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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.22245 |
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| _version_ | 1866909370719141888 |
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| author | Cichacz, Sylwia |
| author_facet | Cichacz, Sylwia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Disjoint zero-sum subsets in Abelian groups and its application -- survey Cichacz, Sylwia Combinatorics 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. |
| title | Disjoint zero-sum subsets in Abelian groups and its application -- survey |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.22245 |