Opposite group

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.

References

References

1. Clark, Alan. "Elements of Abstract Algebra". Dover Publications, Inc..