On the existence of zero-sum subsequences of distinct lengths over certain groups of rank three

Let G be an additive finite abelian group. Denote by disc(G) the smallest positive integer t such that every sequence S over G of length \(|S|\geq t\) $|S|\ge t$ has two nonempty zero-sum subsequences of distinct lengths. In this paper, we focus on the direct and inverse problems associated with disc(G) for certain groups of rank three. Explicitly, we first determine the exact value of disc(G) for \(G\cong C_2\oplus C_{n_1}\oplus C_{n_2}\) $G\cong C_2\oplus C_{n_1}\oplus C_{n_2}$ with \(2\mid n_1\mid n_2\) $2\mid n_1\mid n_2$ and \(G\cong C_3\oplus C_{6n_3}\oplus C_{6n_3}\) $G\cong C_3\oplus C_{6n_3}\oplus C_{6n_3}$ with \(n_3\geq 1\) $n_3\ge 1$ . Then we investigate the inverse problem. Let \(\mathcal {L}_1(G)\) ${\mathcal{L}}_1(G)$ denote the set of all positive integers t satisfying that there is a sequence S over G of length \(|S|=\operatorname{disc}(G)-1\) $|S|=disc(G)-1$ such that every nonempty zero-sum subsequence of S has the same length t. We determine \(\mathcal {L}_1(G)\) ${\mathcal{L}}_1(G)$ completely for certain groups of rank three.