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 be a group under the operation. The opposite group of, denoted, has the same underlying set as, and its group operation is defined by.
If is abelian, then it is equal to its opposite group. Also, every group is naturally isomorphic to its opposite group: An isomorphism is given by. More generally, any antiautomorphism gives rise to a corresponding isomorphism via, sinceLet be an object in some category, and be a right action. Then is a left action defined by, or.
