Abelian Group

An abelian group is a group $(G,\cdot)$ such that for all $a,b\in G$, $a\cdot b = b\cdot a.$ (That is, the group operation is commutative.)

Abelian groups are often written in additive notation, replacing $\cdot$ by $+$, the identity element by $0$, and inverses by negatives. They form the algebraic backbone of linear structures (e.g. every vector space is an abelian group under addition).

Examples:

• $(\mathbb{Z},+)$ and $(\mathbb{R},+)$ are abelian groups.
• $(\mathbb{Z}/n\mathbb{Z},+)$ is an abelian group for each $n\ge 1$.
• The nonzero complex numbers $(\mathbb{C}^{\times},\times)$ form an abelian group.
• (Non-example) The symmetric group $S_3$ is not abelian.