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 $Γ$ is defined as a bijection $φ$ $Γ$ such that the mapping $g \mapsto g^{-1}φ(g)$ is also bijective. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when $Γ$ is Abelian, for any $k \geq 2$ dividing $|Γ| -1$, there exists an orthomorphism of $Γ$ 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 $|Γ|\cong 4\pmod{24}$. We also present some applications of zero-sum sets in graph labeling.