nLab
abelian group

An abelian group is a group where the multiplication satisfies the commutative law:

$$xy=yx.$$x y = y x.

Alternatively, just as a group is a groupoid with one object, an abelian group is a 2-groupoid with one object and one morphism.

The category with abelian groups as objects and group homomorphisms as morphisms is called Ab.