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.

Definition

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 isomorphic to its opposite group: An isomorphism $\varphi: G \to G^{op}$ is given by φ(x) = x ^{− 1}. More generally, any anti-automorphism $\psi: G \to G$ gives rise to a corresponding isomorphism $\psi': G \to G^{op}$ via ψ'(g) = ψ(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 $\rho^{op}: G^{op} \to \mathrm{Aut}(X)$ is a left action defined by ρ^op(g)x = ρ(g)x, or g^opx = xg.

External links

• http://planetmath.org/encyclopedia/OppositeGroup.html