Definition:Abelian Group

Definition
An abelian group is a group $G$ where:
 * $\forall a, b \in G: a b = b a$

That is, every element in $G$ commutes with every other element in $G$.

Equivalently, $G$ is abelian iff $G = Z \left({G}\right)$, where $Z \left({G}\right)$ denotes the center of $G$.

Additive Notation
When discussing abelian groups, it is customary to use additive notation, where:

Abelian Group Axioms
Under this regime, the group axioms read:

This notation gains in importance and usefulness when discussing rings.

Also known as
The usual way of spelling this is without a capital letter, i.e. abelian, but Abelian is frequently seen.

Linguistic Note
The correct pronunciation of abelian is a-bee-lee-an, putting emphasis on the second syllable.

Note that the term abelian has thus phonetically lost the connection to its eponym Abel (correctly pronounced a-bl).