Note on a magic rectangle set on dihedral group
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
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| _version_ | 1866916009317761024 |
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| author | Cichacz, Sylwia |
| author_facet | Cichacz, Sylwia |
| contents | Let $Γ$ be a group of order $mnk$ and $MRS_Γ(m,n;k)=(a_{i,j}^s)_{m\times n}$ be a collection of $k$ arrays $m\times n$ whose entries are all distinct elements of $Γ$. If there exist elements $ρ,σ\inΓ$ such that for every row $i$, there exists an ordering of elements such that
$$
a_{i,j_1}^s a_{i,j_2}^s \dots a_{i,j_{n-1}}^s a_{i,j_n}^s= ρ
$$
and for every column $j$ there exists an ordering of elements such that
$$
a_{i_1,j}^s a_{i_2,j}^s \dots a_{i_{m-1},j}^s a_{i_m,j}^s = σ,
$$
then $MRS_Γ(m,n;k)$ is called a \emph{$Γ$-magic rectangle set}.
We investigate magic rectangle sets over dihedral groups and prove that $\mathrm{MRS}_Γ(m,n;k)$ exists for every dihedral group $Γ$ of order $mnk$, provided that $m$ and $n$ are even. As a consequence, we obtain broad existence results for magic rectangles and magic squares over dihedral groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13393 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Note on a magic rectangle set on dihedral group Cichacz, Sylwia Combinatorics Let $Γ$ be a group of order $mnk$ and $MRS_Γ(m,n;k)=(a_{i,j}^s)_{m\times n}$ be a collection of $k$ arrays $m\times n$ whose entries are all distinct elements of $Γ$. If there exist elements $ρ,σ\inΓ$ such that for every row $i$, there exists an ordering of elements such that $$ a_{i,j_1}^s a_{i,j_2}^s \dots a_{i,j_{n-1}}^s a_{i,j_n}^s= ρ $$ and for every column $j$ there exists an ordering of elements such that $$ a_{i_1,j}^s a_{i_2,j}^s \dots a_{i_{m-1},j}^s a_{i_m,j}^s = σ, $$ then $MRS_Γ(m,n;k)$ is called a \emph{$Γ$-magic rectangle set}. We investigate magic rectangle sets over dihedral groups and prove that $\mathrm{MRS}_Γ(m,n;k)$ exists for every dihedral group $Γ$ of order $mnk$, provided that $m$ and $n$ are even. As a consequence, we obtain broad existence results for magic rectangles and magic squares over dihedral groups. |
| title | Note on a magic rectangle set on dihedral group |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.13393 |