n-Magic Groups and Some New Directions

Abstract

Magic squares are classical mathematical objects characterized by elegant symmetry, where the sum of each row, column, and diagonal is constant. In this paper, we extend this notion from integers to algebraic structures by introducing n-magic groups. An n-magic group is defined as a finite or finitely generated group whose elements can be arranged in an n × n array such that the product (or sum, in the abelian case) of every row, column, and diagonal equals a fixed group element. We present constructions for both abelian and non-abelian groups, including Zn × Zn and S3 × Z3, and show that the magic property is preserved under direct products. For finitely generated abelian groups, we give sufficient and necessary conditions for the existence of 3-magic arrays, emphasizing the role of subgroups isomorphic to Z3 × Z3. In addition, we introduce complex n-magic squares based on roots of unity, highlighting connections between magic groups, polynomial roots on the unit circle, and complex multiplication. These results unify algebraic and analytic perspectives on magic structures and provide explicit constructions that open new directions for further research in algebra and discrete mathematics.