Abelian group

An abelian group is a group $G=(X,\bullet )$ where $\bullet$ is commutative. In other words, the group operation satisfies the five axioms:

1. Closure: For all $x,y$ in $X$, $x\bullet y$ is defined and in $X$. We abbreviate $x\bullet y$ as $xy$.
2. Associativity: $x\left(yz\right)=\left(xy\right)z$ for all $x,y,z$ in $X$.
3. Identity: There is an element $e$ such that for all $x$ in $X$, $xe=ex=x$.
4. Inverses: For each $x$ in $X$ is an element $x^{-1}$ in $X$ such that $x{x}^{-1}=x^{-1}x=e$.
5. Commutativity: For all $x,y$ in $X$, $xy=yx$.

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 $ab{a}^{-1}d{b}^{-1}=d^{-1}$, 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 $ab{a}^{-1}$ 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).