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 ${\displaystyle G}$ be a group under the operation ${\displaystyle *}$. The opposite group of ${\displaystyle G}$, denoted ${\displaystyle G^{\mathrm {op} }}$, has the same underlying set as ${\displaystyle G}$, and its group operation ${\displaystyle {\mathbin {\ast '}}}$ is defined by ${\displaystyle g_{1}{\mathbin {\ast '}}g_{2}=g_{2}*g_{1}}$.

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

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

Group action

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