# Commutative property

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An operation (especially a binary operation) is said to have the commutative property or to be commutative if the order of its arguments does not affect the value.

For example, the operation addition is commutative on the most commonly used number systems (the complex numbers and its subsets such as the real numbers, integers, etc.) because $\displaystyle a + b = b + a$. However, the operation of division is not commutative over these sets because usually $\frac ab \neq \frac ba$.

Formally, an operation $G: S \to S$ is commutative if and only if $\forall a, b \in S, G(a, b) = G(b, a)$.

An operation which is not commutative is said to be noncommutative.

Commutivity is especially important in abstract algebra. The study of groups in which the group operation is commutative (abelian groups) is a very important part of group theory.

## Examples

• The integers commute under both addition and multiplication, but not subtraction or division.
• Some functions commute under composition. For example, the functions $f_n: \mathbb{C} \to \mathbb{C}$, $f_n(z) = z^n$ for all $z$ (with $n$ taking values in the positive integers) commute: $f_n \circ f_m = f_m \circ f_n$.
• Function composition is not, in general, commutative. For example, composing two linear transformations between three vector spaces corresponds to multiplying the corresponding matrices, and matrix multiplication does not commute in general. In particular, let $A$ be a $1 \times 2$ matrix and $B$ a $2 \times 1$ matrix. Then $AB$ and $BA$ are matrices of sizes $1 \times 1$ and $2 \times 2$, respectively.