Note on zero-sum magic squares on Abelian groups

Let $(\Gamma,+)$ be an Abelian group of order $n^2$. A $\Gamma$-magic square of order $n$ is an $n\times n$ array whose entries are pairwise distinct elements of $\Gamma$ such that all row sums, column sums, and the two main diagonal sums are equal to the same element $\mu \in \Gamma$, called the magic constant. A combinatorial design is called $\Gamma$-additive if its point set is a subset of an Abelian group $\Gamma$ and every block has sum zero. If the point set coincides with $\Gamma$, the design is said to be strictly $\Gamma$-additive. Motivated by this notion, we construct $\Gamma$-magic squares with magic constant $\mu=0$ whose rows, columns, and two main diagonals can be used as blocks of a strictly $\Gamma$-additive design. We call such a square zero-sum $\Gamma$-magic square. In this paper, we establish necessary and sufficient conditions for the existence of zero-sum $\Gamma$-magic squares.