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 $\phi :\Gamma \to \Gamma$ 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 $\Gamma$ and a partition $S_1,S_2,\dots S_t$ of elements of $\Gamma$, 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 $\Gamma$-magic rectangle set $MR{S}_{\Gamma }(a,b;c)$ 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 $\{a,b\}\ne \{2k+1,2^{\alpha }\}$, 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.

arXiv admin note: text overlap with arXiv:1804.00321

References in corpus (1)

• A magic rectangle set on Abelian groups

Cited by in corpus (1)

• Existence of magic rectangle sets over finite abelian groups