Magic squares on Abelian groups

Sylwia Cichacz; Dalibor Froncek

doi:10.1016/j.disc.2026.115033

Magic squares on Abelian groups

Sylwia Cichacz
, Dalibor Froncek

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

Let (Γ,+) be an Abelian group of order n² and MS_Γ(n) be an n×n array whose entries are all distinct elements of Γ. Then MS_Γ(n) is a Γ-magic square if all row, column, main diagonal and backward diagonal sums are equal to the same element μ∈Γ. We prove that for every Abelian group Γ of order n², n≠2, there exists a magic square MS_Γ(n) where the square entries are elements of Γ.

Original language	English (US)
Article number	115033
Journal	Discrete Mathematics
Volume	349
Issue number	7
DOIs	https://doi.org/10.1016/j.disc.2026.115033
State	Published - Jul 2026

Bibliographical note

Publisher Copyright:
© 2026 Elsevier B.V.

Keywords

Abelian group
Kotzig array
Latin squares
Magic rectangles
Magic squares
Γ-additive designs

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10.1016/j.disc.2026.115033

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title = "Magic squares on Abelian groups",

abstract = "Let (Γ,+) be an Abelian group of order n2 and MSΓ(n) be an n{\texttimes}n array whose entries are all distinct elements of Γ. Then MSΓ(n) is a Γ-magic square if all row, column, main diagonal and backward diagonal sums are equal to the same element μ∈Γ. We prove that for every Abelian group Γ of order n2, n≠2, there exists a magic square MSΓ(n) where the square entries are elements of Γ.",

keywords = "Abelian group, Kotzig array, Latin squares, Magic rectangles, Magic squares, Γ-additive designs",

author = "Sylwia Cichacz and Dalibor Froncek",

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year = "2026",

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AU - Froncek, Dalibor

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N2 - Let (Γ,+) be an Abelian group of order n2 and MSΓ(n) be an n×n array whose entries are all distinct elements of Γ. Then MSΓ(n) is a Γ-magic square if all row, column, main diagonal and backward diagonal sums are equal to the same element μ∈Γ. We prove that for every Abelian group Γ of order n2, n≠2, there exists a magic square MSΓ(n) where the square entries are elements of Γ.

AB - Let (Γ,+) be an Abelian group of order n2 and MSΓ(n) be an n×n array whose entries are all distinct elements of Γ. Then MSΓ(n) is a Γ-magic square if all row, column, main diagonal and backward diagonal sums are equal to the same element μ∈Γ. We prove that for every Abelian group Γ of order n2, n≠2, there exists a magic square MSΓ(n) where the square entries are elements of Γ.

KW - Abelian group

KW - Kotzig array

KW - Latin squares

KW - Magic rectangles

KW - Magic squares

KW - Γ-additive designs

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DO - 10.1016/j.disc.2026.115033

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VL - 349

JO - Discrete Mathematics

JF - Discrete Mathematics

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ER -

Magic squares on Abelian groups

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