# 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) $\varphi\colon G\to G^{\mathrm{op}}$ is given by $\varphi(x)=x^{-1}$. More generally, any anti-automorphism $\psi\colon G\to G$ gives rise to a corresponding isomorphism $\psi^{\prime}\colon G\to G^{\mathrm{op}}$ 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.

Title opposite group OppositeGroup 2013-03-22 17:09:56 2013-03-22 17:09:56 Wkbj79 (1863) Wkbj79 (1863) 10 Wkbj79 (1863) Definition msc 08A99 msc 20-00