Unusual elementary axiomatizations for abelian groups

TL;DR

This paper explores alternative elementary axiomatizations of Abelian groups, reducing the necessary properties to a minimal set and utilizing rarely mentioned properties to characterize these structures.

Contribution

It introduces new axiomatizations for Abelian groups, simplifying the set of properties needed and employing uncommon properties for characterization.

Findings

Reduced the list of axioms for Abelian groups to two properties.

Provided alternative axiomatizations using rarely mentioned properties.

Enhanced understanding of the foundational properties of Abelian groups.

Abstract

One of the most studied algebraic structures with one operation is the Abelian group, which is defined as a structure whose operation satisfies the associative and commutative properties, has identical element and every element has an inverse element. In this article, we characterize the Abelian groups with other properties and we even reduce it to two the list of properties to be fulfilled by the operation. For this, we make use of properties that, in general, are hardly ever mencioned.