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 Γ is a bijection φ Γ→ Γ for which the mapping x ↦ x+φ(x) is a bijection. In this paper we consider the existence of a complete mapping φ of Γ and a partition S₁,S₂,… Sₜ of elements of Γ, such that ∑s∈ Sᵢs=∑s∈ Sᵢφ(s)=0 for every i, 1 ≤ i ≤ t. A Γ-magic rectangle set MRSΓ(a, b; c) of order $abc$ is a collection of c arrays (a× b) whose entries are elements of group Γ of order $abc$, each appearing once, with all row sums in every rectangle equal to a constant ω∈ Γ and all column sums in every rectangle equal to a constant δ ∈ Γ. While a complete characterization of MRS_Γ(a,b;c) exists for cases where ,b\=\2k+1,2α\, the scenario where ,b\=\2k+1,2α\ remains unsolved for α>1. Using the partition of Γ into zero-sum sets by complete mappings, we give some sufficient conditions that a Γ-magic rectangle set MRSΓ(2k+1, 2α;c) exists.