Switching lemma

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 $\ast$. The opposite group of $G$, denoted $G^{op}$, has the same underlying set as $G$, and its group operation ${\ast }^{\prime }$ is defined by $g_1{\ast }^{\prime }{g}_2=g_2\ast g_1$.

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 $\phi :G\to G^{op}$ is given by $\phi (x)=x^{-1}$. More generally, any anti-automorphism $\psi :G\to G$ gives rise to a corresponding isomorphism ${\psi }^{\prime }:G\to G^{op}$ via ${\psi }^{\prime }(g)=\psi (g)$, since

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

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 ${\rho }^{op}:G^{op}\to \mathrm{A}\mathrm{u}\mathrm{t}(X)$ is a left action defined by ${\rho }^{op}(g)x=\rho (g)x$, or $g^{op}x=xg$.

External links

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