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

TL;DR

This paper introduces the concept of n-magic groups, focusing on 3-magic finitely generated abelian groups, and provides a characterization theorem for these groups, expanding understanding of their structure.

Contribution

It defines n-magic groups, especially characterizes 3-magic finitely generated abelian groups, and explores properties of non-abelian and larger n-magic groups.

Findings

Characterization theorem for 3-magic finitely generated abelian groups

Preliminary results on n-magic groups

Discussion on non-abelian and n>3 cases

Abstract

In this paper, we define an $n$-magic square in a group to be an $(n\times n)$ array of group elements whose rows, columns, and diagonals have the same product. 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 with some preliminary results and focus much of our attention on $3$-magic groups. Through a series of propositions, we ultimately prove a characterization theorem for $3$-magic finitely generated abelian groups. We then discuss some additional results about non-abelian groups as well as $n$-magic groups where $n>3$.