A magic rectangle set on Abelian groups

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

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