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 formula_1 be a group under the operation formula_2. The opposite group of formula_1, denoted formula_4, has the same underlying set as formula_1, and its group operation formula_6 is defined by formula_7.
If formula_1 is abelian, then it is equal to its opposite group. Also, every group formula_1 (not necessarily abelian) is naturally isomorphic to its opposite group: An isomorphism formula_10 is given by formula_11. More generally, any antiautomorphism formula_12 gives rise to a corresponding isomorphism formula_13 via formula_14, since
Group action.
Let formula_16 be an object in some category, and formula_17 be a right action. Then formula_18 is a left action defined by formula_19, or formula_20.