A complete mapping of a group \((\Gamma ,+)\) is a bijection \(\varphi :\Gamma \rightarrow \Gamma \) for which the mapping \(x \mapsto x+\varphi (x)\) is a bijection. In this paper we consider the existence of a complete mapping \(\varphi \) of an Abelian group \(\Gamma \) and a partition \(S_1,S_2,\ldots , S_t\) of elements of \(\Gamma \) such that \(\sum _{s\in S_i}s=\sum _{s\in S_i}\varphi (s)=0\) for every i, \(1 \le i \le t\) . A \(\Gamma \) -magic rectangle set \(\textrm{MRS}_{\Gamma }(a, b; c)\) of order abc is a collection of c arrays \((a\times b)\) whose entries are elements of an Abelian group \(\Gamma \) of order abc, each appearing once, with all row sums in every rectangle equal to the constant \(\omega \in \Gamma \) and all column sums in every rectangle equal to the constant \(\delta \in \Gamma \) . While a complete characterization of \(\textrm{MRS}_\Gamma (a,b;c)\) exists for cases where \(\{a,b\}\not =\{2k+1,2^{\alpha }\}\) , the scenario where \(\{a,b\}=\{2k+1,2^{\alpha }\}\) remains unsolved for \(\alpha >1\) . Using the partition of \(\Gamma \) into zero-sum sets by complete mappings, we give some sufficient conditions that a \(\Gamma \) -magic rectangle set \(\textrm{MRS}_{\Gamma }(2k+1, 2^{\alpha };c)\) exists.
