Note on a magic rectangle set on dihedral group

Abstract

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