−1.