We provide a summary of research on disjoint zero-sum subsets in finite Abelian groups, which is a branch of additive group theory and combinatorial number theory. An orthomorphism of a group \(\Gamma \) is defined as a bijection \(\varphi \) of \(\Gamma \) such that the mapping \(g \mapsto g^{-1}\varphi (g)\) is also bijective. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when \(\Gamma \) is Abelian, for any \(k \ge 2\) dividing \(|\Gamma | -1\) , there exists an orthomorphism of \(\Gamma \) fixing the identity and permuting the remaining elements as products of disjoint k-cycles. Using the idea of disjoint zero-sum subsets, we provide a solution of this conjecture for \(k=3\) and \(|\Gamma |\equiv 4\pmod {24}\) . We also present some applications of zero-sum sets in graph labeling.