Existence of magic rectangle sets over finite abelian groups

TL;DR

This paper determines the exact conditions under which magic rectangle sets over finite abelian groups exist, confirming a previously conjectured characterization.

Contribution

It provides a complete characterization of the existence of G-magic rectangle sets over finite abelian groups, resolving a conjecture by Cichacz and Hinc.

Findings

Established necessary and sufficient conditions for existence

Confirmed the conjecture by Cichacz and Hinc

Unified the theory for all finite abelian groups

Abstract

Let $a$, $b$ and $c$ be positive integers. Let $(G,+)$ be a finite abelian group of order $abc$. A $G$-magic rectangle set MRS$_G(a,b;c)$ 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 MRS$_G(a,b;c)$ for any finite abelian group $G$, thereby confirming a conjecture presented by Cichacz and Hinc.