Let $G$ be a group under the operation $*$ . The opposite group of $G$ , denoted $G^{\mathrm{op}}$ , has the same underlying set as $G$ , and its group operation is $*'$ defined by $g_1*'g_2=g_2*g_1$ .

If $G$ is abelian, then it is equal to its opposite group. Also, every group $G$ (not necessarily abelian) is isomorphic to its opposite group: The isomorphism $\varphi \colon G \to G^{\mathrm{op}}$ is given by $\varphi(x)=x^{-1}$ . More generally, any anti-automorphism $\psi \colon G \to G$ gives rise to a corresponding isomorphism $\psi' \colon G \to G^{\mathrm{op}}$ via $\psi'(g)=\psi(g)$ , since $\psi'(g*h)=\psi(g*h)=\psi(h)*\psi(g)=\psi(g)*'\psi(h)=\psi'(g)*'\psi'(h)$ .

Opposite groups are useful for converting a right action to a left action and vice versa. For example, if $G$ is a group that acts on $X$ on the right, then a left action of $G^{\mathrm{op}}$ on $X$ can be defined by $g^{\mathrm{op}}x=xg$ .

Similar constructions occur in opposite ring and opposite category.