On a conjecture by Sylwia Cichacz and Tomasz Hinc, and a related problem

A Γ-magic rectangle set MRS_Γ(a, b; c) is a collection of c arrays of size a × b whose entries are the elements of an abelian group Γ of order abc, each one appearing once and in a unique array in such a way that the sum of the entries of each row is equal to a constant ω ∈ Γ and the sum of the entries of each column is equal to a constant δ ∈ Γ. In this paper we provide new evidences for the validity of a conjecture proposed by Sylwia Cichacz and Tomasz Hinc on the existence of an MRSΓ(a, b; c). We also generalize this problem describing constructions of Γ-magic rectangle sets whose elements are partially filled arrays.