Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varGamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>Γ</mi> </math></EquationSource> </InlineEquation> be a finite additive Abelian group. A subgroup magic rectangle is an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$m \times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> array <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {S}=(\zeta _{i,j})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>ζ</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\zeta _{i,j} \in \varGamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>Γ</mi> </mrow> </math></EquationSource> </InlineEquation> for all i, j and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|\varGamma |=mn$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>Γ</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi>m</mi> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, each element appears exactly once in such a way that the set of all row-sums forms a subgroup of order m and the set of all column-sums forms a subgroup of order n. This combinatorial object is a generalisation of the constant sum partition of a finite Abelian group and group magic rectangle. In this article, we discuss the structural properties of subgroup magic rectangles over elementary Abelian 2-groups, Abelian groups whose order is the product of a given prime powers, and provide a characterisation for the existence of a subgroup magic rectangle over <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {Z}_{mn}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mrow> <mi mathvariant="italic">mn</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gcd (m,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is odd. Moreover, a recursive backtracking algorithm is given to construct a subgroup magic rectangle over <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {Z}_{mn}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mrow> <mi mathvariant="italic">mn</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>.

Structural properties of subgroup magic rectangles

期刊论文
发表日期：

2026年7月18日

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S. Karthik,
Aruna Venkatesan,
Krishnan Paramasivam

摘要

Let $\varGamma $ $Γ$ be a finite additive Abelian group. A subgroup magic rectangle is an $m \times n$ $m \times n$ array $\mathbb {S}=(\zeta _{i,j})$ $S = (ζ_{i, j})$ , where $\zeta _{i,j} \in \varGamma $ $ζ_{i, j} \in Γ$ for all i, j and $|\varGamma |=mn$ $| Γ | = m n$ , each element appears exactly once in such a way that the set of all row-sums forms a subgroup of order m and the set of all column-sums forms a subgroup of order n. This combinatorial object is a generalisation of the constant sum partition of a finite Abelian group and group magic rectangle. In this article, we discuss the structural properties of subgroup magic rectangles over elementary Abelian 2-groups, Abelian groups whose order is the product of a given prime powers, and provide a characterisation for the existence of a subgroup magic rectangle over $\mathbb {Z}_{mn}$ $Z_{mn}$ , where $\gcd (m,n)$ $gcd (m, n)$ is odd. Moreover, a recursive backtracking algorithm is given to construct a subgroup magic rectangle over $\mathbb {Z}_{mn}$ $Z_{mn}$ .