Abelian Group¶

Definition¶

An Abelian group is a group whose operation is commutative.

Interpretation¶

An Abelian group is a group in which the order of the two elements never matters: $a\star b=b\star a$ for all $a,b$. Identity and inverses still come from the group axioms; commutativity is the only extra condition.

Many familiar groups are Abelian: $(\mathbb{Z},+)$, $(\mathbb{R},+)$ and the nonzero real numbers under multiplication. The symmetric group $S_3$ is not: swapping the first two of three objects and then the last two gives a different permutation from the same swaps in the other order.

Abelian groups are often written additively, with $+$ for the operation, $0$ for the identity and $-a$ for the inverse of $a$. The name honours Niels Henrik Abel; the alias commutative group says the same. Later, the addition of a vector space makes it an Abelian group.

Key Results¶

• Every Abelian group is a group, but not every group is Abelian.

Examples¶

• $(\mathbb{R},+)$ is an Abelian group.
• The permutations of $\{1,2,3\}$ under composition form the symmetric group $S_3$, which is not Abelian: the swaps $(1\,2)$ and $(2\,3)$ do not commute.

Prerequisites¶

• Group
• Commutativity

Related Concepts¶

• Symmetric Group

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: commutative group Symbols: integers, real numbers

Last updated: 20 October 2026 03:02