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

Abstract

A complete mapping of a group $\Gamma$ is a bijection $\varphi\colon \Gamma\to \Gamma$ 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 $\Gamma$ 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 $i$, $1 \leq i \leq t$. 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 $\Gamma$ of order $abc$, 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 $\delta \in \Gamma$. While a complete characterization of 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 MRS$_{\Gamma}(2k+1, 2^{\alpha};c)$ exists.