Commutator

of two elements $a$ and $b$ in a group with multiple operators

The element

$$-a-b+a+b.$$

For groups without multiple operators (here the operation is usually called multiplication), the commutator of the elements $a$ and $b$ is the element $a^{-1}b^{-1}ab$. The set of all commutators in a group $G$ generates a subgroup, called the commutator subgroup of $G$.

In an associative algebra the element $[x,y]=xy-yx$ is called the Lie product, or commutator, of $x$ and $y$.