Opposite group

File:Opposite group nature.svg

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 $G$ be a group under the operation $*$. The opposite group of $G$, denoted $G^{\mathrm{o}\mathrm{p}}$, has the same underlying set as $G$, and its group operation ${\ast }^{\prime }$ is defined by $g_1{\ast }^{\prime }{g}_2=g_2*g_1$. 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 $\phi :G\to G^{\mathrm{o}\mathrm{p}}$ is given by $\phi (x)=x^{-1}$. More generally, any antiautomorphism $\psi :G\to G$ gives rise to a corresponding isomorphism ${\psi }^{\prime }:G\to G^{\mathrm{o}\mathrm{p}}$ via ${\psi }^{\prime }(g)=\psi (g)$, since

${\psi }^{\prime }(g*h)=\psi (g*h)=\psi (h)*\psi (g)=\psi (g){\ast }^{\prime }\psi (h)={\psi }^{\prime }(g){\ast }^{\prime }{\psi }^{\prime }(h).$

Group action

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

See also

• Opposite ring
• Opposite category

External links

• http://planetmath.org/oppositegroup