Existence of magic rectangle sets over finite abelian groups

TL;DR

The paper addresses the problem of when a $G$-magic rectangle set $MRS_G(a,b;c)$ exists for a finite abelian group $G$, establishing a complete characterization that confirms the conjecture of Cichacz and Hinc CH21-1. The authors reduce the problem to the pivotal case $MRS_{\mathbb{Z}_p\oplus S_2}(p,4;|S_2|/4)$ with odd $p$ and a noncyclic $2$-Sylow subgroup $S_2$, and then treat two regimes: $\exp(S_2)\leq 4$ and $\exp(S_2)\geq 8$. They develop two constructive frameworks (Constructions I–II) and a novel $C_p$-based Construction III, along with the concept of incomplete magic rectangle sets (IMRS), to inductively build $MRS$ for larger groups from base cases, covering all admissible group structures. The results resolve CH21-1, provide systematic combinatorial constructions for a broad class of groups, and suggest pathways to higher-dimensional generalizations of magic rectangles over 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 $γ\in G$ and the sum of each column in every array equals a constant $δ\in G$. 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.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.1
• proof
• Lemma 1.2
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 3.1