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 o p , has the same underlying set as G , and its group operation ∗ ′ is defined by g 1 ∗ ′ 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 φ : G → G o p is given by φ ( x ) = x − 1 . More generally, any antiautomorphism ψ : G → G gives rise to a corresponding isomorphism ψ ′ : G → G o p via ψ ′ ( g ) = ψ ( g ) , since
ψ ′ ( g ∗ h ) = ψ ( g ∗ h ) = ψ ( h ) ∗ ψ ( g ) = ψ ( g ) ∗ ′ ψ ( h ) = ψ ′ ( g ) ∗ ′ ψ ′ ( h ) . Let X be an object in some category, and ρ : G → A u t ( X ) be a right action. Then ρ o p : G o p → A u t ( X ) is a left action defined by ρ o p ( g ) x = ρ ( g ) x , or g o p x = x g .
