Commuting elements

Summary

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