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arXiv · 2410.22245

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

Sylwia Cichacz
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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 $\varphi$ $\Gamma$ 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 $k \geq 2$ 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 $|\Gamma|\cong 4\pmod{24}$. We also present some applications of zero-sum sets in graph labeling.

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Sylwia Cichacz. 2024-10-29. Disjoint zero-sum subsets in Abelian groups and its application -- survey. https://arxiv.org/abs/2410.22245

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