Note on a magic rectangle set on dihedral group

Abstract

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.