Opposite group
# Opposite group

In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action.

Monoids, groups, rings, and algebras can be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.

## Definition

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 ${\mathbin {\ast '}}$ is defined by $g_{1}{\mathbin {\ast '}}g_{2}=g_{2}*g_{1}$.

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

    $\psi '(g*h)=\psi (g*h)=\psi (h)*\psi (g)=\psi (g){\mathbin {\ast '}}\psi (h)=\psi '(g){\mathbin {\ast '}}\psi '(h).$

## Group action

Let $X$ be an object in some category, and $\rho :G\to \mathrm {Aut} (X)$ be a right action. Then $\rho ^{\mathrm {op} }:G^{\mathrm {op} }\to \mathrm {Aut} (X)$ is a left action defined by $\rho ^{\mathrm {op} }(g)x=x\rho (g)$, or $g^{\mathrm {op} }x=xg$.

