Existence of Magic Rectangle Sets Over Finite Abelian Groups

Abstract

Let $a,b$, and $c$ be positive integers. Let $\left(G,+\right)$ be a finite abelian group of order $abc$. A $G$-magic rectangle set ${\text{MRS}}_G\left(a,b;c\right)$ is a collection of $c$ arrays of size $a\times b$, whose entries are elements of a group $G$, each appearing exactly once, such that the sum of each row in every array equals a constant $\gamma \in G$, and the sum of each column in every array equals a constant $\delta \in G$. This paper establishes the necessary and sufficient conditions for the existence of an ${\text{MRS}}_G\left(a,b;c\right)$, for any finite abelian group $G$, thereby confirming a conjecture posted by Cichacz and Hinc.