Abelian group

https://arbital.com/p/abelian_group

by Nate Soares May 9 2016 updated Jul 18 2016

A group where the operation commutes. Named after Niels Henrik Abel.

[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]).