Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$
An abelian group is a group where the elements commute. Another term is a commutative group. So as well as satisfying the 4 axioms of a group, they satisfy $a\cdot b=b\cdot a~\forall a,b\in A$ where $A$ is a set.