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.

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 $G^$, has the same underlying set as $G$, and its group operation $\mathbin$ is defined by $g_1 \mathbin g_2 = g_2 * g_1$.

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 $\varphi: G \to G^$ is given by $\varphi(x) = x^$. More generally, any antiautomorphism $\psi: G \to G$ gives rise to a corresponding isomorphism $\psi': G \to G^$ via $\psi'(g)=\psi(g)$, since
: $\psi'(g * h) = \psi(g * h) = \psi(h) * \psi(g) = \psi(g) \mathbin \psi(h)=\psi'(g) \mathbin \psi'(h).$

Group action

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

See also

* Opposite ring
* Opposite category

External links

http://planetmath.org/oppositegroup

Group theory
Representation theory