Opposite group

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This is a natural transformation of binary operation from a group to its opposite. <g1, g2> denotes the ordered pair of the two group elements. *' can be viewed as the naturally induced addition of +.

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.


Let G be a group under the operation *. The opposite group of G, denoted G^{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^{op} is given by \varphi(x) = x^{-1}. More generally, any anti-automorphism \psi: G \to G gives rise to a corresponding isomorphism \psi': G \to G^{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[edit]

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

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