Magic squares on Abelian groups

TL;DR

The paper proves that for every Abelian group $\Gamma$ with $|\Gamma|=n^2$ and $n>2$, there exists an $n\times n$ $\Gamma$-magic square MS$_{\Gamma}(n)$ with all row, column, and both diagonal sums equal to a fixed $\mu\in\Gamma$. It develops a unified constructive framework using Kotzig arrays, generalized Kotzig arrays, Latin squares, and MOLS, combined with a Kronecker-type composition, to handle odd $n$, powers of two, and even sides that are not powers of two. A central technical tool lifts a smaller magic square over a subgroup to a larger one over a direct sum, enabling a comprehensive case-based assembly that covers all $n>2$. The results extend prior partial findings and raise open questions about magic squares over subsets and non-Abelian groups, suggesting natural directions for further research in combinatorial design and algebraic constructions.

Abstract

Let $(Γ,+)$ be an Abelian group of order $n^2$ and MS$_Γ(n)$ be an $n\times n$ array whose entries are all elements of $Γ$. Then MS$_Γ(n)$ is a $Γ$-magic square if all row, column, main and backward main diagonal sums are equal to the same element $μ\inΓ$. We prove that for every Abelian group $Γ$ of order $n^2$, $n>2$, there exists a magic square MS$_Γ(n)$ where the square entries are elements of $Γ$.

Figures (10)

• Figure 1: MR$_{Z_{4}}(2)$
• Figure 2: $\mathrm{MS}_{Z_{36}}(6)$
• Figure 3: $Z_4$- and $Z_2\oplus Z_2$-squares of side 2
• Figure 4: $\mathrm{MS}_{ Z_{3}\oplus Z_3\oplus Z_2\oplus Z_2}(6)$
• Figure 5: $\mathrm{MS}_{Z_{2}\oplus Z_{32}}(8)$

Theorems & Definitions (42)

• Definition 2.1
• Definition 2.2
• Theorem 2.3: Evans Evans
• Theorem 2.4: Cichacz, Hinc Cichacz-Hincz-2
• Theorem 2.5: handbook
• Theorem 2.6
• Claim 2.8: Sun, Yihui Sun-Yihui
• Theorem 2.9: Sun, Yihui Sun-Yihui
• Definition 3.1
• Theorem 3.2: handbook