Magic squares on Abelian groups

TL;DR

This paper proves that for any Abelian group of order n^2 with n>2, there exists an n×n magic square with entries from the group where all rows, columns, and diagonals sum to the same element.

Contribution

It establishes the existence of group-based magic squares for all Abelian groups of order n^2 when n>2, generalizing classical constructions.

Findings

Existence of a-magic squares for all Abelian groups of order n^2

Construction methods for such magic squares

Extension of classical magic square theory to algebraic structures

Abstract

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