Semi-magic dihedral squares

TL;DR

This paper characterizes semi-magic squares with entries from dihedral groups, showing their properties and the possibility of having multiple magic constants depending on product order.

Contribution

It provides a complete characterization of semi-magic squares in dihedral groups and explores the phenomenon of multiple magic constants within these structures.

Findings

Complete characterization of semi-magic squares in dihedral groups

Existence of semi-magic squares with multiple magic constants

Construction methods for such semi-magic squares

Abstract

Let $\Gamma$ be a group of order $n^2$ and $SMS_{\Gamma}(n)=(a_{i,j})_{n\times n}$ be an $n\times n$ array whose entries are all distinct elements of $\Gamma$. If there exists an element $\mu\in\Gamma$ such that for every row $i$, there exists an ordering of elements such that $$ a_{i,j_1} a_{i,j_2} \dots a_{i,j_{n-1}} a_{i,j_n} = \mu $$ and for every column $j$ there exists an ordering of elements such that $$ a_{i_1,j} a_{i_2,j} \dots a_{i_{m-1},j} a_{i_m,j} = \mu, $$ then $SMS_{\Gamma}(n)$ is called a \emph{$\Gamma$-semi-magic square of side $n$} and $\mu$ is called a \emph{magic constant}. We provide a complete characterization of semi-magic squares of side $n$ whose entries belong to a dihedral group $D_k$. Moreover, we show that in our constructions a single semi-magic square may admit two distinct magic constants, depending on the order in which the products are computed.