Partition of Abelian groups into zero-sum sets by complete mappings and its application to the existence of a magic rectangle set

A complete mapping of a group $\mathrm{\Gamma }$ is a bijection $\phi \text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to \mathrm{\Gamma }$ for which the mapping $x\mapsto x+\phi (x)$ is a bijection. In this paper we consider the existence of a complete mapping $\phi$ of $\mathrm{\Gamma }$ and a partition $S_1,S_2,\dots S_t$ of elements of $\mathrm{\Gamma }$, such that ${\sum }_{s\in S_i}s={\sum }_{s\in S_i}\phi (s)=0$ for every $i$, $1\le i\le t$. A $\mathrm{\Gamma }$-magic rectangle set $MR{S}_{\mathrm{\Gamma }}(a,b;c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $\mathrm{\Gamma }$ of order $abc$, each appearing once, with all row sums in every rectangle equal to a constant $\omega \in \mathrm{\Gamma }$ and all column sums in every rectangle equal to a constant $\delta \in \mathrm{\Gamma }$. While a complete characterization of MRS${}_{\mathrm{\Gamma }}(a,b;c)$ exists for cases where $\{a,b\}≠\{2k+1,2^{\alpha }\}$, the scenario where $\{a,b\}=\{2k+1,2^{\alpha }\}$ remains unsolved for $\alpha >1$. Using the partition of $\mathrm{\Gamma }$ into zero-sum sets by complete mappings, we give some sufficient conditions that a $\mathrm{\Gamma }$-magic rectangle set MRS${}_{\mathrm{\Gamma }}(2k+1,2^{\alpha };c)$ exists.