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In this paper, we define an $n$-magic square in a group to be an $(n\timesn)$ array of group elements whose rows, columns, and diagonals have the sameproduct. This definition is akin to the idea of magic squares in the integers.Groups that have an $n$-magic square are said to be $n$-magic. We begin withsome preliminary results and focus much of our attention on $3$-magic groups.Through a series of propositions, we ultimately prove a characterizationtheorem for $3$-magic finitely generated abelian groups. We then discuss someadditional results about non-abelian groups as well as $n$-magic groups where$n>3$.

Introducing $n$-Magic Groups and Characterizing $3$-Magic Finitely Generated Abelian Groups