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 $$g^{\mathrm{op}}x = xg$$.