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.
The fundamental theorem of abelian groups says that all finite abelian groups are direct products of cyclic groups, themselves abelian.
