Note on a magic rectangle set on dihedral group

TL;DR

This paper studies the existence of magic rectangle sets over dihedral groups, proving they exist for all even dimensions and broadening the understanding of magic rectangles and squares in group theory.

Contribution

It establishes the existence of $ ext{MRS}_ ext{Γ}(m,n;k)$ for all dihedral groups of order $mnk$ when $m$ and $n$ are even, expanding prior results.

Findings

Magic rectangle sets exist over dihedral groups of order $mnk$ for even $m$ and $n$.

Broad existence results for magic rectangles and squares over dihedral groups.

Provides constructive proofs for the existence of these structures.

Abstract

Let $\Gamma$ be a group of order $mnk$ and $MRS_{\Gamma}(m,n;k)=(a_{i,j}^s)_{m\times n}$ be a collection of $k$ arrays $m\times n$ whose entries are all distinct elements of $\Gamma$. If there exist elements $\rho,\sigma\in\Gamma$ such that for every row $i$, there exists an ordering of elements such that $$ a_{i,j_1}^s a_{i,j_2}^s \dots a_{i,j_{n-1}}^s a_{i,j_n}^s= \rho $$ and for every column $j$ there exists an ordering of elements such that $$ a_{i_1,j}^s a_{i_2,j}^s \dots a_{i_{m-1},j}^s a_{i_m,j}^s = \sigma, $$ then $MRS_{\Gamma}(m,n;k)$ is called a \emph{$\Gamma$-magic rectangle set}. We investigate magic rectangle sets over dihedral groups and prove that $\mathrm{MRS}_{\Gamma}(m,n;k)$ exists for every dihedral group $\Gamma$ 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.