A magic rectangle set on Abelian groups and its application

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