Definition:Abelian Group

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:
 * the group product of $$a$$ and $$b$$ is denoted $$a + b$$;
 * the identity is denoted $$0$$, and
 * the inverse of $$a$$ denoted $$-a$$.

Under this regime, the group axioms read:


 * An algebraic structure $$\left({G, +}\right)$$ is an abelian group iff the following conditions are satisfied:

This notation gains in importance and usefulness when discussing rings.