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# 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 $\varphi\colon G\to G^{{\mathrm{op}}}$ 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 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.

## Mathematics Subject Classification

08A99*no label found*20-00

*no label found*

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