Disjoint zero-sum subsets in Abelian groups and its application -- survey

Abstract

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 $\phi$ $\Gamma$ such that the mapping $g\mapsto g^{-1}\phi (g)$ 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 subset we provide a solution of this conjecture for $k=3$ and $|\Gamma |\cong 4(\mathrm{mod}24)$. We also present some applications of zero-sum sets in graph labeling.