Magic squares on Abelian groups

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 $\Gamma$.