abelian group
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An abelian group is a mathematical system where you can perform an operation on pairs of elements, and the order in which you combine them doesn't matter—combining A with B gives the same result as combining B with A. This property, called commutativity, makes abelian groups one of the most fundamental and well-understood structures in mathematics, with applications across algebra, number theory, and physics.
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In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel.
The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.