opposite group

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 ${*}^{\prime }$ defined by $g_1{*}^{\prime }{g}_2=g_2*g_1$.

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 (http://planetmath.org/GroupIsomorphism) $\phi :G\to G^{\mathrm{op}}$ is given by $\phi (x)=x^{-1}$. More generally, any anti-automorphism $\psi :G\to G$ gives rise to a corresponding isomorphism ${\psi }^{\prime }:G\to G^{\mathrm{op}}$ via ${\psi }^{\prime }(g)=\psi (g)$, since ${\psi }^{\prime }(g*h)=\psi (g*h)=\psi (h)*\psi (g)=\psi (g){*}^{\prime }\psi (h)={\psi }^{\prime }(g){*}^{\prime }{\psi }^{\prime }(h)$.

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 , then a left action of $G^{\mathrm{op}}$ on $X$ can be defined by $g^{\mathrm{op}}x=xg$.

constructions occur in opposite ring and opposite category.