Abelian group

[summary: An abelian group is a group where the operation is commutative. That is, an abelian group $~$G$~$ is a pair $~$(X, \bullet)$~$ where $~$X$~$ is a set and $~$\bullet$~$ is a binary operation obeying the four group axioms plus an axiom of commutativity:

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(yz) = (xy)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 $~$xx^{-1}=x^{-1}x=e$~$.
5. Commutativity: For all $~$x, y$~$ in $~$X$~$, $~$xy=yx$~$.

Abelian groups are very "well-behaved" groups that are often easier to deal with than their non-commuting counterparts.]

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(yz) = (xy)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 $~$xx^{-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 $~$aba^{-1}db^{-1}=d^{-1}$~$, we can shuffle the elements to get $~$aa^{-1}bb^{-1}d=d^{-1}$~$ and reduce this to the claim $~$d=d^{-1}$~$. This would be invalid for a nonabelian group, because $~$aba^{-1}$~$ doesn't necessarily equal $~$aa^{-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 [group_theory_direct_product direct product] of cyclic groups (the [structure_theorem_for_finitely_generated_abelian_groups structure theorem for finitely generated abelian groups]).