---
tag: YJRBCWQV6B
kind: concept
name: Commuting elements
msc: [08A02, 20A05]
sources:
  - url: https://en.wikipedia.org/wiki/Commutative_property
---

## Summary

Two elements $a$ and $b$ commute for a [[binary operation|0M418FE7BB]] $\ast$ when $a \ast b = b \ast a$.
An operation is [[commutative|099E72KTF8]] exactly when every pair of elements commutes, but in a noncommutative structure some pairs commute and some do not.
In a [[group|G5S1M9]], $a$ and $b$ commute if and only if their [[commutator|0PXW0BN7DW]] is the identity.
[[Commuting matrices|8G3X8MPG7E]] are an important special case.

## Explanations

### L1 human

Two things commute when doing them in either order gives the same result.
Turning a square by $90$ degrees and turning it by $180$ degrees commute: both orders give a turn by $270$ degrees.
A turn by $90$ degrees and a flip across a diagonal do not commute: the two orders give different pictures.
Every element commutes with itself and with the identity element.