For questions about abelian groups, including the basic theory of abelian groups as a topic in elementary group theory as well as more advanced topics (classification, structure theory, theory of $\mathbb{Z}$-modules as related to modules over other rings, homological algebra of abelian groups, etc.). Consider also using the tag (group-theory) or (modules) depending on the perspective of your question.
An abelian or commutative group is a group $$(G,*)$$ in which all elements commute: $$\forall a,b\in G\,\,, a*b=b*a\,.$$ Usually the product is denoted by $$+$$ in an abelian group, and the identity of the group by $$0$$. Abelian groups are also known as modules over the ring $$\mathbb{Z}$$ of integers.
Examples include the integers $$\mathbb{Z}$$ under addition, as well as the rationals $$\mathbb{Q}$$ under addition. In fact, every cyclic group is an abelian group. Non-examples include $$S_3$$, the symmetry group on three elements, as well as $$\mathrm{SO}(3)$$, the rotations in three dimensions.