Abelian

Return to Glossary.

Formal Definition

---

A group $G$ is abelian if its binary operation is commutative.

Informal Definition

---

A group $G$ is abelian if its binary operation can be rearranged and still compute the same result.

Example(s)

---

The familiar multiplicative properties of rational,real,and complex numbers show that the sets $\mathbb{Q^+}$ and $\mathbb{R^+}$ of positive numbers and the sets $\mathbb{Q^*}$,$\mathbb{R^*}$,and $\mathbb{C^*}$ of nonzero numbers under multiplication are abelian groups.

Non-example(s)

---

The set $\mathbb{Z^+}$ under addition is not a group.There is no identity element for + in $\mathbb{Z^+}$.

Additional Comments

---

Add any other comments you have about the term here