# Definition:Commutative/Set

Jump to navigation
Jump to search

## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $X \subseteq S$ be a subset of $S$ such that:

- $\forall a, b \in X: a \circ b = b \circ a$

That is, every element of $X$ commutes with every other element.

Then $X$ is a **commuting set of elements** of $S$.

## Also defined as

Some treatments use this definition only when the algebraic structure $S$ is a group.

## Also known as

The terms **permute** and **permutable** can sometimes be seen instead of **commute** and **commutative**.

## Sources

- 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts