An abelian group is a group where $∙$ is commutative. In other words, the group operation satisfies the five axioms:

Closure: For all $x, y$ in $X$ , $x ∙ y$ is defined and in $X$ . We abbreviate $x ∙ y$ as $x y$ .

Associativity: $x (y z) = (x y) z$ for all $x, y, z$ in $X$ .

Identity: There is an element $e$ such that for all $x$ in $X$ , $x e = e x = x$ .

Inverses: For each $x$ in $X$ is an element $x^{- 1}$ in $X$ such that $x x^{- 1} = x^{- 1} x = e$ .

Commutativity: For all $x, y$ in $X$ , $x y = y x$ .

The first four are the standard group axioms; the fifth is what distinguishes abelian groups from groups.

Commutativity gives us license to re-arrange chains of elements in formulas about commutative groups. For example, if in a commutative group with elements ${1, a, a^{- 1}, b, b^{- 1}, c, c^{- 1}, d}$ , we have the claim $a b a^{- 1} d b^{- 1} = d^{- 1}$ , we can shuffle the elements to get $a a^{- 1} b b^{- 1} d = d^{- 1}$ and reduce this to the claim $d = d^{- 1}$ . This would be invalid for a nonabelian group, because $a b a^{- 1}$ doesn't necessarily equal $a a^{- 1} b$ in general.

Abelian groups are very well-behaved groups, and they are often much easier to deal with than their non-commutative counterparts. For example, every Subgroup of an abelian group is normal, and all finitely generated abelian groups are a direct product of cyclic groups (the structure theorem for finitely generated abelian groups).