A magic rectangle set on Abelian groups

Abstract: A $\Gamma$-magic rectangle set $MRS_{\Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $\Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $\omega\in \Gamma$ and all column sums in every rectangle equal to a constant $\delta \in \Gamma$. In this paper we prove that for ${a,b}\neq{2^{\alpha},2k+1}$ where $\alpha$ and $k$ are some natural numbers, a $\Gamma$-magic rectangle set MRS${\Gamma}(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|\Gamma|$ is odd or $\Gamma$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $\Gamma$-magic rectangle MRS${\Gamma}(a, b)$=MRS$_{\Gamma}(a, b;1)$.