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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.

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 {\displaystyle G} be a group under the operation ∗ {\displaystyle *} . The opposite group of G {\displaystyle G} , denoted G o p {\displaystyle G^{\mathrm {op} }} , has the same underlying set as G {\displaystyle G} , and its group operation ∗ ′ {\displaystyle {\mathbin {\ast '}}} is defined by g 1 ∗ ′ g 2 = g 2 ∗ g 1 {\displaystyle g_{1}{\mathbin {\ast '}}g_{2}=g_{2}*g_{1}} .

If G {\displaystyle G} is abelian, then it is equal to its opposite group. Also, every group G {\displaystyle G} (not necessarily abelian) is naturally isomorphic to its opposite group: An isomorphism φ : G → G o p {\displaystyle \varphi :G\to G^{\mathrm {op} }} is given by φ ( x ) = x − 1 {\displaystyle \varphi (x)=x^{-1}} . More generally, any antiautomorphism ψ : G → G {\displaystyle \psi :G\to G} gives rise to a corresponding isomorphism ψ ′ : G → G o p {\displaystyle \psi ':G\to G^{\mathrm {op} }} via ψ ′ ( g ) = ψ ( g ) {\displaystyle \psi '(g)=\psi (g)} , since

ψ ′ ( g ∗ h ) = ψ ( g ∗ h ) = ψ ( h ) ∗ ψ ( g ) = ψ ( g ) ∗ ′ ψ ( h ) = ψ ′ ( g ) ∗ ′ ψ ′ ( h ) . {\displaystyle \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 {\displaystyle X} be an object in some category, and ρ : G → A u t ( X ) {\displaystyle \rho :G\to \mathrm {Aut} (X)} be a right action. Then ρ o p : G o p → A u t ( X ) {\displaystyle \rho ^{\mathrm {op} }:G^{\mathrm {op} }\to \mathrm {Aut} (X)} is a left action defined by ρ o p ( g ) x = x ρ ( g ) {\displaystyle \rho ^{\mathrm {op} }(g)x=x\rho (g)} , or g o p x = x g {\displaystyle g^{\mathrm {op} }x=xg} .

See also

External links

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